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Motivated by resent work of Agarwal [1], the author is establish the new theorem associated with the H - function (Generalized Mellin-Barnes type of Contour Integral), which was introduced and study in a series of papers by by Inayat -Hussain ([12], [13]).Theorem involves a product of the H -function, Generalized hypergeometric functions and Srivastava polynomials. The convergence and existence condition, basic properties of H -function were given by Buschman and Srivastava ([7]). Next, we obtain certain new integrals by the application of our theorem. These results, besides being of very general character have been put in a compact form avoiding the occurrence of infinite series and thus making them useful in applications. Our findings provide interesting unifications and extensions of a number of new results

On a new Theorem Involving Generalized Mellin-Barnes Type of Contour Integral and Srivastava Polynomials