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 toUn(x)=∑k=0nPk(x)Pn−k(x), where Un(x)Un(x) are the Chebyshev polynomials of the second kind and Pk(x)Pk(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 expansionHn(x+y)=2−n/2∑k=0n(nk)Hk(2x)Hn−k(2y) for the Hermite polynomials, due to Appell and Kampé de Fériet, to obtain new interesting inequalities about the zeros of the corresponding orthogonal polynomials.