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

The system of Gegenbauer or ultraspherical polynomials {C-n(lambda)(x);n = 0,1,...} is a classical family of polynomials orthogonal with respect to th e weight function omega(lambda)(x) = (1 - x(2))(lambda - 1/2) on the suppor t interval [-1,+1]. Integral functionals of Gegenbauer polynomials with int egrand 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. The n, the explicit expression for some specific functionals of this type is fo und in a closed and compact form; namely, for the functionals with f(x) equ al to (1 - x)(alpha)(1 + x)(beta), log(1 - x(2)), and (1 + x)log(1 + x), wh ich appear in numerous physico-mathematical problems. Finally, these functi onals are used in the explicit evaluation of the momentum expectation value s [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 u nit argument, which is a considerable improvement with respect to Hey's exp ression (the only one existing up to now) which requires a double sum. (C) 2000 American Institute of Physics. [S0022-2488(00)01509-7].