A few identities and integrals which involve Pochhammer symbols, the Jacobi polynomials and the hypergeometric function

Abstract: We find some identities involving the Pochhammer symbol (rising factorial) and apply them to find a closed form for some integrals of the Jacobi polynomials, as well as a hypergeometric function multiplied by the product of the Jacobi polynomial and the Beta density that makes this polynomial a member of the family of orthogonal polynomials. In other words, we expand a hypergeometric function in an orthogonal series of Jacobi polynomials.