Convolutions and zeros of orthogonal polynomials

In an attempt to answer a long standing open question of Al-Salam we generate various beautiful formulae for convolutions of orthogonal polynomials similar to U n ( x ) = ∑ k = 0 n P k ( x ) P n − k ( x ) , where U n ( x ) are the Chebyshev polynomials of the second kind and P k ( x ) are the Legendre polynomials. The results are derived both via the generating functions approach and a new convolution formulae for hypergeometric functions. We apply some addition formulae similar to the well-known expansion H n ( x + y ) = 2 − n / 2 ∑ k = 0 n ( n k ) H k ( 2 x ) H n − k ( 2 y ) for the Hermite polynomials, due to Appell and Kampé de Fériet, to obtain new interesting inequalities about the zeros of the corresponding orthogonal polynomials.