Title: Functionals of Gegenbauer polynomials and D-dimensional hydrogenic momentum expectation values
Authors: Van Assche, Walter ×
Yanez, RJ
Gonzalez-Ferez, R
Dehesa, JS #
Issue Date: 2000
Publisher: Amer inst physics
Series Title: Journal of mathematical physics vol:41 issue:9 pages:6600-6613
Abstract: The system of Gegenbauer or ultraspherical polynomials {C-n(lambda)(x);n = 0,1,...} is a classical family of polynomials orthogonal with respect to the weight function omega(lambda)(x) = (1 - x(2))(lambda - 1/2) on the support interval [-1,+1]. Integral functionals of Gegenbauer polynomials with integrand f(x)[C-n(lambda)(x)](2)omega(lambda)(x), where f(x) is an arbitrary function which does not depend on n or lambda, are considered in this paper. First, a general recursion formula for these functionals is obtained. Then, the explicit expression for some specific functionals of this type is found in a closed and compact form; namely, for the functionals with f(x) equal to (1 - x)(alpha)(1 + x)(beta), log(1 - x(2)), and (1 + x)log(1 + x), which appear in numerous physico-mathematical problems. Finally, these functionals are used in the explicit evaluation of the momentum expectation values [p(alpha)] and [log p] of the D-dimensional hydrogenic atom with nuclear charge Z greater than or equal to 1. The power expectation values [p(alpha)] are given by means of a terminating F-5(4) hypergeometric function with unit argument, which is a considerable improvement with respect to Hey's expression (the only one existing up to now) which requires a double sum. (C) 2000 American Institute of Physics. [S0022-2488(00)01509-7].
ISSN: 0022-2488
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Analysis Section
× corresponding author
# (joint) last author

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