Closed-form sums for some perturbation series involving hypergeometric functions

Infinite series of the type Sigma(n=1)(infinity)(alpha/2)n/n 1/n!F-2(1)(-n,b;gamma;y) are investigated. Closed-form sums are obtained for alpha a positive integer, alpha = 1, 2, 3,.... The limiting case of b --> infinity, after gamma is replaced with x(2)/b, leads to Sigma(n=1)(infinity)(alpha/2)(n)/n 1/n! F-1(1)(-n,gamma,x(2)) This type of Series appears in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = -d(2)/dx(2) + Bx(2) + A/x(2) + lambda/x(alpha) 0 less than or equal to x less than or equal to infinity, alpha, lambda > 0, A greater than or equal to 0 These results have immediate applications to perturbation series for the energy and wavefunction of the spiked harmonic oscillator Hamiltonian H = -d(2)/dx(2) + Bx(2) + lambda/x(alpha) 0 less than or equal to x less than or equal to infinity, alpha, lambda > 0.