Some families of generating functions for the Jacobi polynomials

The authors derive a unification and generalization of several families of linear, bilinear, and bilateral generating functions for the classical Jacobi polynomials, which were given in a number of earlier works on the subject. They also show how the results considered here can be extended further to hold true for the product of two Jacobi polynomials of different arguments. Relevant historical remarks and observations, and connections with other works on applications of some of these generating functions, are presented rather briefly.