Kober fractional q-integral of basic analogue of multivariable H-function and
basic analogue of multivariable Meijer-function
1 Teacher in High School, France E-mail : [email protected]
ABSTRACT
The Kober fractional q-integral of the basic analogue of multivariable H-function and basic analogue of multivariable Meijer variables have been evaluated in this paper. At the end we shall give particular cases.
Keywords : Kober fractional q-integral operator, basic integration, basic analogue of multivariable H-function, basic analogue of multivariable Meijer function.
2010 Mathematics Subject Classification. 33C99, 33C60, 44A20
1. Introduction and preliminaries.
Recently the authors, see ([8],[11]-[14]) have been investigated the applications of the Riemann-Liouville and Kober fractional q-integral operators to various basic hypergeometric function of one variable including the basic analogue of Fox’s H-function. These results are a new contribution to the theory of q-fractional calculus. Motivated by these works, and a possible scope for their application in evaluation and solution of the q-integral equations, we further explore the possibility of evaluation of Kober type fractional q-integrals involving basic analogue of multivariable H-function and.basic analogue of multivariable Meijer-function and.
Also, the basic analogue of the Kober fractional integral operator, see Agarwal [1] is defined by
(1.1)
where is an arbitrary order of integration with and being real or complex quantity.
Following Agarwal [1], Al-Salam [2] and Jackson [4] the basic integration is defined as
(1.2)
By virtue of the result (2), the operator (1) can be expressed as :
(1.3)
where and being real or complex quantity.
In the theory of q-series, for real or complex and , the -shifted factorial is defined as :
(1.4)
so that
or equivalently
(1.5)
Kober Fractional Q-Integral of Basic Analogue
of Multivariable H-Function and Basic Analogue
The q-gamma function [4] is given by
(1.6)
also
(1.7)
2.Basic analogue of multivariable H-function.
In this section, we introduce the basic analogue of multivariable H-function defined by Srivastava and Panda [9,10]
We note
(2.1)
(2.2)
where
(2.3)
(2.4)
where the integers are constrained by the inequalities and
. The poles of the integrand are assumed to be simple.
The quantities, are complex
numbers and the following quantities ,
are positive real numbers.
The contour in the complex -plane is of the Mellin-Barnes type which runs from to with indentations, if
necessary to ensure that all the poles of are separated from those of
, , . For large values of the integrals converge if
If the quantities , then the basic analogue of multivariable H-function reduces in basic analogue of multivariable Meijer function defined by Khadia and Goyal [5], we obtain
(2.5)
where
(2.6)
(2.7)
where the integers are constrained by the inequalities and
. The poles of the integrand are assumed to be simple.
The quantities, are complex
numbers.
The contour in the complex -plane is of the Mellin-Barnes type which runs from to with indentations, if necessary to ensure that all the poles of are separated from those of
, , . For large values of the integrals converge if
.
3. Main results.
Let
; (3.1)
(3.2)
(3.3)
In this section, the following result involving the Kober fractional q-integral operator [8] of the multiple basic analogue of multivariable H-function
(3.4)
where Re ,
Proof
Replace the basic analogue of multivariable H-function by this integral contour, we obtain by using (1.3) and (2.2)
(3.4)
On Interchanging the order of integrations and then on summing the inner series with the help of the Heine summation theorem, Gasper and Rahman [3] namely
(3.5)
the left hand side, after algebraic manipulations, we obtain
Interpreting in view of the definition (2.2), we obtain the result
Corollary 1.
(3.6)
where Re ,
4. Riemann-Liouville fractional q-integral.
It is intereting to observe that several result similar to be above results can be deduced from the other result (3.3). Further, the results proved in this paper may find certain applications to the solution of the q- differintegral equations associated with the aforementioned function. If , we obtain
(4.1)
where denotes the Riemann-Liouville fractional q-integral operator defined as :
We have the following corollary
Corollary.
(4.3)
where and
Corollary 1.
(4.4)
where Re ,
If the basic analogue of multivariable H-function reduces to basic of Srivastava-Daoust function, see the work of Yadav et al. [13].
We obtain the same relations with basic analogue of H-function of two variables and one variable and basic analogue of Meijer-function of two variables and one variable. The basic analogue of H-function of two variables and the basic analogue of H-function of one variables are defined by Saxena et al. [7] and [6] respectively.
5. Conclusion.
The importance of our all the results lies in their manifold generality. By specialising the various parameters as well as variables in the basic analogue of multivariable H-function and the basic analogue of multivariable Meijer-function, we obtain a large number of results involving remarkably wide variety of useful basic functions ( or product of such basic functions) which are expressible in terms of basic H-function, Basic Meijer’s G-function, Basic E-function, basic hypergeometric function of one and two variables and simpler special basic functions of one and two variables. Hence the formulae derived in this paper are most general in character and may prove to be useful in several interesting cases appearing in literature of Pure and Applied Mathematics and Mathematical Physics.
References
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[2] W.A. Al-Salam, Some fractional q-integrals and q-derivatives, Proc. Edin. Math. Soc., 15(1966), 135-140.[4] G.
[3] Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cam-bridge, 1990.
[5] S.S. Khadia and A.N. Goyal, On a generalized function of n variables, Vijnana Parishad Anusandhan Patrika 13 (1970), 191-201.
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[12] R.K. Yadav and S.D. Purohit, On applications of Kober fractional q-integral to basic hypergeometric function. Rajasthan Acad. Phys.Sci. (accepted).
[13]R.K. Yadav, S.D. Purohit and S.L. Kalla, On the Riemann-Liouville fractional q-integral operator involving a basic a basic analogue of the H-function. Frac.Cal.Appl.Anal. 8(3)(2005), 313-322.
