Applications of a Convolution Theorem to Jacobi Polynomials

In this paper we use a convolution theorem to study certain polynomials in the unit disk whose coefficients are constant multiples of Jacobi polynomials. We show these polynomials have no zeros in the unit disk. This result extends a theorem of Ruscheweyh on Gegenbauer polynomials, and is related to results of Askey and Gasper on Jacobi polynomials.