New product and linearization formulae of Jacobi polynomials of certain parameters

In this research article, a new product formula of Jacobi polynomials of certain parameters is established. This formula is expressed in terms of a terminating hypergeometric function of the type 6F5(1) and it generalizes a formula which connects explicitly the squares of two ultraspherical polynomials with different parameters. Thanks to symbolic algebraic computation, and in particular, the celebrated algorithms of Zeilberger and Petkovsek-van Hoeij, several reduction formulae for summing certain terminating hypergeometric functions of unit argument are given, and hence some new product and linearization formulae of Jacobi polynomials of certain parameters are deduced. The latter formulae are used to obtain new formulae for some definite integrals.