Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients

The answers to many important questions in the harmonic analysis of orthogonal polynomials are known to depend on the determination of when formulas of the types and their duals hold, where p n ( x ) and q n ( x ) are suitably normalized orthogonal polynomials or orthogonal polynomials multiplied by certain functions; e.g. e − px L n ( x ).