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

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Abstract
Infinite series ∑n = 1∞ [(α/2)n/n] [1/n!] 1F1(−n, γ, x2), where 1F1(−n, γ, x2) = [n!/(γ)n] Ln(γ−1)(x2), appear in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = −d2/dx2 + Bx2 + A/x2 + λ/xα, 0 ≤ x < ∞, α, λ > 0, A ≥ 0. It is proved that the series is convergent for all x > 0 and 2γ > α where γ = 1 + ½√(1 + 4A). Closed-form sums are presented for these series for the cases α = 2, 4 and 6. A general formula for finding the sum for α/2 = 2 + m, m = 0, 1, 2, ... in terms of associated Laguerre polynomials is also provided.
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