Resumen de Convolutions and zeros of orthogonal polynomials

Iván Carlos Area Carracedo , Dimitar K. Dimitrov, Eduardo Godoy Malvar

• 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 View the MathML source where Un(x) are the Chebyshev polynomials of the second kind and 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 expansion View the MathML source for the Hermite polynomials, due to Appell and Kampé de Fériet, to obtain new interesting inequalities about the zeros of the corresponding orthogonal polynomials.