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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 thefirst-order perturbation correction for the wavefunction of the generalizedspiked 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 theseries 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 casesalpha = 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.Comment: 16 page

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