Finite dimensional irreducible representations of Lie superalgebra D (2, 1; &#x03B1;)

Xi Chen(&#x09648;&#x066E6;); Wen-Li Yang(&#x06768;&#x06587;&#x0529B;); Xiang-Mao Ding(&#x04E01;&#x07965;&#x08302;); Yao-Zhong Zhang(&#x05F20;&#x08000;&#x04E2D;)

doi:10.1088/1572-9494/ad1588

Communications in Theoretical Physics >

2024 , Vol. 76 >Issue 2: 25002

DOI: https://doi.org/10.1088/1572-9494/ad1588

Mathematical Physics

Finite dimensional irreducible representations of Lie superalgebra D (2, 1; α)

Xi Chen(陈曦) ^,¹^,^* ,
Wen-Li Yang(杨文力) ^,²^,^* ,
Xiang-Mao Ding(丁祥茂) ³ ,
Yao-Zhong Zhang(张耀中) ⁴

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¹College of Intelligent Systems Science and Engineering, Hubei Minzu University 445000, China
²Institute of Modern Physics, Northwest University, Xian 710069, China
³Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
⁴School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

^*Authors to whom any correspondence should be addressed.

Received date: 2023-10-10

Revised date: 2023-12-07

Accepted date: 2023-12-14

Online published: 2024-02-01

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Abstract

This paper focuses on the finite dimensional irreducible representations of Lie superalgebra D(2, 1; α). We explicitly construct the finite dimensional representations of the superalgebra D(2, 1; α) by using the shift operator and differential operator representations. Unlike ordinary Lie algebra representation, there are typical and atypical representations for most superalgebras. Therefore, its typical and atypical representation conditions are also given. Our results are expected to be useful for the construction of primary fields of the corresponding current superalgebra of D(2, 1; α).

Key words： superalgebra; representations; shift operator; conformal field theory

Cite this article

Xi Chen(陈曦) , Wen-Li Yang(杨文力) , Xiang-Mao Ding(丁祥茂) , Yao-Zhong Zhang(张耀中) . Finite dimensional irreducible representations of Lie superalgebra D (2, 1; α)[J]. Communications in Theoretical Physics, 2024 , 76(2) : 025002 . DOI: 10.1088/1572-9494/ad1588

1. Introduction

Affine Lie algebras and their corresponding conformal field theories (CFTs) have essential applications in many subfields of physics [1]. Supersymmetry is the superalgebra associated with the symmetry generator. The concepts of supersymmetry relate to bosonic and fermionic degrees of freedom [2]. Supersymmetry theory is a uniform framework for the systems of bosons and fermions. The conformal field theories are based on current algebras. Current superalgebras and their corresponding two-dimensional conformal field theory have played a fundamental role in the high-energy physics and statistical physics at critical point, such as logarithmic CFTs [3], topological field theory [4], disordered systems and integer quantum Hall effects [5–11]. In most applications of conformal field theories, one needs to construct the finite-dimensional representations of a superalgebra explicitly.

Unlike ordinary bosonic algebra representation, there are typical and atypical representations for most superalgebras. The typical representation is similar to the representation that appeared in bosonic algebra. The atypical representation can be irreducible or not fully reducible. There is no atypical representation's counterpart in ordinary bosonic algebra representation [12, 13]. This makes the study of the representations of superalegbras extremely difficult. The superalgebras psl(n∣n) and osp(2n + 2∣2n) stand out as a most interesting class due to the fact that the corresponding sigma models with their supergroups have a vanishing super-dimension or vanishing dual Coexter number. The nonlinear sigma models based on the supergroups have a vanishing one-loop β function, which are expected to be conformal invariant without adding the Wess–Zumino terms [14]. Finite-dimensional typical and atypical representations of osp(2∣2) and gl(2∣2) have been studied in several papers [15, 16].

The sigma model associated with psl(4∣4) (or su(2, 2∣4)) is related to the string theory on the AdS₅ × S⁵ background. Recent studies show that the superalgebra D(2, 1; α) is the one-parameter deformation of Lie superalgebra D(2, 1) = osp(4∣2) and has a vanishing dual Coexter number. It has played an important role in describing the origin of the Yangian symmetry of AdS/CFT [17, 18] and the symmetry of string theory on AdS₃ × S³ × S³ × S¹. There are two types of AdS₃ geometries which preserve superconformal symmetry; the finite-dimensional subalgebras of these superconformal algebras are psu(1, 1∣2) and D(2, 1; α) [19]. The parameter α is related to the relative size of the radius of geometry [20]. Thus, the study of the D(2, 1; α) model would provide essential insight into the quantization of the string theory on the AdS₃ × S³ × S³ × S¹ background.

This paper is organized as follows. In section 2, we review the definition of finite-dimensional exceptional superalgebra D(2, 1; α) and its commutation relations. In section 3, we explicitly give the differential operator representations of all the generators. In section 4, we give the shift operators. In section 5, we construct the finite-dimensional representation of superalgebra D(2, 1; α). In section 6, we give four atypical conditions. If none of the four atypical conditions are satisfied, then the representation is a typical representation. Section 7 is devoted to our conclusions.

2. D(2, 1; α) superalgebra

The exceptional Lie superalgebra D(2, 1; α) with α forms a continuous one-parameter family of superalgebras of rank 3 and dimension 17 [2]. It is a deformation of the Lie superalgebra osp(4∣2) with the parameter α ≠ 0, − 1, ∞ . The bosonic (or even) part is a su(2) ⊕ su(2) ⊕ su(2) of dimension 9, and the fermionic (or odd) part is a spinor representation (2, 2, 2) of the bosonic part of dimension 8. In terms of the orthogonal basis vector ε₁, ε₂, ε₃ with the inner product

(2.1)$\begin{eqnarray}\begin{array}{l}({\epsilon }_{1},{\epsilon }_{1})=-\displaystyle \frac{1+\alpha }{2},\ \ ({\epsilon }_{2},{\epsilon }_{2})=\displaystyle \frac{1}{2},\\ ({\epsilon }_{3},{\epsilon }_{3})=\displaystyle \frac{\alpha }{2},\ \ ({\epsilon }_{i},{\epsilon }_{j})=0\ \ {for}\ i\ne j.\end{array}\end{eqnarray}$

The even roots Δ₀ and the odd roots Δ₁ of D(2, 1; α) are given by

(2.2)$\begin{eqnarray}{{\rm{\Delta }}}_{0}=\{\pm 2{\epsilon }_{i}\},\ \ {{\rm{\Delta }}}_{1}=\{\pm {\epsilon }_{1}\pm {\epsilon }_{2}\pm {\epsilon }_{3}\}.\end{eqnarray}$

Let Π = {α₁ = ε₁ − ε₂ − ε₃, α₂ = 2ε₂, α₃ = ε₃} be the simple root system, with α₁ being fermionic and α₂, α₃ being bosonic. The positive roots system ${{\rm{\Delta }}}^{+}={{\rm{\Delta }}}_{0}^{+}\cup {{\rm{\Delta }}}_{1}^{+}$ is a union of the positive even and odd roots. The positive even roots set ${{\rm{\Delta }}}_{0}^{+}$ and positive odd roots set ${{\rm{\Delta }}}_{1}^{+}$ are given by ${{\rm{\Delta }}}_{0}^{+}=\{{\alpha }_{2},{\alpha }_{3},2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}\}$, ${{\rm{\Delta }}}_{1}^{+}=\{{\alpha }_{1},{\alpha }_{1}+{\alpha }_{2},{\alpha }_{1}+{\alpha }_{3},{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}\}$. The Cartan matrix a_ij is given by

(2.3)$\begin{eqnarray}a=\left(\begin{array}{ccc}0 & 1 & \alpha \\ -1 & 2 & 0\\ -1 & 0 & 2\end{array}\right),\end{eqnarray}$

and with each positive root δ, there are generators E_δ (raising operator), F_δ ≡ E_−δ (lowering operator) and H_δ(Cartan generator). These operators have definite Z₂-gradings:

(2.4)$\begin{eqnarray}[{H}_{\delta }]=0,\quad \quad [{E}_{\delta }]=[{F}_{\delta }]=\left\{\begin{array}{ll}0, & \delta \in {{\rm{\Delta }}}_{0}^{+},\\ 1, & \delta \in {{\rm{\Delta }}}_{1}^{+}.\end{array}\right.\end{eqnarray}$

For any two generators a, b ∈ D(2, 1; α), the (anti)commutator is defined by

(2.5)$\begin{eqnarray}[a,b]={ab}-{(-)}^{[a][b]}{ba},\end{eqnarray}$

the commutation relations of D(2, 1; α) are

(2.6)$\begin{eqnarray}[{E}_{{\alpha }_{i}},{F}_{{\alpha }_{j}}]={\delta }_{{ij}}\,{H}_{{\alpha }_{i}},\,[{H}_{{\alpha }_{i}},{H}_{{\alpha }_{j}}]=0,\,i,j=1,2,3,\end{eqnarray}$

$\begin{eqnarray*}\begin{array}{l}[{H}_{{\alpha }_{i}},{E}_{{\alpha }_{j}}]={a}_{{ij}}{E}_{{\alpha }_{j}},\quad [{H}_{{\alpha }_{i}},{F}_{{\alpha }_{j}}]=-{a}_{{ij}}{F}_{{\alpha }_{j}},,\quad i,j=1,2,3,\\ [{E}_{{\alpha }_{1}},{E}_{{\alpha }_{2}}]=-{E}_{{\alpha }_{1}+{\alpha }_{2}},\quad \quad [{E}_{{\alpha }_{1}},{E}_{{\alpha }_{3}}]=-{E}_{{\alpha }_{1}+{\alpha }_{3}},\\ [{E}_{{\alpha }_{1}},{E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]=-(1+\alpha ){E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\\ [{E}_{{\alpha }_{1}},{H}_{{\alpha }_{2}}]=[{E}_{{\alpha }_{1}},{H}_{{\alpha }_{3}}]={E}_{{\alpha }_{1}},\\ [{E}_{{\alpha }_{1}},{F}_{{\alpha }_{1}+{\alpha }_{2}}]={F}_{{\alpha }_{2}},\quad [{E}_{{\alpha }_{1}},{F}_{{\alpha }_{1}+{\alpha }_{3}}]=\alpha {F}_{{\alpha }_{3}},\\ [{E}_{{\alpha }_{1}},{F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]=-{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\quad [{E}_{{\alpha }_{2}},{E}_{{\alpha }_{1}+{\alpha }_{3}}]\\ ={E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\\ [{E}_{{\alpha }_{2}},{H}_{{\alpha }_{1}}]=-{E}_{{\alpha }_{2}},\quad [{E}_{{\alpha }_{2}},{H}_{{\alpha }_{2}}]=-2{E}_{{\alpha }_{2}},\\ [{E}_{{\alpha }_{2}},{F}_{{\alpha }_{1}+{\alpha }_{2}}]={F}_{{\alpha }_{1}},\quad [{E}_{{\alpha }_{2}},{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]={F}_{{\alpha }_{1}+{\alpha }_{3}},\\ [{E}_{{\alpha }_{3}},{E}_{{\alpha }_{1}+{\alpha }_{2}}]={E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\quad [{E}_{{\alpha }_{3}},{H}_{{\alpha }_{1}}]=-\alpha {E}_{{\alpha }_{3}},\\ [{E}_{{\alpha }_{3}},{H}_{{\alpha }_{3}}]=-2{E}_{{\alpha }_{3}},\quad [{E}_{{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{3}}]={F}_{{\alpha }_{1}},\\ [{E}_{{\alpha }_{1}},{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]={F}_{{\alpha }_{1}+{\alpha }_{2}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{2}},{H}_{{\alpha }_{1}}]=[{E}_{{\alpha }_{1}+{\alpha }_{2}},{H}_{{\alpha }_{2}}]\\ =-{E}_{{\alpha }_{1}+{\alpha }_{2}},[{E}_{{\alpha }_{1}+{\alpha }_{2}},{H}_{{\alpha }_{3}}]={E}_{{\alpha }_{1}+{\alpha }_{2}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{2}},{E}_{{\alpha }_{1}+{\alpha }_{3}}]=(1+\alpha ){E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{2}},{F}_{{\alpha }_{1}}]=-{E}_{{\alpha }_{2}},\quad [{E}_{{\alpha }_{1}+{\alpha }_{2}},{F}_{{\alpha }_{1}+{\alpha }_{2}}]=-{H}_{{\alpha }_{1}}+{H}_{{\alpha }_{2}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{2}},{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]=-\alpha {F}_{{\alpha }_{3}},\quad [{E}_{{\alpha }_{1}+{\alpha }_{2}},{F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]\\ =-{F}_{{\alpha }_{1}+{\alpha }_{3}},[{E}_{{\alpha }_{1}+{\alpha }_{3}},{H}_{{\alpha }_{1}}]=-\alpha {E}_{{\alpha }_{1}+{\alpha }_{3}},\quad [{E}_{{\alpha }_{1}+{\alpha }_{3}},{H}_{{\alpha }_{2}}]\\ ={E}_{{\alpha }_{1}+{\alpha }_{3}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{3}},{H}_{{\alpha }_{3}}]=-{E}_{{\alpha }_{1}+{\alpha }_{3}},\quad [{E}_{{\alpha }_{1}+{\alpha }_{3}},{F}_{{\alpha }_{1}}]\\ =-\alpha {E}_{{\alpha }_{3}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{3}},{F}_{{\alpha }_{3}}]=-{E}_{{\alpha }_{1}},\quad [{E}_{{\alpha }_{1}+{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{3}}]\\ =-{H}_{{\alpha }_{1}}+\alpha {H}_{{\alpha }_{3}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]=-{F}_{{\alpha }_{2}},\quad [{E}_{{\alpha }_{1}+{\alpha }_{3}},{F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]\\ =-{F}_{{\alpha }_{1}+{\alpha }_{2}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{H}_{{\alpha }_{1}}]=-(1+\alpha ){E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\quad [{E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{H}_{{\alpha }_{2}}]\\ ={E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{H}_{{\alpha }_{3}}]=-{E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\quad [{E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{F}_{{\alpha }_{2}}]\\ =-{E}_{{\alpha }_{1}+{\alpha }_{3}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{F}_{{\alpha }_{3}}]=-{E}_{{\alpha }_{1}+{\alpha }_{2}},\quad [{E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{2}}]=\alpha {E}_{{\alpha }_{3}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{2}}]={E}_{{\alpha }_{2}},\quad [{E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]\\ ={H}_{{\alpha }_{1}}-{H}_{{\alpha }_{2}}-\alpha {H}_{{\alpha }_{3}},\\ [{E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]=-{F}_{{\alpha }_{1}},\quad [{E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{H}_{{\alpha }_{1}}]\\ =-(1+\alpha ){E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\\ [{E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{F}_{{\alpha }_{1}}]={E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\quad [{E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{2}}]\\ ={E}_{{\alpha }_{1}+{\alpha }_{3}},\\ [{E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{3}}]={E}_{{\alpha }_{1}+{\alpha }_{2}},\quad [{E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]\\ ={E}_{{\alpha }_{1}},[{E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},{F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]=\displaystyle \frac{2}{1+\alpha }{H}_{{\alpha }_{1}}-\displaystyle \frac{1}{1+\alpha }{H}_{{\alpha }_{2}}\\ +\displaystyle \frac{\alpha }{1+\alpha }{H}_{{\alpha }_{3}},[{H}_{{\alpha }_{1}},{F}_{{\alpha }_{2}}]=-{F}_{{\alpha }_{2}},\quad [{H}_{{\alpha }_{1}},{F}_{{\alpha }_{3}}]=-\alpha {F}_{{\alpha }_{3}},\\ [{H}_{{\alpha }_{1}},{F}_{{\alpha }_{1}+{\alpha }_{2}}]=-{F}_{{\alpha }_{1}+{\alpha }_{2}},\quad [{H}_{{\alpha }_{1}},{F}_{{\alpha }_{1}+{\alpha }_{3}}]=-\alpha {F}_{{\alpha }_{1}+{\alpha }_{2}},\\ [{H}_{{\alpha }_{1}},{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]=-(1+\alpha ){F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\quad [{H}_{{\alpha }_{1}},{F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]\\ =-(1+\alpha ){F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\\ [{H}_{{\alpha }_{2}},{F}_{{\alpha }_{1}}]={F}_{{\alpha }_{1}},\quad [{H}_{{\alpha }_{2}},{F}_{{\alpha }_{2}}]=-2{F}_{{\alpha }_{2}},\\ [{H}_{{\alpha }_{2}},{F}_{{\alpha }_{1}+{\alpha }_{2}}]=-{F}_{{\alpha }_{1}+{\alpha }_{2}},\quad [{H}_{{\alpha }_{2}},{F}_{{\alpha }_{1}+{\alpha }_{3}}]={F}_{{\alpha }_{1}+{\alpha }_{3}},\\ [{H}_{{\alpha }_{2}},{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]=-{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\quad [{H}_{{\alpha }_{3}},{F}_{{\alpha }_{1}}]={F}_{{\alpha }_{1}},\\ [{H}_{{\alpha }_{3}},{F}_{{\alpha }_{2}}]=-2{F}_{{\alpha }_{3}},\quad [{H}_{{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{2}}]={F}_{{\alpha }_{1}+{\alpha }_{2}},\\ [{H}_{{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{3}}]=-{F}_{{\alpha }_{1}+{\alpha }_{3}},\quad [{H}_{{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]=-{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\\ [{F}_{{\alpha }_{1}},{F}_{{\alpha }_{2}}]=-{F}_{{\alpha }_{1}+{\alpha }_{2}},\quad [{F}_{{\alpha }_{1}},{F}_{{\alpha }_{3}}]=-{F}_{{\alpha }_{1}+{\alpha }_{3}},\\ [{F}_{{\alpha }_{1}},{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}]=(1+\alpha ){F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\quad \quad \\ [{F}_{{\alpha }_{2}},{F}_{{\alpha }_{1}+{\alpha }_{3}}]={F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\,[{F}_{{\alpha }_{3}},{F}_{{\alpha }_{1}+{\alpha }_{2}}]\end{array}\end{eqnarray*}$

(2.7)$\begin{eqnarray}=\,{F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\quad [{F}_{{\alpha }_{1}+{\alpha }_{2}},{E}_{{\alpha }_{1}+{\alpha }_{3}}]=-(1+\alpha ){F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\end{eqnarray}$

and all the other commutators are zero.

3. Differential operator representation of D(2, 1; α)

To obtain a shift operator [22] of D(2, 1; α), one needs to construct the differential operator representations [23–31] of the Lie superalgebra D(2, 1; α). Let ⟨Λ∣ be the highest weight vector in the representation of D(2, 1; α) with the highest weights λ_i, satisfying the following conditions:

(3.1)$\begin{eqnarray}\langle {\rm{\Lambda }}| {F}_{{\alpha }_{i}}=0,\end{eqnarray}$

(3.2)$\begin{eqnarray}\langle {\rm{\Lambda }}| {H}_{{\alpha }_{i}}={\lambda }_{i}\langle {\rm{\Lambda }}| .\end{eqnarray}$

An arbitrary vector in the representation space is parametrized by the bosonic coordinate variables ${x}_{{\alpha }_{i}}$ and fermionic coordinate variables ${\theta }_{{\alpha }_{i}}$,

(3.3)$\begin{eqnarray}\langle {\rm{\Lambda }},x,\theta | =\langle {\rm{\Lambda }}| {G}_{+}(x,\theta ).\end{eqnarray}$

We constructed the corresponding G₊(x, θ) as follows:

(3.4)$\begin{eqnarray}{G}_{+}(x,\theta )={G}_{{\alpha }_{3}}{G}_{{\alpha }_{2}}{G}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{G}_{{\alpha }_{1}+{\alpha }_{3}}{G}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{G}_{{\alpha }_{1}+{\alpha }_{2}}{G}_{{\alpha }_{1}},\end{eqnarray}$

and the associated G_δ are given by (e is Euler's number)

(3.5)$\begin{eqnarray}{G}_{\delta }=\Space{0ex}{2.5ex}{0ex}\{\begin{array}{ll}{e}^{{x}_{{\alpha }_{i}}{E}_{{\alpha }_{i}}}, & \mathrm{if}\quad [{E}_{{\alpha }_{i}}]=0,\\ {e}^{{\theta }_{{\alpha }_{i}}{E}_{{\alpha }_{i}}}, & \mathrm{if}\quad [{E}_{{\alpha }_{i}}]=1.\end{array}\end{eqnarray}$

One can define a differential operator realization ρ^(d) of the generators of Lie superalgebra D(2, 1; α) by the following relation

(3.6)$\begin{eqnarray}{\rho }^{(d)}(g)\langle {\rm{\Lambda }},x,\theta | \equiv \langle {\rm{\Lambda }},x,\theta | g,\quad \forall g\in D(2,1;\alpha ).\end{eqnarray}$

Here, ρ^(d)(g) is a differential operator of the bosonic and fermionic coordinate variables $\{{x}_{{\alpha }_{i}},{\theta }_{{\alpha }_{i}}\}$ associated with the generator g. After some manipulations, we obtain the following differential operator representations of all generators of Lie superalgebra D(2, 1; α):

(3.7)$\begin{eqnarray}{\rho }^{(d)}({E}_{{\alpha }_{1}})={\partial }_{{\theta }_{{\alpha }_{1}}},\end{eqnarray}$

(3.8)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({E}_{{\alpha }_{2}})=-{\theta }_{{\alpha }_{1}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}}}\\ -{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}+{\partial }_{{x}_{{\alpha }_{2}}},\end{array}\end{eqnarray}$

(3.9)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({E}_{{\alpha }_{3}})=-{\theta }_{{\alpha }_{1}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}-(1+\alpha ){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ \,-{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}+{\partial }_{{x}_{{\alpha }_{3}}},\end{array}\end{eqnarray}$

(3.10)$\begin{eqnarray}{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}})={\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}}},\end{eqnarray}$

(3.11)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{3}})={\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}\\ +(1+\alpha ){\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}},\end{array}\end{eqnarray}$

(3.12)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})={\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ -(1+\alpha ){\theta }_{{\alpha }_{1}}{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}},\end{array}\end{eqnarray}$

(3.13)$\begin{eqnarray}{\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})={\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}},\end{eqnarray}$

(3.14)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({H}_{{\alpha }_{1}})=-{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}}}\\ -(1+\alpha ){x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}-\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}\\ -(1+\alpha ){\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ -{x}_{{\alpha }_{2}}{\partial }_{{x}_{{\alpha }_{2}}}-\alpha {x}_{{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{3}}}+{\lambda }_{{\alpha }_{1}},\end{array}\end{eqnarray}$

(3.15)$\begin{eqnarray}\begin{array}{rcl}{\rho }^{(d)}({H}_{{\alpha }_{2}}) & = & {\theta }_{{\alpha }_{1}}{\partial }_{{\theta }_{{\alpha }_{1}}}-{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}}}+{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}\\ & & -{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}-2{x}_{{\alpha }_{2}}\partial {x}_{{\alpha }_{2}}+{\lambda }_{{\alpha }_{2}},\end{array}\end{eqnarray}$

(3.16)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({H}_{{\alpha }_{3}})={\theta }_{{\alpha }_{1}}{\partial }_{{\theta }_{{\alpha }_{1}}}+{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}}}\\ -{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}\\ -{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ -2{x}_{{\alpha }_{3}}{\partial }_{{\alpha }_{3}}+{\lambda }_{{\alpha }_{3}},\end{array}\end{eqnarray}$

(3.17)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({F}_{{\alpha }_{1}})={\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}-{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{x}_{{\alpha }_{2}}}\\ +{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ -\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{3}}}-{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}}}\\ -(1+\alpha ){\theta }_{{\alpha }_{1}}{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ -\alpha {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{a}_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}-(1+\alpha ){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ -{\theta }_{{\alpha }_{1}}{x}_{{\alpha }_{2}}{\partial }_{{x}_{{\alpha }_{2}}}\\ -\alpha {\theta }_{{\alpha }_{1}}{x}_{{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{3}}}\\ +{\theta }_{{\alpha }_{1}}{\lambda }_{{\alpha }_{1}},\end{array}\end{eqnarray}$

(3.18)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({F}_{{\alpha }_{2}})=-{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{\theta }_{{\alpha }_{1}}}-{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}\\ -{x}_{{\alpha }_{2}}^{2}{\partial }_{{x}_{{\alpha }_{2}}}+{x}_{{\alpha }_{2}}{\lambda }_{{\alpha }_{2}},\end{array}\end{eqnarray}$

(3.19)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({F}_{{\alpha }_{3}})=-{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}}}+(1+\alpha ){\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}\\ \times \,{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}-{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}}}-{x}_{{\alpha }_{3}}^{2}{\partial }_{{x}_{{\alpha }_{3}}}+{x}_{{\alpha }_{3}}{\lambda }_{{\alpha }_{3}},\end{array}\end{eqnarray}$

(3.20)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}})=(1+\alpha ){\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}\\ +\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ -{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{x}_{{\alpha }_{2}}{\partial }_{{x}_{{\alpha }_{2}}}+{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}\\ +\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{3}}}-{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{\theta }_{{\alpha }_{1}}}\\ +(1+\alpha ){\theta }_{{\alpha }_{1}+{\alpha }_{2}}{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ -{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}\\ +\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{x}_{{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{3}}}\\ +{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\lambda }_{{\alpha }_{2}}-{\theta }_{\alpha 1+{\alpha }_{2}}{\lambda }_{{\alpha }_{1}}\\ +{\theta }_{{\alpha }_{1}}{x}_{{\alpha }_{2}}{\lambda }_{{\alpha }_{2}}\\ -{\theta }_{{\alpha }_{1}}{x}_{{\alpha }_{2}}^{2}{\partial }_{{x}_{{\alpha }_{2}}},\end{array}\end{eqnarray}$

(3.21)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{3}})=-{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ +{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{2}}}\\ -\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{3}}{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ +{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}}}\\ -\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{3}}{x}_{{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{3}}}\\ -\alpha {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}}}\\ +\alpha (1+\alpha ){\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}}{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ -\alpha {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}}}\\ +{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{x}_{{\alpha }_{2}}{\partial }_{{x}_{{\alpha }_{2}}}-\alpha {\theta }_{{\alpha }_{1}}{x}_{{\alpha }_{3}}^{2}{\partial }_{{x}_{{\alpha }_{3}}}\\ +\alpha {\theta }_{{\alpha }_{1}}{x}_{{\alpha }_{3}}{\lambda }_{{\alpha }_{3}}-{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\lambda }_{{\alpha }_{1}}\\ +\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\lambda }_{{\alpha }_{3}},\end{array}\end{eqnarray}$

(3.22)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})=\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}}}+{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}}}\\ +{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}\\ +{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{x}_{{\alpha }_{2}}^{2}{\partial }_{{x}_{{\alpha }_{2}}}\\ -\alpha (1+\alpha ){\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ +\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}}}\\ +{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{x}_{{\alpha }_{2}}{\partial }_{{x}_{{\alpha }_{2}}}\\ +\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{x}_{{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{3}}}+\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{x}_{{\alpha }_{3}}^{2}{\partial }_{{x}_{{\alpha }_{3}}}\\ -\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\lambda }_{{\alpha }_{3}}-\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{x}_{{\alpha }_{3}}{\lambda }_{{\alpha }_{3}}\\ -{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{x}_{{\alpha }_{2}}{\lambda }_{{\alpha }_{2}}-{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\lambda }_{{\alpha }_{2}}+{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\lambda }_{{\alpha }_{1}},\end{array}\end{eqnarray}$

(3.23)$\begin{eqnarray}\begin{array}{l}{\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})={\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ -{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{x}_{{\alpha }_{2}}{\partial }_{{x}_{{\alpha }_{2}}}\\ +\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{x}_{{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{3}}}\\ -{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{2}}}\\ -{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}\\ -{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{\alpha }_{1}+{\alpha }_{2}}\\ -\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{3}}}\\ -\alpha {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{\alpha }_{1}}\\ +\alpha (1+\alpha ){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ +\alpha {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\partial }_{{\alpha }_{1}+{\alpha }_{2}}\\ -\alpha {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{x}_{{\alpha }_{3}}^{2}{\partial }_{{x}_{{\alpha }_{3}}}\\ -{\theta }_{{\alpha }_{1}}{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}}}\\ -{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{2}{\partial }_{{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ +{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{3}}}\\ -{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{x}_{{\alpha }_{2}}^{2}{\partial }_{{x}_{{\alpha }_{2}}}\\ -{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\partial }_{{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}}\\ -{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{x}_{{\alpha }_{2}}{\partial }_{{x}_{{\alpha }_{2}}}\\ -\alpha {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{x}_{{\alpha }_{3}}{\partial }_{{x}_{{\alpha }_{3}}}+{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\lambda }_{{\alpha }_{1}}\\ +\displaystyle \frac{2}{1+\alpha }{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\lambda }_{{\alpha }_{1}}-{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\lambda }_{{\alpha }_{1}}\\ +\alpha {\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\lambda }_{{\alpha }_{3}}-\displaystyle \frac{\alpha }{1+\alpha }{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\lambda }_{{\alpha }_{3}}\\ +\alpha {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{x}_{{\alpha }_{3}}{\lambda }_{{\alpha }_{3}}+\alpha {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\lambda }_{{\alpha }_{3}}\\ -\displaystyle \frac{1}{1+\alpha }{x}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\lambda }_{{\alpha }_{2}}\\ +{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{x}_{{\alpha }_{2}}{\lambda }_{{\alpha }_{2}}+{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\lambda }_{{\alpha }_{2}}.\end{array}\end{eqnarray}$

One can directly check that the differential operator realizations satisfy the commutation relations of Lie superalgebra D(2, 1; α) [21].

4. Shift operator of D(2, 1; a)

The even part of Lie superalgebra D(2, 1; α) is su(2) ⊕ su(2) ⊕ su(2), with the basis s_i, t_i, u_i(i = 0, ± ), satisfying the relations

(4.1)$\begin{eqnarray}\begin{array}{l}[{s}_{0},{s}_{\pm }]=\pm {s}_{\pm },\quad [{t}_{0},{t}_{\pm }]=\pm {t}_{\pm },\\ [{u}_{0},{u}_{\pm }]=\pm {u}_{\pm },\quad [{s}_{+},{s}_{-}]=2{s}_{0},\\ [{t}_{+},{t}_{-}]=2{t}_{0},\quad [{u}_{+},{u}_{-}]=2{u}_{0}.\end{array}\end{eqnarray}$

The odd part of Lie superalgebras D(2, 1; α) is a spinor representation (2, 2, 2) of the even part, with components ${R}_{{ijk}}(i,j,k=\pm \tfrac{1}{2})$ [22]. In our assumption, the elements of D(2, 1; α) are given by

(4.2)$\begin{eqnarray}\begin{array}{l}{s}_{0}=\displaystyle \frac{1}{2(1+\alpha )}(2{H}_{{\alpha }_{1}}-{H}_{{\alpha }_{2}}-\alpha {H}_{{\alpha }_{3}}),\\ {s}_{+}=-{{iE}}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\quad {s}_{-}={{iF}}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\\ {t}_{0}=\displaystyle \frac{1}{2}{H}_{{\alpha }_{2}},\quad {t}_{+}={E}_{{\alpha }_{2}},\\ {t}_{-}={F}_{{\alpha }_{2}},\quad {u}_{0}=\displaystyle \frac{1}{2}{H}_{{\alpha }_{3}},\\ {u}_{+}={E}_{{\alpha }_{3}},\quad {u}_{-}={F}_{{\alpha }_{3}},\\ {R}_{\displaystyle \frac{1}{2},\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}}={E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\quad {R}_{-\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2}}={{iF}}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}},\\ {R}_{\displaystyle \frac{1}{2},\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2}}={E}_{{\alpha }_{1}+{\alpha }_{2}},\quad {R}_{-\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}}={{iF}}_{{\alpha }_{1}+{\alpha }_{2}},\\ {R}_{\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}}={E}_{{\alpha }_{1}+{\alpha }_{3}},\quad {R}_{-\displaystyle \frac{1}{2},\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2}}={F}_{{\alpha }_{1}+{\alpha }_{3}},\\ {R}_{\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2}}={E}_{{\alpha }_{1}},\quad {R}_{-\displaystyle \frac{1}{2},\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}}={{iF}}_{{\alpha }_{1}}.\end{array}\end{eqnarray}$

The invariant scalars of the Lie subalgebra of D(2, 1; α) are given by

(4.3)$\begin{eqnarray}\begin{array}{l}{S}^{2}={s}_{+}{s}_{-}+{s}_{0}^{2}-{s}_{0},\quad {T}^{2}={t}_{+}{t}_{-}+{t}_{0}^{2}-{t}_{0},\\ {U}^{2}={u}_{+}{u}_{-}+{u}_{0}^{2}-{u}_{0}.\end{array}\end{eqnarray}$

Irreducible representations of Lie superalgebra can be reduced into the direct sum of a set of irreducible representations of subalgebra. The representation of su(2) ⊕ su(2) ⊕ su(2) can be labeled by (s, t, u), where s(s + 1), t(t + 1), u(u + 1) are the eigenvalues of the subalgebra invariants S², T², U². And the representations of D(2, 1; α) are labeled by ∣s, m_s; t, m_t; u, m_u; λ⟩, where m_s, m_t, m_u are eigenvalues of the s₀, t₀, u₀. The degeneracy representations can be labeled by λ. The operator $\hat{s}$ is defined by

(4.4)$\begin{eqnarray}\hat{s}| s,{m}_{s};t,{m}_{t};u,{m}_{u};\lambda \rangle =s| s,{m}_{s};t,{m}_{t};u,{m}_{u};\lambda \rangle .\end{eqnarray}$

The operators $\hat{t}$ and $\hat{u}$ are defined in the same way. Let (p, q, r) be the corresponding (s, t, u) values, and p be the maximum s value in the reduction of a D(2, 1; α) representation. Therefore, the decomposition into su(2) ⊕ su(2) ⊕ su(2) is given by

(4.5)$\begin{eqnarray}\begin{array}{l}{\mathbb{F}}=\{(p,q,r),\left(p-\displaystyle \frac{1}{2},q\pm \displaystyle \frac{1}{2},r\pm \displaystyle \frac{1}{2}\right),\left(p-\displaystyle \frac{1}{2},q\pm \displaystyle \frac{1}{2},r\mp \displaystyle \frac{1}{2}\right),\\ (p-1,q\pm 1,r),(p-1,q,r\pm 1),(p-1,q,r;1),(p-1,q,r;2),\\ \left(p-\displaystyle \frac{3}{2},q\pm \displaystyle \frac{1}{2},r\pm \displaystyle \frac{1}{2}\right),\left(p-\displaystyle \frac{3}{2},q\pm \displaystyle \frac{1}{2},r\mp \displaystyle \frac{1}{2}\right),(p-2,q,r)\}.\end{array}\end{eqnarray}$

The (s, t, u) = (p − 1, q, r) is a twofold degeneracy. Therefore, the multiplicity of the (s, t, u) representation is denoted as ∣p − 1, m_p; q, m_q; r, m_r; λ⟩(λ = 1, 2).

The shift operators ${O}^{i,j,k}\left(i,j,k=\tfrac{1}{2}\right)$ have been studied by Hughes and Yadegar [32]. The shift operators for D(2, 1; α) are given by Van der Jeugt [22]

(4.6)$\begin{eqnarray}\begin{array}{l}{O}^{\displaystyle \frac{1}{2},\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}}=-{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})(\hat{s}+{s}_{0}+1)(\hat{t}+{t}_{0}+1)\\ \times (\hat{u}+{u}_{0}+1)-{\rho }^{(d)}({F}_{{\alpha }_{1}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{t}+{t}_{0}+1)(\hat{u}+{u}_{0}+1)\\ -{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{3}})(\hat{s}+{s}_{0}+1){\rho }^{(d)}({E}_{{\alpha }_{2}})(\hat{u}+{u}_{0}+1)\\ -{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}})(\hat{s}+{s}_{0}+1)(\hat{t}+{t}_{0}+1){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({E}_{{\alpha }_{2}})(\hat{u}+{u}_{0}+1)\\ -{\rho }^{(d)}({E}_{{\alpha }_{1}})(\hat{s}+{s}_{0}+1){\rho }^{(d)}({E}_{{\alpha }_{2}}){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{3}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})(\hat{t}+{t}_{0}+1){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}\times \,({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({E}_{{\alpha }_{2}}){\rho }^{(d)}({E}_{{\alpha }_{3}}),\end{array}\end{eqnarray}$

(4.7)$\begin{eqnarray}\begin{array}{l}{O}^{\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}}=-{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{s}+{s}_{0}+1){\rho }^{(d)}({F}_{{\alpha }_{2}})(\hat{u}+{u}_{0}+1)\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({F}_{{\alpha }_{2}})(\hat{u}+{u}_{0}+1)\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{3}})(\hat{s}+{s}_{0}+1)(\hat{t}+{t}_{0})(\hat{u}+{u}_{0}+1)\\ -{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}})(\hat{s}+{s}_{0}+1){\rho }^{(d)}({F}_{{\alpha }_{2}}){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{t}+{t}_{0})(\hat{u}+{u}_{0}+1)\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}})(\hat{s}+{s}_{0}+1)(\hat{t}+{t}_{0}){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{3}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}\\ \times \,({F}_{{\alpha }_{2}}){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{t}+{t}_{0}){\rho }^{(d)}({E}_{{\alpha }_{3}}),\end{array}\end{eqnarray}$

(4.8)$\begin{eqnarray}\begin{array}{l}{O}^{\displaystyle \frac{1}{2},\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2}}=-{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{s}+{s}_{0}+1)(\hat{t}+{t}_{0}+1){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{t}+{t}_{0}+1){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{3}})(\hat{s}+{s}_{0}+1){\rho }^{(d)}\\ \times \,({E}_{{\alpha }_{2}}){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}})(\hat{s}+{s}_{0}+1)\\ \times \,(\hat{t}+{t}_{0}+1)(\hat{u}+{u}_{0})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}\\ \times \,({E}_{{\alpha }_{2}}){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}})(\hat{s}+{s}_{0}+1){\rho }^{(d)}\\ \times \,({E}_{{\alpha }_{2}})(\hat{u}+{u}_{0})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{3}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{t}+{t}_{0}+1)(\hat{u}+{u}_{0})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\times \,{\rho }^{(d)}({E}_{{\alpha }_{2}})(\hat{u}+{u}_{0}),\end{array}\end{eqnarray}$

(4.9)$\begin{eqnarray}\begin{array}{l}{O}^{\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2}}=-{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{s}+{s}_{0}+1){\rho }^{(d)}({F}_{{\alpha }_{2}}){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({F}_{{\alpha }_{2}}){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{3}})(\hat{s}+{s}_{0}+1)(\hat{t}+{t}_{0}){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}})(\hat{s}+{s}_{0}+1){\rho }^{(d)}\\ \times \,({F}_{{\alpha }_{2}})(\hat{u}+{u}_{0})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{t}+{t}_{0}){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({E}_{{\alpha }_{1}})(\hat{s}+{s}_{0}+1)(\hat{t}+{t}_{0})(\hat{u}+{u}_{0})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{3}}){\rho }^{(d)}({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,{\rho }^{(d)}({F}_{{\alpha }_{2}})(\hat{u}+{u}_{0})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}\\ \times \,({E}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})(\hat{t}+{t}_{0})(\hat{u}+{u}_{0}),\end{array}\end{eqnarray}$

(4.10)$\begin{eqnarray}\begin{array}{l}{O}^{-\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2}}=-{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,{\rho }^{(d)}({F}_{{\alpha }_{2}}){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,{\rho }^{(d)}({F}_{{\alpha }_{2}})(\hat{u}+{u}_{0})\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{3}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{t}+{t}_{0}){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{t}+{t}_{0})(\hat{u}+{u}_{0})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}})(\hat{s}+{s}_{0}){\rho }^{(d)}\\ \times \,({F}_{{\alpha }_{2}}{\rho }^{(d)}({F}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{3}})(\hat{s}+{s}_{0})\\ \times \,{\rho }^{(d)}({F}_{{\alpha }_{2}})(\hat{u}+{u}_{0})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}})(\hat{s}+{s}_{0})\\ \times \,(\hat{t}+{t}_{0}){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})(\hat{s}+{s}_{0})\\ \times \,(\hat{t}+{t}_{0})(\hat{u}+{u}_{0}),\end{array}\end{eqnarray}$

(4.11)$\begin{eqnarray}\begin{array}{l}{O}^{-\displaystyle \frac{1}{2},\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}}=-{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,{\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})(\hat{t}+{t}_{0}+1)(\hat{u}+{u}_{0}+1)\\ -{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{t}+{t}_{0}+1){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{3}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,{\rho }^{(d)}({E}_{{\alpha }_{2}})(\hat{u}+{u}_{0}+1)\\ -{\rho }^{(d)}({E}_{{\alpha }_{1}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}\\ \times \,({E}_{{\alpha }_{2}}){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}})(\hat{s}+{s}_{0})(\hat{t}+{t}_{0}+1)(\hat{u}+{u}_{0}+1)\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{3}})(\hat{s}+{s}_{0})\\ \times \,(\hat{t}+{t}_{0}+1){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}})(\hat{s}+{s}_{0}){\rho }^{(d)}\\ \times \,({E}_{{\alpha }_{2}})(\hat{u}+{u}_{0}+1)\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})(\hat{s}+{s}_{0})\\ \times \,{\rho }^{(d)}({E}_{{\alpha }_{2}}){\rho }^{(d)}({E}_{{\alpha }_{3}}),\end{array}\end{eqnarray}$

(4.12)$\begin{eqnarray}\begin{array}{l}{O}^{-\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}}=-{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,{\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({F}_{{\alpha }_{2}})(\hat{u}+{u}_{0}+1)\\ -{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({F}_{{\alpha }_{2}}){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{3}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})(\hat{t}+{t}_{0})(\hat{u}+{u}_{0}+1)\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})(\hat{t}+{t}_{0}){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}})(\hat{s}+{s}_{0}){\rho }^{(d)}({F}_{{\alpha }_{2}})(\hat{u}+{u}_{0}+1)\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{3}})(\hat{s}+{s}_{0}){\rho }^{(d)}({F}_{{\alpha }_{2}}){\rho }^{(d)}({E}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}})(\hat{s}+{s}_{0})(\hat{t}+{t}_{0})(\hat{u}+{u}_{0}+1)\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})(\hat{s}+{s}_{0})(\hat{t}+{t}_{0}){\rho }^{(d)}({E}_{{\alpha }_{3}}),\end{array}\end{eqnarray}$

(4.13)$\begin{eqnarray}\begin{array}{l}{O}^{-\displaystyle \frac{1}{2},\displaystyle \frac{1}{2},-\displaystyle \frac{1}{2}}=-{\rho }^{(d)}\\ \times \,({E}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})(\hat{t}+{t}_{0}+1){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{2}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})(\hat{t}+{t}_{0}+1)(\hat{u}+{u}_{0})\\ -{\rho }^{(d)}({E}_{{\alpha }_{1}+{\alpha }_{3}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({E}_{{\alpha }_{2}}){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ +{\rho }^{(d)}({E}_{{\alpha }_{1}}){\rho }^{(d)}({F}_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}){\rho }^{(d)}({E}_{{\alpha }_{2}})(\hat{u}+{u}_{0})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}})(\hat{s}+{s}_{0})\\ \times \,(\hat{t}+{t}_{0}+1){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{3}})(\hat{s}+{s}_{0})\\ \times \,(\hat{t}+{t}_{0}+1)(\hat{u}+{u}_{0})\\ +{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}})(\hat{s}+{s}_{0}){\rho }^{(d)}\\ \times \,({E}_{{\alpha }_{2}}){\rho }^{(d)}({F}_{{\alpha }_{3}})\\ -{\rho }^{(d)}({F}_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}})\\ \times \,(\hat{s}+{s}_{0}){\rho }^{(d)}({E}_{{\alpha }_{2}})(\hat{u}+{u}_{0}).\end{array}\end{eqnarray}$

The shift operators O^i,j,k shift an eigenstate into one or two eigenstates (for the twofold degenerate case),

(4.14)$\begin{eqnarray}\begin{array}{l}{O}^{i,j,k}| s,{m}_{s};t,{m}_{t};u,{m}_{u};\lambda \rangle \\ \propto \displaystyle \sum _{{\lambda }^{{\prime} }}| s+i,{m}_{s}+i;t+j,{m}_{t}+j;u+k,{m}_{u}+k;{\lambda }^{{\prime} }\rangle .\end{array}\end{eqnarray}$

The normalized shift operator A^i,j,k is

(4.15)$\begin{array}{l} A^{i, j, k}=O^{i, j, k}\left[( \hat { s } + s _ { 0 } + \frac { 1 } { 2 } + i ) \left(\hat{t}+t_{0}\right.\right. \\ \left.\left.+\frac{1}{2}+j\right)\left(\hat{u}+u_{0}+\frac{1}{2}+k\right)\right]^{-\frac{1}{2}} \end{array}$

5. Representations of D(2, 1; α)

The exceptional Lie superalgebra D(2, 1; α) (α ≠ 0, − 1) forms

(5.1)$\begin{eqnarray}\begin{array}{l}| p-2,{m}_{p};q,{m}_{q};r,{m}_{r}\rangle ={\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}\\ \times {\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-2}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}},\end{array}\end{eqnarray}$

(5.2)$\begin{eqnarray}\begin{array}{l}\left|p-\displaystyle \frac{3}{2},{m}_{p};q+\displaystyle \frac{1}{2},{m}_{q};r+\displaystyle \frac{1}{2},{m}_{r}\right\rangle \\ =\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ -(q-{m}_{q}+\displaystyle \frac{1}{2})\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ -\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\\ \times \,{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}},\end{array}\end{eqnarray}$

(5.3)$\begin{eqnarray}\begin{array}{l}\left|p-\displaystyle \frac{3}{2},{m}_{p};q-\displaystyle \frac{1}{2},{m}_{q};r+\displaystyle \frac{1}{2},{m}_{r}\right\rangle \\ =(r+{m}_{r}+\displaystyle \frac{1}{2}){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ +\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}},\end{array}\end{eqnarray}$

(5.4)$\begin{eqnarray}\begin{array}{l}\left|p-\displaystyle \frac{3}{2},{m}_{p};q+\displaystyle \frac{1}{2},{m}_{q};r-\displaystyle \frac{1}{2},{m}_{r}\right\rangle \\ =\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}\\ \times \,{\chi }_{{\alpha }_{3}}^{r+\displaystyle \frac{1}{2}-{m}_{r}}-\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}-\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ +(q-{m}_{q}+\displaystyle \frac{1}{2}){\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}\\ {\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}},\end{array}\end{eqnarray}$

(5.5)$\begin{eqnarray}\begin{array}{l}\left|p-\displaystyle \frac{3}{2},{m}_{p};q-\displaystyle \frac{1}{2},{m}_{q};r-\displaystyle \frac{1}{2},{m}_{r}\right\rangle \\ ={\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}\,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}\,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}\,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}\,\times \,{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ -{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}\\ \times \,{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}-{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}},\end{array}\end{eqnarray}$

(5.6)$\begin{eqnarray}\begin{array}{l}\left|p-1,{m}_{p};q,{m}_{q};r-1,{m}_{r}\right\rangle \\ =-4(p-1)q{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ -4(p-1)q{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+1}\\ +4(p-1)q{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ -4(p-1)q{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-1}-(p-{m}_{p}-1)\\ [4(1+\alpha )(p-1)q+\alpha (p-{m}_{p}-2)(q+{m}_{q})]\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{2}}{\theta }_{{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-2}\,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}},\end{array}\end{eqnarray}$

(5.7)$\begin{eqnarray}\begin{array}{l}| p-1,{m}_{p};q-1,{m}_{q};r,{m}_{r}\rangle \\ =4(p-1)r{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +4(p-1)r{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+1}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +4(p-1)r{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ -4(p-1)r{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-1}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +(p-{m}_{p}-1)\alpha (p-{m}_{p}-2)(r-{m}_{r})\\ \times \,{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{2}}{\theta }_{{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-2}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}},\end{array}\end{eqnarray}$

(5.8)$\begin{eqnarray}\begin{array}{l}\left|p-1,{m}_{p};q+1,{m}_{q};r,{m}_{r}\right\rangle \\ =4(p-1)r(q+{m}_{q}+1)(q+{m}_{q})r{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+1}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ -4(p-1)r(q+{m}_{q}+1)(q-{m}_{q}+1)r{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}\times \,{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ -4(p-1)r(q+{m}_{q}+1)(q-{m}_{q}+1)r{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +4(p-1)r(q-{m}_{q}+1)(q-{m}_{q})r{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-1}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}-(p-{m}_{p}-1)\\ (q+1-{m}_{q})(q+{m}_{q}+1)\alpha (p-{m}_{p}-2)(r-{m}_{r})\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{2}}{\theta }_{{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-2}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}},\end{array}\end{eqnarray}$

(5.9)$\begin{eqnarray}\begin{array}{l}| p-1,{m}_{p};q,{m}_{q};r+1,{m}_{r}\rangle \\ =-2(r+{m}_{r}+1)(r+{m}_{r})q{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+1}\\ -2(r-{m}_{r}+1)(r+{m}_{r}+1)q{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +2(r-{m}_{r}+1)(r+{m}_{r}+1)q{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ -2(r-{m}_{r}+1)(r-{m}_{r})q{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-1}+(r+{m}_{r}+1)\\ [(q+{m}_{q})\alpha (p-{m}_{p}-2)+4(1+\alpha )(p-1)q]\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{2}}{\theta }_{{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-2}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}},\end{array}\end{eqnarray}$

(5.10)$\begin{eqnarray}\begin{array}{l}\left|p-1,{m}_{p};q,{m}_{q};r,{m}_{r};1\right\rangle \\ =4(p-1)(q+1)(r+{m}_{r})(q+{m}_{q}+1){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+1}+2(p-1)(q+{m}_{q}+1)\\ [(q+{m}_{q}+1)(r+{m}_{r})-(q-{m}_{q}+1)(r-{m}_{r})]\\ \times \,{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ {\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +4(p-1)r(q+{m}_{q}+1)(q+{m}_{q}){\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+1}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}+2(p-1)(q+{m}_{q}+1)\\ [(q+{m}_{q}+1)(r-{m}_{r})-(r+{m}_{r})(q-{m}_{q}+1)]\\ {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ -4(p-1)r(q+{m}_{q}+1)(q-{m}_{q}){\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-1}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ -4(p-1)(q+1)(q+{m}_{q}+1)(r-{m}_{r}){\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-1}+(p-{m}_{p}-1)(q+{m}_{q}+1)\\ [-4\alpha r(q+1)(r+1)+4(1+\alpha )(p-1)(q+1)m-r\\ -\alpha (p-{m}_{p}-2)(r-{m}_{r})(q+{m}_{q}+2)+4{qr}(q+1)]\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{2}}{\theta }_{{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}\\ \times \,{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-2}\\ \times \,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}},\end{array}\end{eqnarray}$

(5.11)$\begin{eqnarray}\begin{array}{l}| p-1,{m}_{p};q,{m}_{q};r,{m}_{r};2\rangle =-4(p-1)(r+{m}_{r})q{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+1}\\ +4(p-1)(q+{m}_{q})r{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}\,{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+1}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +2(p-1)[(q+{m}_{q})(r+{m}_{r})-(q-{m}_{q})(r-{m}_{r})]\\ {\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +2(p-1)[-(q-{m}_{q})(r+{m}_{r})+(q+{m}_{q})(r-{m}_{r})]\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ -4(p-1)(q-{m}_{q})r{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-1}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ -4(p-1)(r-{m}_{r})q{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-1}\\ +(p-{m}_{p}-1)\left\{2(1+\alpha )(1-p)\left[{m}_{r}(2q+1)-\displaystyle \frac{1}{2}\right]+4{qr}(q+\alpha r+1+\alpha )\right.\\ \left.+{m}_{q}(2{m}_{r}-1)+\alpha {m}_{r}(2{m}_{q}-1)+\alpha (p-2-{m}_{p})(q+{m}_{q})(r-{m}_{r})\right\}\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{2}}{\theta }_{{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-2}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}},\end{array}\end{eqnarray}$

(5.12)$\begin{eqnarray}\begin{array}{l}\left|p-\displaystyle \frac{1}{2},{m}_{p};q-\displaystyle \frac{1}{2},{m}_{q};r-\displaystyle \frac{1}{2},{m}_{r}\right\rangle =-4(2p-1)(p-1)q(2r-1){\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}\\ {\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}-4(2p-1)(p-1)q(2r-1){\theta }_{{\alpha }_{1}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ -4(2p-1)(p-1)q(2r-1){\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}\\ {\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}-4(2p-1)(p-1)q(2r-1){\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ +\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left\{4(p-1)q\left[-\left(r+{m}_{r}-\displaystyle \frac{1}{2}\right)\left(-(1+\alpha )(p-2)-{m}_{q}-\displaystyle \frac{1}{2}+\alpha r\right)\right.\right.\\ -\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\left(\alpha \left(r-{m}_{r}-\displaystyle \frac{1}{2}\right)+(1+\alpha )(2p-3)\right)+\left(r-{m}_{r}-\displaystyle \frac{5}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\\ \left.\left.+\left(r-{m}_{r}+\displaystyle \frac{3}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{3}{2}\right)\right]-\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)(2p-3)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\right\}\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +(p-{m}_{p}-\displaystyle \frac{1}{2})\left\{4(p-1)q\left[-\left(r+{m}_{r}-\displaystyle \frac{1}{2}\right)(-(1+\alpha )(p-1)-q+\alpha (r+1))\right.\right.\\ \left.+\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\left(-3(1+\alpha )(p-1)-\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)-\alpha r+1-(1+\alpha )p+q\right)\right]\\ \left.-\alpha \left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)(2p-3)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\right\}\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left\{4(p-1)q\left[-(2r-1)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)+\left(r-{m}_{r}-\displaystyle \frac{1}{2}\right)\right.\right.\\ \left(\left(1+\alpha )(p-1)-{m}_{q}-\displaystyle \frac{1}{2}+\alpha (r+1\right)\right)+\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\left(\alpha \left(r+{m}_{r}-\displaystyle \frac{3}{2}\right)-(1+\alpha )(2p-3)\right)\\ \left.-\left(r+{m}_{r}-\displaystyle \frac{1}{2}\right)\left((1+\alpha )(p-1)+{m}_{q}+\displaystyle \frac{1}{2}-\alpha {m}_{r}+\displaystyle \frac{1}{2}\alpha \right)+2(p-1)(1+\alpha )+\alpha \right]\\ \left.-\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)(2p-3)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\right\}\\ {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ +\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\{4(p-1)q\left[\left(r+{m}_{r}-\displaystyle \frac{1}{2}\right)(\alpha r-(1+\alpha )(p-1)-(q+1))+\left(r-{m}_{r}-\displaystyle \frac{1}{2}\right)\right.\\ \left.\left(\left(1+\alpha )(p-1)+\alpha (p-{m}_{p}-\displaystyle \frac{3}{2})+\alpha r-(q+1\right)\right)-(1+\alpha )(2p-3)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\right]\\ \left.-\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)(2p-3)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\right\}\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}},\end{array}\end{eqnarray}$

(5.13)$\begin{eqnarray}\begin{array}{l}\left|p-\displaystyle \frac{1}{2},{m}_{p};q+\displaystyle \frac{1}{2},{m}_{q};r-\displaystyle \frac{1}{2},{m}_{r}\right\rangle \\ =-4(2p-1)(p-1)q(2r-1)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ -4(2p-1)(p-1)q(2r-1)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ -4(2p-1)(p-1)q(2r-1)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ -4(2p-1)(p-1)q(2r-1)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ +\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left\{4(p-1)q\left[\left(r-{m}_{r}-\displaystyle \frac{1}{2}\right)\left(-\left(p+{m}_{p}+\displaystyle \frac{3}{2}\right)-2\alpha r-2q-2\alpha p\right)\right.\right.\\ \left.+1+(1+\alpha )\left(p+{m}_{p}-\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}-\displaystyle \frac{1}{2}\right)-\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)+\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\right]\\ -\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2})\left(q+{m}_{q}-\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\right\}{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}+\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)\\ \left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left\{4(p-1)q\left[\left(r-{m}_{r}-\displaystyle \frac{1}{2}\right)\left((1+\alpha )(p-1)+\alpha (p-{m}_{p}-\displaystyle \frac{3}{2}\right)+\alpha r+q\right)+\left(r+{m}_{r}-\displaystyle \frac{1}{2}\right)\right.\\ (-(1+\alpha )(p-1)+q+\alpha r)+q+\alpha r)+(1+\alpha )(2p-1)\\ \left.\left.-(1+\alpha )\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\right]-\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)\left(q+{m}_{q}-\displaystyle \frac{1}{2}\right)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\right\}\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ +\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left\{4(p-1)q\left[\left(r+{m}_{r}-\displaystyle \frac{1}{2})(-(1+\alpha )(p-2)+q+\alpha r\right)\right.\right.\\ +\left(r-{m}_{r}-\displaystyle \frac{1}{2}\right)\left(\alpha \left(r-{m}_{r}-\displaystyle \frac{1}{2}\right)+\alpha r+3q-(1+\alpha )p)+\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\right]\\ \left.+\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)\left(q-{m}_{q}-\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\right\}\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\left\{4(p-1)q\left[-\left(r-{m}_{r}-\displaystyle \frac{1}{2}\right)\left((1+\alpha )(p-1)+q+\alpha (r+1)\right)\right.\right.\\ +\left(r+{m}_{r}-\displaystyle \frac{1}{2}\right)\left((1+\alpha )(p-1)-q-\alpha r\right)+\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)(1+2\alpha )\\ \left.-2(1+\alpha )\left(p+{m}_{p}-\displaystyle \frac{3}{2}\right)\right]+\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}(q+{m}_{q}-\displaystyle \frac{1}{2})(r-{m}_{r}+\displaystyle \frac{1}{2})\right\}\\ {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}},\end{array}\end{eqnarray}$

(5.14)$\begin{eqnarray}\begin{array}{l}\left|p-\displaystyle \frac{1}{2},{m}_{p};q-\displaystyle \frac{1}{2},{m}_{q};r+\displaystyle \frac{1}{2},{m}_{r}\right\rangle \\ =4(2p-1)(p-1)r(2q-1)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +4(2p-1)(p-1)r(2q-1)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ -4(2p-1)(p-1)r(2q-1)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ -4(2p-1)(p-1)r(2q-1)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right){\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ +\left\{4(p-1)r\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\left[-\left(q+{m}_{q}-\displaystyle \frac{1}{2}\right)(-(1+\alpha )(p-1)+q+\alpha r)\right.\right.\\ \left.-\left(q-{m}_{q}-\displaystyle \frac{1}{2}\right)\left((1+\alpha )\left({m}_{p}+\displaystyle \frac{1}{2}\right)+q+\alpha r\right)\right]+\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)\\ \left.\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\right\}{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +\{4(p-1)r\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)(2q-1)((1+\alpha )(p-1)-\alpha r-q)\\ \left.-\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\alpha (p-{m}_{p}-\displaystyle \frac{3}{2})\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\right\}\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +\left\{4(p-1)r\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\left(\left(q+{m}_{q}-\displaystyle \frac{1}{2}\right)\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)\right.\right.\\ \left.+(2q-1)\left[(1+\alpha )p+q+\alpha r\right]\right)+\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)\\ \left.{\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)}^{2}\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)\right\}{\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ +\left\{4(p-1)r\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\left(-\left(q-{m}_{q}-\displaystyle \frac{1}{2}\right)\left[-\alpha \left({m}_{r}+\displaystyle \frac{1}{2}\right)-q+(1+\alpha )p\right]\right.\right.\\ -\left.\left.\alpha \left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)(2q-1)\right]-\left(q+{m}_{q}-\displaystyle \frac{1}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\right)-\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\\ \left.\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right){\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)}^{2}\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\right\}\\ {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}},\end{array}\end{eqnarray}$

(5.15)$\begin{eqnarray}\begin{array}{l}\left|p-\displaystyle \frac{1}{2},{m}_{p};q+\displaystyle \frac{1}{2},{m}_{q};r+\displaystyle \frac{1}{2},{m}_{r}\right\rangle =-4(2p-1)(p-1)r(2q+3)\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\\ {\theta }_{{\alpha }_{1}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}+4(2p-1)(p-1)r(2q+3)\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\\ {\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ +4(2p-1)(p-1)r(2q+3)\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\\ {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ -4(2p-1)(p-1)r(2q+3)\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\\ {\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ +\left[4(p-1)r\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left\{\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\right.\right.\\ \left[(2q+3)(-(1+\alpha )(p-1)-(1+q))+\alpha r\left(q-{m}_{q}+\displaystyle \frac{3}{2}\right)+\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\right.\\ \left.\left(\alpha {m}_{r}-\displaystyle \frac{1}{2}\alpha -2\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\right)\right]+\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\\ \left.\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\right]\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +\left[4(p-1)r\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\right.\\ \left\{\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)(-1-q+(1+\alpha )p+\alpha (r+1))+\alpha \left(r+{m}_{r}+\displaystyle \frac{3}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{3}{2}\right)\right.\\ +\left((1+\alpha )(p-1)+\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)-\left(-{m}_{q}+\displaystyle \frac{1}{2}\right)-\alpha \left({m}_{r}+\displaystyle \frac{1}{2})\right)\right.\\ \left.-\left(q+{m}_{q}-\displaystyle \frac{1}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{3}{2}\right)\right\}-\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)(q+{m}_{q}+\displaystyle \frac{3}{2})\\ \left.\alpha (p-{m}_{p}-\displaystyle \frac{3}{2}){\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)}^{2}\right]\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}}\\ +\left[4(p-1)r\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\left\{\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)(2q+3)\right.\right.\\ ((1+\alpha )(p-1)+1+q-\alpha r)+\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{1}{2}\right)\left(\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)\right.\\ -\alpha \left.\left.\left({m}_{r}+\displaystyle \frac{1}{2}\right)+\left({m}_{q}-\displaystyle \frac{1}{2}\right)+(1+\alpha )(p-1)\right)\right\}-\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\\ \left.\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\left(r+{m}_{r}+\displaystyle \frac{1}{2}\right)\right]\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+\displaystyle \frac{1}{2}}\\ +\left[4(p-1)r\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\left\{-(q-{m}_{q}+\displaystyle \frac{1}{2})\right.\right.\\ \left\{-\left(q-{m}_{q}+\displaystyle \frac{3}{2}\right)\left(-1+\alpha \left(p-{m}_{p}-\displaystyle \frac{3}{2}\right)-\alpha \left({m}_{r}+\displaystyle \frac{1}{2}\right)-q+(1+\alpha )(p-1)\right.\right.\\ \left.\left.-\alpha \left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\right)\right\}-\left(p-{m}_{p}-\displaystyle \frac{1}{2}\right)\left(q-{m}_{q}+\displaystyle \frac{1}{2}\right)\left(q+{m}_{q}+\displaystyle \frac{3}{2}\right)\\ \left.\alpha (p-{m}_{p}-\displaystyle \frac{3}{2}){\left(r-{m}_{r}+\displaystyle \frac{1}{2}\right)}^{2}\right]\\ {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-\displaystyle \frac{3}{2}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-\displaystyle \frac{1}{2}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-\displaystyle \frac{1}{2}},\end{array}\end{eqnarray}$

(5.16)$\begin{eqnarray}\begin{array}{l}\left|p,{m}_{p};q,{m}_{q};r,{m}_{r}\right\rangle \\ =32{{pq}}^{2}r(p-1)(2p-1)(2r-1){\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +(2p-1)(r+{m}_{r})[-(q+{m}_{q}-1)V-(q-{m}_{q}+1)W+4(p-{m}_{p})\\ (p-1)q(2r-1)(q-{m}_{q})(-(1+\alpha )(p-1)+\alpha (r-1)+q)]\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}+1}\\ +(2p-1)[(q+{m}_{q})(r-{m}_{r}+1)V+(-(r+{m}_{r})+(q+{m}_{q}))Y\\ +(r+{m}_{r})(q-{m}_{q}+1)Z+4(p-{m}_{p})(p-1)q(2r-1)\{(q-{m}_{q})\\ [(r-{m}_{r})({m}_{q}+\alpha {m}_{r}-(1+\alpha )p)-(r+{m}_{r})(\alpha (r-{m}_{r}+1)-(q+{m}_{q}))]\\ +(q+{m}_{q})[(1+\alpha )(2p-1)-(q-{m}_{q}+1)(r-{m}_{r})+(r+{m}_{r})\\ (-(1+\alpha )p+{m}_{q}+\alpha r)\}]\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +(2p-1)(q+{m}_{q})[(r-{m}_{r}+1)W-(r-{m}_{r}+1)Z+4(p-{m}_{p})(p-1)\\ q(2r-1)\{(r-{m}_{r}+1)(\alpha (p-{m}_{p})+\alpha r+(1+\alpha )(p-1)+q)\\ +(r+{m}_{r})(-(1+\alpha )p+\alpha r+q)+(1+\alpha )p-\alpha (p-{m}_{p})-q+\alpha r)\}]\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}+1}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +(2p-1)[(r-{m}_{r}+1)V-(r-{m}_{r}+1)(q-{m}_{q}+1)W-(q-{m}_{q}+1)Z\\ -((q+{m}_{q})(r+{m}_{r})-1)Y+4(p-{m}_{p})(p-1)q(2r-1)\\ \{(q+{m}_{q})[(r+{m}_{r})\alpha (r-{m}_{r}+1)+(r-{m}_{r})(\alpha (p-{m}_{p}-1)\\ -\alpha {m}_{r}+{m}_{q}+(1+\alpha )p)]+(q-{m}_{q})[\alpha (r-{m}_{r}+1)-(q+{m}_{q})(r+{m}_{r})\\ +(r+{m}_{r})({m}_{q}+(1+\alpha )p-\alpha {m}_{r})-(1+\alpha )(2p-1)-\alpha -(q+{m}_{q}+1)\\ -(r-{m}_{r})\alpha (r+{m}_{r}+1)+(r-{m}_{r}+1)(r+{m}_{r})\}]\\ {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +(2p-1)(q-{m}_{q})[(r-{m}_{r}+1)V+(r+{m}_{r}-1)Y+4(p-1)q(2r-1)\\ (p-{m}_{p})\{2r(-(q+1)+(1+\alpha (p-\alpha r)-(2p-1)(1+\alpha )\}]\\ {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}-1}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}}\\ +(2p-1)(r-{m}_{r})[-(q+{m}_{q}-1)Y-(q-{m}_{q}+1)Z+4(p-1)q(2r-1)\\ (p-{m}_{p})\{(q-{m}_{q})(-\alpha (p-{m}_{p}-1)+\alpha -(1+\alpha )p-q-\alpha r)\\ -(r+m-r)\alpha (q-{m}_{q})+(q-{m}_{q})(\alpha ({m}_{r}+1)-(1+\alpha )p-q)\}\\ {\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-1}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}-1}\\ +(p-{m}_{p}-1)[4(p-{m}_{p})(p-1)(2p-1)q(2r-1)\{-2q(r-{m}_{r}+1)\\ \alpha (1+\alpha )-(q+m-q)(1+\alpha )(-\alpha (p-{m}_{p}-2)+\alpha (r-{m}_{r})\\ -(1+\alpha )p-q+\alpha r)-(1+\alpha )(q-{m}_{q})(\alpha {m}_{r}-(1+\alpha )p-q)\}\\ +(r-{m}_{r}+1)V\{-(q+{m}_{q})(-\alpha (p-{m}_{p}-2)-(1+\alpha )(p-1)\\ -{m}_{q}-\alpha r)+({m}_{q}-\alpha r-(1+\alpha )(p-1))-(q-{m}_{q})(q+{m}_{q}+1)\}\\ +(q-{m}_{q}+1)W(r-{m}_{r}+1)(q+(1+\alpha )(p-1)+\alpha r)\\ +Y\{((1+\alpha )p-{m}_{q}-\alpha {m}_{r})((r+{m}_{r})(q+{m}_{q}-1)-1)\\ -(r+{m}_{r}+1)((q-{m}_{q})(q+{m}_{q}+1)+(q+{m}_{q}-1)\alpha (r-{m}_{r}))\\ -(q+{m}_{q})(-(1+\alpha )(p-1)-\alpha (p-{m}_{p}-2)+\alpha {m}_{r}-{m}_{q})\}\\ +Z(q-{m}_{q}+1)\{-(r+{m}_{r})(-(1+\alpha )p+q+\alpha {m}_{r})\\ +(r-{m}_{r})((q-{m}_{q}+1)-\alpha (r+{m}_{r}+1))+((1+\alpha )(p-1)+{m}_{q}-\alpha {m}_{r})\\ {\theta }_{{\alpha }_{1}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}}{\theta }_{{\alpha }_{1}+{\alpha }_{3}}{\theta }_{{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}{\chi }_{2{\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}}^{p-{m}_{p}-2}{\chi }_{{\alpha }_{2}}^{q-{m}_{q}}{\chi }_{{\alpha }_{3}}^{r-{m}_{r}},\end{array}\end{eqnarray}$

(5.17)$\begin{eqnarray}\begin{array}{l}V=4(p-1)q(p-{m}_{p})[-(r+{m}_{r}-1)(-(1+\alpha )(p-2)-{m}_{q}+\alpha r)\\ -(r+{m}_{r})(\alpha (r-{m}_{r})+(1+\alpha )(2p-3))+(r+{m}_{r}-3)\\ (q-{m}_{q}+1)(r-{m}_{r}+2)(q+{m}_{q}+1)]-\alpha (p-{m}_{p}-1)\\ (q+{m}_{q})(2p-3)(r+{m}_{r}),\end{array}\end{eqnarray}$

(5.18)$\begin{eqnarray}\begin{array}{l}W=4(p-1)q(p-{m}_{p})[-(r+{m}_{r}-1)(-(1+\alpha )(p-1)-q+\alpha (r+1))\\ +(r+{m}_{r})(-3(1+\alpha )(p-1)-\alpha (p-{m}_{p}-1)-\alpha r+1+q)]\\ -\alpha (p-{m}_{p}-1)(q+{m}_{q})(2p-3)(r+{m}_{r}),\end{array}\end{eqnarray}$

(5.19)$\begin{eqnarray}\begin{array}{l}Y=4(p-1)q(p-{m}_{p})[-(2r-1)(q-{m}_{q}+1)+(r-{m}_{r})((1+\alpha )(p-1)\\ -{m}_{q}+\alpha (r+1))+(r-{m}_{r}+1)(\alpha (r+{m}_{r}-2)-(1+\alpha )(2p-3))\\ -(r+{m}_{r}-1)((1+\alpha )(p-1)+{m}_{q}-\alpha {m}_{r}+\alpha )+2(p-1)(1+\alpha )\\ +\alpha ]-\alpha (p-{m}_{p}-1)(q+{m}_{q})(2p-3)(r-{m}_{r}-1),\end{array}\end{eqnarray}$

(5.20)$\begin{eqnarray}\begin{array}{l}Z=4(p-1)q(p-{m}_{p})[(r+{m}_{r}+1)(\alpha r-(1+\alpha )(p-1)-(q+1))\\ +(r-{m}_{r})((1+\alpha )9p-1)-\alpha (p-{m}_{p}-1)+\alpha r(q+1))-(1+\alpha )(2p-3)\\ (r-{m}_{r}+1)]-\alpha (p-{m}_{p}-1)(q+{m}_{q})(2p-3)(r-{m}_{r}-1).\end{array}\end{eqnarray}$

6. The typical and atypical representation of D(2, 1; α)

The (s, t, u) components must satisfy

(6.1)$\begin{eqnarray}s,t,u\in \displaystyle \frac{1}{2}N=\left\{0,\displaystyle \frac{1}{2},1,\ldots \right\},\end{eqnarray}$

and the (p, q, r) also belongs to this set. If p ≥ 2, q ≥ 1, r ≥ 1, there are four atypical conditions [22] given by

(6.2)$\begin{eqnarray}(1+\alpha )p+q+\alpha r=0,\end{eqnarray}$

(6.3)$\begin{eqnarray}(1+\alpha )p-(q+1)-\alpha r=0,\end{eqnarray}$

(6.4)$\begin{eqnarray}(1+\alpha )p+q-\alpha (r+1)=0,\end{eqnarray}$

(6.5)$\begin{eqnarray}(1+\alpha )p-(q+1)-\alpha (r+1)=0.\end{eqnarray}$

If none of the four atypical conditions are satisfied, then the representation is a typical representation, which decomposes into 16 subalgebra irreducible representations. If one of the conditions is satisfied, the representation is reducible but indecomposable generally. The shift operator will separate the 16-dimensional lattice into two 8-dimensional lattices. Since

(6.6)$\begin{eqnarray}{O}^{\tfrac{1}{2},-j,-k}{O}^{-\tfrac{1}{2},j,k}| s,t,u\gt =\ 0.\end{eqnarray}$

For (6.2), $j=\tfrac{1}{2},k=\tfrac{1}{2}$, the two 8-dimensional lattices are given by:

(6.7)$\begin{eqnarray}\begin{array}{l}{{\mathbb{F}}}_{1}=\{(p,q,r),\left(p-\displaystyle \frac{1}{2},q+\displaystyle \frac{1}{2},r-\displaystyle \frac{1}{2}\right),\left(p-\displaystyle \frac{1}{2},q-\displaystyle \frac{1}{2},r+\displaystyle \frac{1}{2}\right),\\ \left(p-\displaystyle \frac{1}{2},q-\displaystyle \frac{1}{2},r-\displaystyle \frac{1}{2}\right),(p-1,q,r;1),(p-1,q,r-1),\\ (p-1,q-1,r),\left(p-\displaystyle \frac{3}{2},q-\displaystyle \frac{1}{2},r-\displaystyle \frac{1}{2}\right)\},\end{array}\end{eqnarray}$

(6.8)$\begin{eqnarray}\begin{array}{c}{{\mathbb{F}}}_{2}=\{\left(p-\frac{1}{2},q+\frac{1}{2},r+\frac{1}{2}\right),(p-1,q+1,r),(p-1,q,r+1)\\ (p-1,q,r;2),\left(p-\frac{3}{2},q+\frac{1}{2},r+\frac{1}{2}\right),\left(p-\frac{3}{2},q+\frac{1}{2},r-\frac{1}{2}\right),\\ \left(p-\frac{3}{2},q-\frac{1}{2},r+\frac{1}{2}\right),(p-2,q,r)\}.\end{array}\end{eqnarray}$

For (6.3), $j=-\tfrac{1}{2},k=\tfrac{1}{2}$, the two 8-dimensional lattices are given by:

(6.9)$\begin{eqnarray}\begin{array}{l}{{\mathbb{F}}}_{1}=\{(p,q,r),\left(p-\displaystyle \frac{1}{2},q+\displaystyle \frac{1}{2},r-\displaystyle \frac{1}{2}\right),\left(p-\displaystyle \frac{1}{2},q+\displaystyle \frac{1}{2},r+\displaystyle \frac{1}{2}\right),\\ \left(p-\displaystyle \frac{1}{2},q-\displaystyle \frac{1}{2},r-\displaystyle \frac{1}{2}\right),(p-1,q,r;1),(p-1,q,r-1),\\ (p-1,q,r-1);\left(p-\displaystyle \frac{3}{2},q-\displaystyle \frac{1}{2},r-\displaystyle \frac{1}{2}\right)\},\end{array}\end{eqnarray}$

(6.10)$\begin{eqnarray}\begin{array}{c}{{\mathbb{F}}}_{2}=\Space{0ex}{2.5ex}{0ex}\{\left(p-\frac{1}{2},q-\frac{1}{2},r+\frac{1}{2}\right),(p-1,q-1,r),(p-1,q,r+1)\\ (p-1,q,r;2),\left(p-\frac{3}{2},q-\frac{1}{2},r+\frac{1}{2}\right),\left(p-\frac{3}{2},q-\frac{1}{2},r-\frac{1}{2}\right),\\ \left(p-\frac{3}{2},q+\frac{1}{2},r+\frac{1}{2}\right),(p-2,q,r)\Space{0ex}{2.5ex}{0ex}\}.\end{array}\end{eqnarray}$

For (6.4), $j=\tfrac{1}{2},k=-\tfrac{1}{2}$, the two 8-dimensional lattices are given by:

(6.11)$\begin{eqnarray}\begin{array}{l}{{\mathbb{F}}}_{1}=\{(p,q,r),\left(p-\displaystyle \frac{1}{2},q+\displaystyle \frac{1}{2},r+\displaystyle \frac{1}{2}\right),\left(p-\displaystyle \frac{1}{2},q-\displaystyle \frac{1}{2},r-\displaystyle \frac{1}{2}\right),\\ \left(p-\displaystyle \frac{1}{2},q-\displaystyle \frac{1}{2},r+\displaystyle \frac{1}{2}\right),(p-1,q,r;1),(p-1,q,r-1),\\ (p-1,q,r-1);\left(p-\displaystyle \frac{3}{2},q-\displaystyle \frac{1}{2},r+\displaystyle \frac{1}{2}\right)\},\end{array}\end{eqnarray}$

(6.12)$\begin{eqnarray}\begin{array}{c}{{\mathbb{F}}}_{2}=\{\left(p-\frac{1}{2},q+\frac{1}{2},r-\frac{1}{2}\right),(p-1,q,r-1),(p-1,q+1,r)\\ (p-1,q,r;2),\left(p-\frac{3}{2},q-\frac{1}{2},r-\frac{1}{2}\right),\left(p-\frac{3}{2},q+\frac{1}{2},r-\frac{1}{2}\right),\\ \left(p-\frac{3}{2},q+\frac{1}{2},r+\frac{1}{2}\right),(p-2,q,r)\}.\end{array}\end{eqnarray}$

For (6.5), $j=-\tfrac{1}{2},k=-\tfrac{1}{2}$, the two 8-dimensional lattices are given by:

(6.13)$\begin{eqnarray}\begin{array}{l}{{\mathbb{F}}}_{1}=\{(p,q,r),\left(p-\displaystyle \frac{1}{2},q-\displaystyle \frac{1}{2},r+\displaystyle \frac{1}{2}\right),\left(p-\displaystyle \frac{1}{2},q+\displaystyle \frac{1}{2},r+\displaystyle \frac{1}{2}\right),\\ \left(p-\displaystyle \frac{1}{2},q+\displaystyle \frac{1}{2},r-\displaystyle \frac{1}{2}\right),(p-1,q,r;1),(p-1,q,r+1),\\ (p-1,q+1,r);\left(p-\displaystyle \frac{3}{2},q+\displaystyle \frac{1}{2},r+\displaystyle \frac{1}{2}\right)\},\end{array}\end{eqnarray}$

(6.14)$\begin{eqnarray}\begin{array}{l}{{\mathbb{F}}}_{2}=\{\left(p-\displaystyle \frac{1}{2},q-\displaystyle \frac{1}{2},r-\displaystyle \frac{1}{2}\right),(p-1,q-1,r),(p-1,q,r-1)\\ (p-1,q,r;2),\left(p-\displaystyle \frac{3}{2},q-\displaystyle \frac{1}{2},r+\displaystyle \frac{1}{2}\right),\left(p-\displaystyle \frac{3}{2},q-\displaystyle \frac{1}{2},r-\displaystyle \frac{1}{2}\right),\\ \left(p-\displaystyle \frac{3}{2},q+\displaystyle \frac{1}{2},r-\displaystyle \frac{1}{2}\right),(p-2,q,r)\}.\end{array}\end{eqnarray}$

If p < 2, q < 1, r < 1, only none-negative value elements appear in the decomposition of the (s, t, u) lattice.

7. Conclusions

First, we have reviewed the explicit differential operator representations for Lie superalgebra D(2, 1; α). Based on the shift operator and differential operator representations, we have constructed the explicitly finite-dimensional representations of superalgebra D(2, 1; α) by using bosonic and fermionic coordinates. Our results are expected to be useful for the construction of primary fields of the corresponding current superalgebra of D(2, 1; α), which play an important role in the computation of quantization of the string theory on the AdS₃ × S³ × S³ × S¹ background.

This work received financial support from the National Natural Science Foundation of China (Grant No. 11 405 051). Yao-Zhong Zhang was supported by the Australian Research Council Discovery Project DP190101529. Xiang-Mao Ding was supported by NSFC Grant 11 775 299.

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