Closed-form sums for some perturbation series involving hypergeometric functions
Abstract
Infinite series of the type Sumn=1,infinity(alpha/2)n2F1(-n, b; gamma; y)/(n n!) are investigated. Closed-form sums are obtained for alpha a positive integer alpha=1,2,3, ... The limiting case of b --> infinity, after y is replaced with x2/b, leads to Sumn=1,infinity(alpha/2)n1F1(-n; gamma; x2)/(n n!). This type of series appears in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = -(d/dx)2 + Bx2 + A/x2 + lambda/xalpha, x >=0, alpha, lambda > 0, A >= 0. These results have immediate applications to perturbation series for the energy and wave function of the spiked harmonic oscillator Hamiltonian H = -(d/dx)2 + Bx2 + lambda/xalpha, x >= 0, alpha, lambda > 0.
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