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

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

2001-10-31

arxiv.org/abs/math-ph/0110042v1

J. Phys. A 34, 11287-11300 (2001)

Physics

Mathematical Physics

16 pages

Scientific paper

10.1088/0305-4470/34/50/310

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.

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