Representation of the quantum mechanical wavefunction by orthogonal polynomials in the energy and physical parameters

A D Alhaidari

Table 1. Asymptotics ($n\to \infty $) of most of the continuous polynomials in the Askey scheme of hypergeometric orthogonal polynomials. The polynomials are shown in the second column in their orthonormal version. The asymptotics of the Wilson polynomial is obtained here in section 2 whereas the asymptotics of the continuous Hahn polynomial is derived in [9]. The asymptotics of the Meixner–Pollaczek and continuous dual Hahn polynomials are obtained in [1]. The rest are well known.

Polynomial Orthonormal version x Asymptotics [1, 9] τ ξ $\theta (x)$ $\varphi (x)$ Laguerre $\sqrt{\displaystyle \frac{n!}{{(\nu +1)}_{n}}}\,{L}_{n}^{\nu }(x)$ $x\geqslant 0$ ${n}^{-1/4}{A}_{L}(x)\cos \left[2\sqrt{nx}-\tfrac{\pi }{2}\left(\nu +\tfrac{1}{2}\right)\right]$ $1/4$ $1/2$ $2\sqrt{x}$ 0 Jacobi $\begin{array}{c}\sqrt{\tfrac{2n+\mu +\nu +1}{\mu +\nu +1}\tfrac{n\,!{(\mu +\nu +1)}_{n}}{{(\mu +1)}_{n}{(\nu +1)}_{n}}}\\ \times {P}_{n}^{(\mu ,\nu )}(\cos \,x)\end{array}$ $\pi \geqslant x\geqslant 0$ ${A}_{J}(x)\cos \left[\left(n+\tfrac{\mu +\nu +1}{2}\right)x-\tfrac{\pi }{2}\left(\mu +\tfrac{1}{2}\right)\right]$ 0 1 x 0 Meixner–Pollaczek ${P}_{n}^{\mu }(x;\theta )$ $x\in {\mathbb{R}}$ ${n}^{-1/2}{A}_{MP}(x)\cos \left[n\theta -x\,\mathrm{log}\,n+{\delta }_{MP}(x)\right]$ $1/2$ 1 θ −x Continuous Hahn ${{\mathscr{P}}}_{n}^{\mu }(x;\nu ;a,b)$ $x\in {\mathbb{R}}$ ${n}^{-1/2}{A}_{H}(x)\cos \left[n\tfrac{\pi }{2}-(2x+a-b)\mathrm{log}\,n+{\delta }_{H}(x)\right]$ $1/2$ 1 $\pi /2$ $-(2x+a-b)$ Continuous dual Hahn ${S}_{n}^{\mu }({x}^{2};a,b)$ $x\geqslant 0$ ${n}^{-1/2}{A}_{dH}(x)\cos \left[x\,\mathrm{log}\,n+{\delta }_{dH}(x)\right]$ $1/2$ ** 0 x Wilson ${W}_{n}^{\mu }({x}^{2};\nu ;a,b)$ $x\geqslant 0$ ${n}^{-1/2}{A}_{W}(x)\cos \left[2x\,\mathrm{log}\,n+{\delta }_{W}(x)\right]$ $1/2$ ** 0 2x