  In this paper we use a contour integral method to derive a generating
function in the form of a double series involving the product of two Chebyshev
polynomials over generalized independent indices expressed in terms of the
incomplete gamma function. The generating function represents a more
generalized form relative to current literature. A possible application of this
function to solving partial differential equations is discussed and some
special cases of this generating function are derived. The work involved in the
computation of this generating function is easier relative to previous methods
as we have a closed form solution as opposed to numerical methods.
