一类包含相关Laguerre多项式的扰动级数的闭型和
Closed-form sums for some perturbation series involving associated Laguerre polynomials
摘要：
无穷级数sum_{n=1} ^infty {（alpha/2）_n /（n n！）}_ 1F_1（-n，gamma，x^2），其中_1F_1（-n，gamma，x^2）={n！（gamma）_n}L_n^（gamma-1）（x^2），出现在广义尖峰谐振子哈密顿量H = -d^2/dx^2 + Bx^2 + A/x^2 + lambda/x^alpha 0 <= x < infty，alpha，lambda > 0，A >= 0的波函数的一阶微扰修正中.证明了该级数对所有x > 0和2 gamma > alpha收敛，其中gamma = 1 +（1/2）sqrt（1+4A）.对于α = 2、4和6的情况，给出了这些序列的闭合形式和。求α/2 = 2 + m，m = 0，1，2，.和的一般公式在相关的Laguerre多项式，也提供。
英文摘要：
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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