Functionals of Gegenbauer polynomials and D-dimensional hydrogenic momentum expectation values

The system of Gegenbauer or ultraspherical polynomials $Cn (x);n50,1,.. .% is a classical family of polynomials orthogonal with respect to the weight function vl(x)5(12x ) on the support interval @21,11# . Integral functionals of Gegenbauer polynomials with integrand f (x)@Cn (x)#vl(x), where f (x) is an arbitrary function which does not depend on n or l, 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 (12x)(11x), log(12x), and (11x)log(11x), which appear in numerous physico-mathematical problems. Finally, these functionals are used in the explicit evaluation of the momentum expectation values ^p& and ^log p& of the D-dimensional hydrogenic atom with nuclear charge Z>1. The power expectation values ^p& are given by means of a terminating 5F4 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. © 2000 American Institute of Physics. @S0022-2488~00!01509-7#