Hall, R. L., Saad, N., & Von Keviczky, A. B. (2001). Closed-form sums for some perturbation series involving associated Laguerre polynomials. Journal Of Physics A-Mathematical And General, 34(50), 11287-11300.

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

Infinite series, Sigma(n=1)(infinity) (alpha/2)(n)/n 1/n! F-1(1)(-n, gamma, x(2)), where F-1(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) + Bx(2) + A/x(2) + lambda/x(alpha), 0 less than or equal to x 0, A greater than or equal to 0. It is proved that the series is convergent for all x > 0 and 2gamma > alpha where gamma = 1 + 1/2root1 Show moreInfinite series, Sigma(n=1)(infinity) (alpha/2)(n)/n 1/n! F-1(1)(-n, gamma, x(2)), where F-1(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) + Bx(2) + A/x(2) + lambda/x(alpha), 0 less than or equal to x 0, A greater than or equal to 0. It is proved that the series is convergent for all x > 0 and 2gamma > alpha where gamma = 1 + 1/2root1+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. Show less