Research Article

# Some Transformation Formulae of Basic Analogue of I-Function Farooq Ahmad, Renu Jain and D. K. Jain

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:  (1)

where, αj, βj, αji, βji, are real and positive, aj, bj, aji, bji are complex numbers 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:  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:  (2)

Proof: To prove (2) we consider the expression: On multiplying above equation by: and making use of the following identity given by Askey (1978): the left hand side takes the form: Hence, we have: If we take r = 1, Ai = A, Bi = B, we get following well know basic analogue of Fox's H function : Transformation formulae of Iq -Function: In this section we derive number of transformation formulae for basic analogue of i-function.

Theorem 3: (3)

Proof: Consider the L.H.S. of (3): By definition of Iq- function, we get: This completes the proof of theorem (3).

Theorem 4: (4)

Theorem 5: (5)

Theorem 6: (6)

Theorem 7: (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.

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. 