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

  • H. M. Srivastava
  • N. P. Singh
Article

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

Keywords

Complex Variable Integral Formula Contour Integral Hermite Polynomial Jacobi Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer 1983

Authors and Affiliations

  • H. M. Srivastava
    • 1
    • 2
  • N. P. Singh
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of VictoriaVictoriaCanada
  2. 2.Department of MathematicsMotilal Vigyan Mahavidyalaya (Bhopal University)BhopalIndia

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