Functionals of Gegenbauer Polynomials and D-dimensional Hydrogenic Momentum Expectation Values 1

The system of Gegenbauer polynomials fC n (x); n = 0; 1; : : :g is a classical family of polynomials orthogonal with respect to the weight function ! (x) = (1?x 2) ? 1 2 on the support interval ?1; +1]. Integral functionals of Gegenbauer polynomials with kernel f(x))C n (x)] 2 ! (x), where f(x) is an arbitrary function which does not depend on n or , are considered in this paper. Firstly, a general recursion formula for these functionals is obtained. Then, the explicit expression for some speciic functionals of this type is found in a closed and compact form; namelly, for the functionals whith f(x) equal to (1?x) (1+x) , log(1 ? x 2) and (1 + x) log(1 + x), which appear in numerous physico-mathematical problems. Finally, usefulness of these functionals is 1 1 illustrated in the explicit evaluation of the momentum expectation values hp i and hlog pi of the D-dimensional hydrogenic atom with nuclear charge Z 1.