The integration of certain products of the multivariableH-function with a general class of polynomials

The authors present six general integral formulas (four definite integrals and two contour inegrals) for theH-function of several complex variables, which was introduced and studied in a series of earlier papers by H. M. Srivastava and R. Panda (cf., e.g., 25] through 29]; see also 14] through 18], 20], 24], 32], 34], 35], 37], and 38]). Each of these integral formulas involves a product of the multivariableH-function and a general class of polynomials with essentially arbitrary coefficients which were considered elsewhere by H. M. Srivastava 21]. By assigning suiatble special values to these coefficients, the main results (contained in Theorems 1, 2 and 3 below) can be reduced to integrals involving the classical orthogonal polynomials including, for example, Hermite, Jacobi and, of course, Gegenbauer (or ultraspherical), Legendre, and Tchebycheff], and Laguerre polynomials, the Bessel polynomials considered by H. L. Krall and O. Frink 9], and such other classes of generalized hypergeometric polynomials as those studied earlier by F. Brafman 3] and by H. W. Gould and A. T. Hopper 8]. On the other hand, the multivariableH-functions occurring in each of our main results can be reduced, under various special cases, to such simpler functions as the generalized Lauricella hypergeometric functions of several complex variables due to H. M. Srivastava and M. C. Daoust (cf. 22] and 23])] which indeed include a great many of the useful functions (or the products of several such functions) of hypergeometric type (in one and more variables) as their particular cases (see,e. g., 1], 10] and 39]). Many of the aforementioned applications of our integral formulas (contained in Theorems 1, 2 and 3 below) are considered briefly. Further usefulness of some of these consequences of Theorems 1 and 2 in terms of the classical orthogonal polynomials is illustrated by considering a simple problem involving the orthogonal expansion of the multivariableH-function in series of Jacobi polynomials. It is also shown how these general integrals are related to a number of results scattered in the literature. 0261 0262 V