Closed-form sums for some perturbation series involving hypergeometric functions

TL;DR

This paper derives closed-form sums for specific hypergeometric series related to perturbation series in quantum mechanics, particularly for the spiked harmonic oscillator, providing explicit formulas for energy and wavefunction corrections.

Contribution

It presents new closed-form expressions for hypergeometric series relevant to perturbation theory in quantum mechanics, especially for the spiked harmonic oscillator.

Findings

Closed-form sums for hypergeometric series involving (alpha/2)_n_2F_1 and (alpha/2)_n_1F_1 functions.

Application of these sums to perturbation series for the energy and wavefunction of the spiked harmonic oscillator.

Explicit formulas facilitate analytical calculations in quantum perturbation problems.

Abstract

Infinite series of the type Sum{n=1,infinity}(alpha/2)_n_2F_1(-n, b; gamma; y)/(n n!) are investigated. Closed-form sums are obtained for alpha a positive integer alpha=1,2,3, ... The limiting case of b --> infinity, after y is replaced with x^2/b, leads to Sum{n=1,infinity}(alpha/2)_n_1F_1(-n; gamma; x^2)/(n n!). This type of series appears in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = -(d/dx)^2 + Bx^2 + A/x^2 + lambda/x^alpha, x >=0, alpha, lambda > 0, A >= 0. These results have immediate applications to perturbation series for the energy and wave function of the spiked harmonic oscillator Hamiltonian H = -(d/dx)^2 + Bx^2 + lambda/x^alpha, x >= 0, alpha, lambda > 0.