The system of Gegenbauer or ultraspherical polynomials {C[sub n][supλ](x);n=0,1,...} is a classical family of polynomialsorthogonal with respect to the weight function ω[sub λ](x)=(1-x[sup 2])[sup λ-1/2] on the support interval [-1,+1]. Integral functionals of Gegenbauer polynomials with integrand f(x)[C[sub n][sup λ](x)][sup 2]ω[sub λ](x), where f(x) is an arbitrary function which does not depend on n or λ, 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)[sup α](1+x)[sup β], log(1-x[sup 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 and of the D-dimensional hydrogenic atom with nuclear charge Z>=1. The power expectation values are given by means of a terminating [sub 5]F[sub 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. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]