Closed-form sums for some perturbation series involving hypergeometric functions

Infinite series of the type ∞ ∑ n=1 ( 2 )n n 1 n! 2F1(−n, b; γ; y) are investigated. Closed-form sums are obtained for α a positive integer α = 1, 2, 3, . . . . The limiting case of b → ∞, after y is replaced with x2/b, leads to ∞ ∑ n=1 ( 2 )n n 1 n! 1F1(−n, γ, x2). This type of series appears in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = − d dx + Bx2 + A x + λ x 0 ≤ x < ∞, α, λ > 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 +Bx2 + λ x 0 ≤ x < ∞, α, λ > 0. PACS 03.65.Ge