The integration of certain products of the multivariableH-function with a general class of polynomials
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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.,  through ; see also  through , , , , , , , and ). 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 . 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 , and such other classes of generalized hypergeometric polynomials as those studied earlier by F. Brafman  and by H. W. Gould and A. T. Hopper . 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.  and )] 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., ,  and ).
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
KeywordsComplex Variable Integral Formula Contour Integral Hermite Polynomial Jacobi Polynomial
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- Erdélyi A., Magnus W., Oberhettinger F. and Tricomi F. G.,Higher Transcendental Functions, Vols. I and II, McGraw-Hill, New York, Toronto and London, 1953.Google Scholar
- Erdélyi A., Magnus W., Oberhettinger F., and Tricomi F. G.,Tables of Integral Transforms, Vol., I and II, McGraw-Hill, New York, Toronto and London, 1954.Google Scholar
- Singh N. P.,A study of H-Function and Its Generalizations, Ph. D. thesis, Bhopal Univ., India, 1980.Google Scholar
- Srivastava H. M. and Panda R.,Some expansion theorems and generating relations for the H function of several complex variables. I and II, Comment. Math. Univ. St. Paul.24 (1975), fasc. 2, 119–137;ibid. Srivastava H. M. and Panda R.,Some expansion theorems and generating relations for the H function of several complex variables. I and II, Comment. Math. Univ. St. Paul.25 (1976), fasc. 2, 167–197.MathSciNetGoogle Scholar
- Srivastava H. M. and Pathan M. A.,Some bilateral generating functions for the extended Jacobi polynomials. I and II, Comment. Math. Univ. St. Paul.28 (1979), fasc. 1, 23–30;ibid. Srivastava H. M. and Pathan M. A.,Some bilateral generating functions for the extended Jacobi polynomials. I and II, Comment. Math. Univ. St. Paul.29 (1980), fasc. 2, 105–114.MATHMathSciNetGoogle Scholar