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Research Article
 

Some Transformation Formulae of Basic Analogue of I-Function



Farooq Ahmad, Renu Jain and D. K. Jain
 
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ABSTRACT

In this study we have introduced an alternative definition of the basic analogue of a generalization of well-known Fox's H-function in terms of I-function using q-Gamma function. This definition has been employed to obtain several transformation formulae. Some special cases have also been discussed.

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  How to cite this article:

Farooq Ahmad, Renu Jain and D. K. Jain, 2012. Some Transformation Formulae of Basic Analogue of I-Function. Asian Journal of Mathematics & Statistics, 5: 158-162.

DOI: 10.3923/ajms.2012.158.162

URL: https://scialert.net/abstract/?doi=ajms.2012.158.162
 
Received: November 04, 2011; Accepted: March 27, 2012; Published: May 18, 2012



INTRODUCTION

Saxena and Kumar (1995), introduced the following basic analogue of I-function in terms of the Mellin-Barnes type basic contour integral as:

Image for - Some Transformation Formulae of Basic Analogue of I-Function

Image for - Some Transformation Formulae of Basic Analogue of I-Function
(1)

where, αj, βj, αji, βji, are real and positive, aj, bj, aji, bji are complex numbers

Image for - Some Transformation Formulae of Basic Analogue of I-Function

where, L is contour of integration running from - i∞ to i∞ in such a manner that all poles of G (qbj-βjs) lie to right of the path and those of G (q1-aj+αjs) are to the left of the path.

Setting r = 1, Ai = A, Bi = B, we get q-analogue of H-function defined by Saxena et al. (1983) as follows:

Image for - Some Transformation Formulae of Basic Analogue of I-Function

Image for - Some Transformation Formulae of Basic Analogue of I-Function

Further, for r = 1, Ai = A, Bi = B, αj = βi = 1, j = 1, 2, 3, ---- A, i = 1, 2, 3, --- B, Eq. 1. reduces to the basic analogue of Meijer's G-function given by Saxena et al. (1983).

MAIN RESULTS

In this we establish an alternative definition of basic analogue of I-function by using q-gamma function:

We shall make use of Iq (.) notation for basic analogue of I-function and the same is defined as:

Image for - Some Transformation Formulae of Basic Analogue of I-Function

Image for - Some Transformation Formulae of Basic Analogue of I-Function
(2)

Proof: To prove (2) we consider the expression:

Image for - Some Transformation Formulae of Basic Analogue of I-Function

On multiplying above equation by:

Image for - Some Transformation Formulae of Basic Analogue of I-Function

and making use of the following identity given by Askey (1978):

Image for - Some Transformation Formulae of Basic Analogue of I-Function

the left hand side takes the form:

Image for - Some Transformation Formulae of Basic Analogue of I-Function

Hence, we have:

Image for - Some Transformation Formulae of Basic Analogue of I-Function

If we take r = 1, Ai = A, Bi = B, we get following well know basic analogue of Fox's H function [3]:

Image for - Some Transformation Formulae of Basic Analogue of I-Function

Transformation formulae of Iq -Function: In this section we derive number of transformation formulae for basic analogue of i-function.

Theorem 3:

Image for - Some Transformation Formulae of Basic Analogue of I-Function
(3)

Proof: Consider the L.H.S. of (3):

Image for - Some Transformation Formulae of Basic Analogue of I-Function

By definition of Iq- function, we get:

Image for - Some Transformation Formulae of Basic Analogue of I-Function

This completes the proof of theorem (3).

Theorem 4:

Image for - Some Transformation Formulae of Basic Analogue of I-Function
(4)

Theorem 5:

Image for - Some Transformation Formulae of Basic Analogue of I-Function
(5)

Theorem 6:

Image for - Some Transformation Formulae of Basic Analogue of I-Function
(6)

Theorem 7:

Image for - Some Transformation Formulae of Basic Analogue of I-Function
(7)

The proofs of theorem (4) to (7) are similar to that of theorem (3).

Special cases: If we take r = 1, Ai = A, Bi = B in theorems(3) to (7), we get the well-known results of basic analogue of Fox’s H-function[2].

CONCLUSION

In this study we have obtained some transformation formulae for basic analogues for I- function. These results are quite general in nature and reduce to corresponding results for G and H functions and their several special cases. Thus these results can be applied to various problems of mathematical physics.

REFERENCES

1:  Askey, R., 1978. The q-γ and q-β functions. Applicable Anal. Int. J., 87: 125-141.
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2:  Saxena, R.K. and R. Kumar, 1995. A basic analogue of the generalized H-function. Le Math., 50: 263-271.
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3:  Saxena, R.K., G.C. Modi and S.L. Kalla, 1983. A basic analogue of Fox's H-function. Rev. Tec. Ing. Univ., Zulin, 6: 139-143.
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