New linearization formulae for the products of Chebyshev polynomials of third and fourth kinds

Abstract

This paper deals with the problem of finding two new closed formulae for linearization coefficients of two special nonsymmetric cases for Jacobi polynomials $P^{(\alpha ,\beta )}_n(x)$ corresponding to the parameters' values $\beta =-\alpha =\pm 1/2$. From these two formulae, the linearization coefficients of the products of Chebyshev polynomials of the third and fourth kinds are established. Based on using algorithmic methods, such as the algorithms by Zeilberger, Petkovsek and Van-Hoeij, and two certain Whipple's transformations, six new closed formulae for summing certain terminating hyper\-geometric functions of unit argument are given.