Closed-form sums for some perturbation series involving associated Laguerre polynomials

TL;DR

This paper derives closed-form expressions for certain infinite series involving associated Laguerre polynomials, which appear in quantum physics perturbation calculations, providing new formulas for specific parameter cases.

Contribution

It introduces closed-form sums for perturbation series involving associated Laguerre polynomials, including a general formula for a class of these series.

Findings

Series converge for all x > 0 when 2 gamma > alpha.

Closed-form sums are obtained for alpha = 2, 4, 6.

A general formula for sums with alpha/2 = 2 + m is provided.

Abstract

Infinite series sum_{n=1}^infty {(alpha/2)_n / (n n!)}_1F_1(-n, gamma, x^2), where_1F_1(-n, gamma, x^2)={n!_(gamma)_n}L_n^(gamma-1)(x^2), appear in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = -d^2/dx^2 + B x^2 + A/x^2 + lambda/x^alpha 0 <= x < infty, alpha, lambda > 0, A >= 0. It is proved that the series is convergent for all x > 0 and 2 gamma > alpha, where gamma = 1 + (1/2)sqrt(1+4A). Closed-form sums are presented for these series for the cases alpha = 2, 4, and 6. A general formula for finding the sum for alpha/2 = 2 + m, m = 0,1,2, ..., in terms of associated Laguerre polynomials, is also provided.