Eulerian Integral Associated with Product of Two Prasad's Multivariable H-Functions, the Classes of Multivariable Polynomials

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Eulerian integral associated with product of two Prasad’s multivariable H-functions,

the classes of multivariable polynomials

1 Teacher in High School , France E-mail : [email protected]

ABSTRACT

Recently, Raina and Srivastava [2] and Srivastava and Hussain [7] have provided closed-form expressions for a number of a Eulerian integral about the multivariable H-functions.The present paper is evaluated a new Eulerian integral associated with the product of two modified multivariable I-functions defined by Prasad [1], a generalized Lauricella function an the classes of multivariable polynomials with general arguments . We shall study the case concerning the Srivastava-Daoust polynomial [4] and we shall give few remarks..

Keywords: Eulerian integral, modified multivariable H-function, generalized Lauricella function of several variables, generalized hypergeometric function, class of polynomials

2010 Mathematics Subject Classification. 33C60, 82C31

1. Introduction

The well-known Eulerian Beta integral

(1.1)

is a basic result for evaluation of numerous other potentially useful integrals involving various special functions and polynomials. The authors Raina and Srivastava [2], Saigo and Saxena [3], Srivastava and Hussain [7], Srivastava and Garg [6] and other have studied the Eulerian integral. In this paper, we consider a general class of Eulerian integral concerning the product of two multivariable I-functions, defined by Prasad [1], the generalized hypergeometric function and the classes of multivariable polynomials.

The generalized polynomials of multivariables defined by Srivastava [4], is given in the following manner :

(1.1)

where are arbitrary positive integers and the coefficients are arbitrary constants, real or complex. Srivastava and Garg [6] introduced and defined a general class of multivariable polynomials as follows

= (1.2)

The coefficients are arbitrary constants, real or complex.

We shall note and

(1.3)

Eulerian Integral Associated with Product of Two

Prasad's Multivariable H-Functions, the Classes

of Multivariable Polynomials

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The multivariable I-function of r-variables defined by Prasad [1] is an extension of the multivariable H-function defined by Srivastava and Panda [8,9]. It is defined in term of multiple Mellin-Barnes type integral :

=

(1.8)

(1.9)

The defined integral of the above function, the existence and convergence conditions, see Y,N Prasad [1]. Throughout the present document, we assume that the existence and convergence conditions of the multivariable I-function.

The condition for absolute convergence of multiple Mellin-Barnes type contour (1.9) can be obtained by extension of the corresponding conditions for multivariable H-function given by as :

, where

(1.10)

where

The complex numbers are not zero.Throughout this document , we assume the existence and absolute convergence conditions of the multivariable I-function.

We may establish the the asymptotic expansion in the following convenient form :

,

where : and

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=

(1.11)

(1.12)

The defined integral of the above function, the existence and convergence conditions, see Y,N Prasad [1]. Throughout the present document, we assume that the existence and convergence conditions of the multivariable I-function.

The condition for absolute convergence of multiple Mellin-Barnes type contour (1.9) can be obtained by extension of the corresponding conditions for multivariable H-function given by as :

where ,

(1.13)

where

The complex numbers are not zero.Throughout this document , we assume the existence and absolute convergence conditions of the multivariable I-function.

We may establish the the asymptotic expansion in the following convenient form :

,

where : and

If we shall this function.

2 Rquired result

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(2.1)

where ; and

is a Lauricella's function of -variables, see Srivastava and Manocha ([8], page 60)

The formula (2.2) can be establish by expanding by means of the formula :

(2.2)

integrating term by term with the help of the integral given by Saigo and Saxena [3, page 93, eq.(3.2)] and applying the definition of the Lauricella function [4, page 454].

2. Eulerian integral

(3.1)

(3.2)

(3.3)

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(3.5)

j

(3.6)

(3.7)

(3.8)

(3.9)

We have the general Eulerian integral.

Theorem.

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We obtain the I-function of

We obtain the I-function of variables. variables.

Provided that

(A ) ,

(B) ;

;

( ;

(C)

(D)

and and

(E)

(F)

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(G) , ,

(G) . The equality holds, when , in addition,

either and

or and

Proof

To prove (3.14), first, we express in serie the class of multivariable polynomials with the help of (1.1), the class of multivariable polynomials in serie with the help of (1.2) and we interchange the order of summations and t-integral (which is permissible under the conditions stated). Expressing the I-functions of r-variables and s-variables defined by Prasad [1] in terms of Mellin-Barnes type contour integral with the help of (1.9) and (1.12) respectively and interchange the order of integrations which is justifiable due to absolute convergence of the integral involved in the process . Now collect the power of Now collect the power of with with . Changing the order of integrations. Changing the order of integrations and summations (which is easily seen to be justified due to the absolute convergence of the integral and the summations

and summations (which is easily seen to be justified due to the absolute convergence of the integral and the summations

involved in the process) , using the lemma and interpreting

involved in the process) , using the lemma and interpreting Mellin-barnes integrals contour inMellin-barnes integrals contour in

modified multivariable -function I defined by Prasad. [1], we obtain the desired result (3.10).

b)

b) If If = = (4.1) (4.1)

then the general class of multivariable polynomial

then the general class of multivariable polynomial reduces to generalized Lauricella function reduces to generalized Lauricella function defined by Srivastava and Daoust [5], it’s a particular case of generalized hypergeometric function defined by

defined by Srivastava and Daoust [5], it’s a particular case of generalized hypergeometric function defined by

Srivastava and Panda [9,10].

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We have the following integral.

Corollary.

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(4.3) (4.3)

under the same conditions and notations that (3.10)

where

where , , is defined by (4.1) is defined by (4.1)

Remarks:

By the following similar procedure, the results of this document can be extented to product of any finite number of

multivariable I-functions defined by Prasad [1], classes of multivariable polynomials defined by Srivastava and Garg

[5] and class of multivariable polynomials defined by Srivastava [4].

[5] and class of multivariable polynomials defined by Srivastava [4]. We have similar integrals concerning other multivariables special functions.

5. Conclusion.

In this paper we have evaluated a generalized Eulerian integral involving the product of two multivariable I-functions defined by Prasad [1], a class of multivariable polynomials defined by Srivastava and Garg [5] and a class of multivariable polynomials defined by Srivastava [3] with general arguments. The formulae established in this paper is very general nature. Thus, the results established in this research work would serve as a key formula from which, upon specializing the parameters, as many as desired results involving the special functions of one and several variables can be obtained.

REFERENCES

[1] Y.N. Prasad , Multivariable I-function , Vijnana Parishad Anusandhan Patrika 29 ( 1986 ) , page 231-237.

[2] R.K. Raina and H.M. Srivastava, Evaluation of certain class of Eulerian integrals. J. phys. A: Math.Gen. 26 (1993), 691-696.

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Calculus 15 (1999), 91-107.

[4] H.M. Srivastava H.M. A multilinear generating function for the Konhauser set of biorthogonal polynomials suggested by Laguerre polynomial, Pacific. J. Math. 177(1985), page183-191.

[5] H.M. Srivastava and M.C. Daoust, Certain generalized Neumann expansions associated with Kampé de Fériet function. Nederl. Akad. Wetensch. Proc. Ser A72 = Indag. Math. 31(1969) page 449-457.

[6] H.M. Srivastava and M. M. Garg, Some integral involving a general class of polynomials and multivariable H-function. Rev. Roumaine Phys. 32(1987), page 685-692.

[7] H.M. Srivastava and M.A. Hussain, Fractional integration of the H-function of several variables. Comput. Math. Appl. 30 (9) (1995), 73-85

[8] H.M. Srivastava and H.L. Manocha, A treatise of generating functions. Ellis. Horwood.Series. Mathematics and Applications 1984, page 60.

[9] H.M. Srivastava and R. Panda, Some expansion theorems and generating relations for the H-function of several complex variables, Comment. Math. Univ. St. Paul. 24(1975), page.119-137.

[10] H.M. Srivastava and R. Panda, Some expansion theorems and generating relations for the H-function of several complex variables II. Comment. Math. Univ. St. Paul. 25(1976), page.167-197