Linearization coefficients of some particular Jacobi polynomials via hypergeometric functions

The main purpose of the present paper is to establish two new linearization formulas for certain Jacobi polynomials. The new established formulas are expressed in terms of terminating hypergeometric functions of the type ${}_{4}F_{3}(1)$ . In virtue of the well-known Pfaff-Saalschütz identity, or by using some computer algebra algorithms, and in particular, the algorithms of Zeilberger, Petkovsek and van Hoeij, the resulting ${}_{4}F_{3}(1)$ can be reduced for particular choices of the involved parameters. This reduction leads to obtaining several simple linearization formulas of some particular Jacobi polynomials free of any hypergeometric functions.