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

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.

16 pages

References in corpus (2)

• Generalized spiked harmonic oscillator
• Perurbation expansions for the spiked harmonic oscillator and related series involving the gamma function

Cited by in corpus (6)

• Relation between chiral symmetry breaking and confinement in YM-theories
• Coherent states associated to the wavefunctions and the spectrum of the isotonic oscillator
• Energy bounds for a class of singular potentials and some related series
• Closed-form sums for some perturbation series involving hypergeometric functions
• Phase coherent states with circular Jacobi polynomials for the pseudoharmonic oscillator
• A new class of coherent states with Meixner-Pollaczek polynomials for the Gol'dman-Krivchenkov Hamiltonian