Eulerian integral associated with product of two multivariable I-functions,

a class of polynomials and the multivariable A-function

1 Teacher in High School , FranceE-mail : fredericayant@gmail.com

ABSTRACTThe present paper is evaluated a new Eulerian integral associated with the product of two multivariable I-functions defined by Nambisan et al [3] ageneralized Lauricella function , a class of multivariable polynomials and multivariable A-function with general arguments . We will study the caseconcerning the multivariable H-function defined by Srivastava et al [10] and Srivastava-Daoust polynomial [6].

Keywords: Eulerian integral, multivariable I-function, generalized Lauricella function of several variables, multivariable A-function, generalizedhypergeometric function, class of polynomials, Srivastava-Daoust polynomials

2010 Mathematics Subject Classification. 33C45, 33C60, 26D20

1. Introduction

In this paper, we evaluate a new Eulerian integral of most general characters associated with the products of twomultivariable I-functions defined by Nambisan et al [3], the A-function of several variables defined by Gautam et al [2]and a class of polynomials with general arguments. The A-function of several variables is an extension of themultivariable H-function defined by Srivastava et al [10].

The serie representation of the multivariable A-function is given by Gautam [2] as

(1.1)

where

(1.2)

(1.3)

and (1.4)

which is valid under the following conditions : (1.5)and

(1.6)

Here ;

The I-function is defined and represented in the following manner.

=

(1.7)

(1.8)

where , , are given by :

(1.9)

(1.10)

For more details, see Nambisan et al [3].

Following the result of Braaksma [1] the I-function of r variables is analytic if

(1.11)

The integral (2.1) converges absolutely if

where

(1.12)

Consider the second multivariable I-function.

=

(1.13)

(1.14)

where , , are given by :

(1.15)

(1.16)

For more details, see Nambisan et al [3].

Following the result of Braaksma [1] the I-function of r variables is analytic if :

(1.17)

The integral (2.1) converges absolutely if

where

(1.18)

Srivastava and Garg [7] introduced and defined a general class of multivariable polynomials as follows

= (1.19)

The coefficients are arbitrary constants, real or complex.

2. Integral representation of generalized Lauricella function of several variables

The following generalized hypergeometric function in terms of multiple contour integrals is also required [8 ,page 39eq .30]

(2.1)

where the contours are of Barnes type with indentations, if necessary, to ensure that the poles of are separated from those of . The above result (1.23) can be easily established by an appeal to thecalculus of residues by calculating the residues at the poles of In order to evaluate a number of integrals of multivariable I-function, we first establish the formula

(2.2)

where

and is a particular case of the generalized Lauricella function introduced by Srivastava-Daoust[6,page454] given by :

(2.3)

Here the contour are defined by starting at the point and terminating at the point with and each of the remaining contour run from to

(2.2) can be easily established by expanding by means of the formula :

(2.4)

integrating term by term with the help of the integral given by Saigo and Saxena [4, page 93, eq.(3.2)] and applying the definition of the generalized Lauricella function [6, page 454].

3. Eulerian integral

In this section , we note :

;

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

; (3.8)

(3.9)

(3.14)

(3.15)

(3.16)

j

(3.17) j

( (1.18)

(3.19)

(3.20)

(3.21)

(3.22)

where

for are defined respectively by (1.2) and (1.3)

(3.23)

(3.24)

We have the general Eulerian integral.

(3.25)(3.25)

Provided that

(A ) ,

(B) ;

The exposants

of various gamma function involved in (1.10) and (1.11) may take non integer values.

;

The exposants

of various gamma function involved in (1.15) and (1.16) may take non integer values.

(C)

(D)

(E)

(F)

;

;

(G)

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

either and

or and

(I)The multiple series occuring on the right-hand side of (3.25) is absolutely and uniformly convergent.

Proof

To prove (3.25), first, we express in serie the multivariable A-function with the help of (1.1), a class of multivariablepolynomials defined by Srivastava et al [7] in serie with the help of (1.19) and we interchange the orderof summations and t-integral (which is permissible under the conditions stated). Expressing the I-functions of r-variables and s-variables defined by Nambisan et al [3] in terms of Mellin-Barnes type contour integral with the help of(1.8) and (1.14) respectivel and interchange the order of integrations which is justifiable due to absolute convergence ofthe integral involved in the process. Now collect the power of Now collect the power of with with

and collect the power of and collect the power of with with . Use the equations (2.2). Use the equations (2.2)and (2.3) and express the result in Mellin-Barnes contour integral. Interpreting the and (2.3) and express the result in Mellin-Barnes contour integral. Interpreting the dimensional dimensionalMellin-Barnes integral in multivariable I-function defined by Nambisan et al [3], we obtain the equation (3.25).Mellin-Barnes integral in multivariable I-function defined by Nambisan et al [3], we obtain the equation (3.25).

RemarksRemarks

If a) If a) ; b) ; b) , we obtain the similar formulas that, we obtain the similar formulas that(3.25) with the corresponding simplifications.(3.25) with the corresponding simplifications.

4.Particular case

a) , the multivariable I-functions defined by Nambisanreduces to multivariable H-function defined by Srivastava et al [10]. We have.

(4.1)(4.1)

under the same conditions and notations that (3.25) with

b) b) If If = = (4.2) (4.2)

then the general class of multivariable polynomial then the general class of multivariable polynomial reduces to generalized Lauricella function reduces to generalized Lauricella functiondefined by Srivastava et al [6]. We have the following integral.defined by Srivastava et al [6]. We have the following integral.

(4.3)(4.3)

under the same conditions and notations that (3.25)under the same conditions and notations that (3.25)

where where , , is defined by (4.2) is defined by (4.2)

Remark:

By the following similar procedure, the results of this document can be extented to product of any finite number ofBy the following similar procedure, the results of this document can be extented to product of any finite number ofmultivariable I-functions defined by Nambisan et al [3] and a class of multivariable polynomials defined by Srivastavamultivariable I-functions defined by Nambisan et al [3] and a class of multivariable polynomials defined by Srivastavaet al [7].et al [7].

5. Conclusion

In this paper we have evaluated a generalized Eulerian integral involving the product of two multivariable I-functionsdefined by Nambisan et al [3], a expansion of multivariable A-function and a class of multivariable polynomialsdefined by Srivastava et al [7] 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 theparameters, as many as desired results involving the special functions of one and several variables can be obtained.

REFERENCES

[1] B. L. J. Braaksma, “Asymptotic expansions and analytic continuations for a class of Barnes integrals,”CompositioMathematical, vol. 15, pp. 239–341, 1964.

[2] Gautam B.P., Asgar A.S. and Goyal A.N. On the multivariable A-function. Vijnana Parishas Anusandhan Patrika Vol29(4) 1986, page 67-81.

[3] Prathima J. Nambisan V. and Kurumujji S.K. A Study of I-function of Several Complex Variables, InternationalJournalof Engineering Mathematics Vol(2014) , 2014 page 1-12

[4] Saigo M. and Saxena R.K. Unified fractional integral formulas for the multivariable H-function I. J.FractionalCalculus 15 (1999), page 91-107.

[5] Sharma C.K.and Ahmad S.S.: On the multivariable I-function. Acta ciencia Indica Math , 1994 vol 20,no2, p 113-116.

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

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

[8] Srivastava H.M. and Karlsson P.W. Multiple Gaussian Hypergeometric series. Ellis.Horwood. Limited. New-York,Chichester. Brisbane. Toronto , 1985.

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

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

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