Functionals of Gegenbauer polynomials and \(D\)-dimensional hydrogenic momentum expectation values (Q2738206)

The paper is devoted to the study of integral functionals of Gegenbauer polynomials \(C_n^\lambda(x)\) of the form \(f(x)[C_n^\lambda (x)]^2 \omega_\lambda(x)\), where \(\omega_\lambda\) denotes the corresponding weight function of the Gegenbauer polynomials. A general recursion formula for these functionals is obtained and an explicit expression for some specific functionals of this type is given in a closed and compact form. Finally, these functionals are used in the explicit evaluation of the momentum expectation values \(\langle p^\alpha\rangle\) and \(\langle\log p\rangle\) of the \(D\)-dimensional hydrogenic atom with nuclear charge \(Z\geq 1\).