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

Abstract

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