Linearization of the products of the Carlitz-Srivastava polynomials of the first and second kinds via their integral representations

In this paper, the authors investigate two families of generalized Lauricella polynomials which are known as the Carlitz-Srivastava polynomials of the first and second kinds. By means of their multiple integral representations, it is shown how one can linearize the product of two different members of each of these two families of the Carlitz-Srivastava polynomials. Upon suitable specialization of the main results presented in this paper, the corresponding integral representations are deduced for such familiar classes of multivariable hypergeometric polynomials as (for example) the Lauricella polynomials FD(r) in r variables and the Appell polynomials F1 in two variables. Each of these integral representations, which are derived as special cases of the main results in this paper, may also be viewed as a linearization relationship for the product of two different members of the associated family of multivariable hypergeometric polynomials.