ABSTRACT
For ultraspherical polynomials and in particular for Legendre polynomials; and for Bessel functions there is a product formula which has as a consequence a type of convolution. This was introduced in 1950 by I.M. Gelfand [55], next by S. Bochner [20] [21] and I.I. Hirchman Jr [71], and has a number of very interesting consequences. It has been used to obtain the analogue of the Hardy-Littlewood theorem on fractional integration [8] [102] and to study the heat equation associated with Bessel functions and ultraspherical polynomials. For Jacobi polynomials product formulas have been found by G. Gasper [52][53][54], and for Jacobi functions by M. Flensted. Jensen and T.H. Koornwinder [44]. These product formulas have given convolution structures which have been used to obtain the above results as well as other applications.
