Asian Journal of Mathematics & Statistics

Year: 2012 | Volume: 5 | Issue: 3 | Page No.: 99-103
DOI: 10.3923/ajms.2012.99.103

Q-integral and 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 Fox's H-function in terms of I-function using q-Gamma function. This definition has been employed to obtain several results based on q-Integral. Also some special cases have also been discussed.

Keywords: q-integral, Basic analogue and I-function

INTRODUCTION

Saxena et al. (1983) introduced the following basic analogue of I-function in terms of the Mellin-Barnes type basic contour integral as:

where, α_j, β_j, α_ji, β_ji, are real and positive, a_j, b_j, a_ji, b_ji are complex numbers and:

where, L is contour of integration running from -i∞ to i∞ in such a manner that all poles of G (q^bj-βjs) lie to right of the path and those of G (q^1-aj+αjs) are to the left of the path.

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

Further, for r = 1, A_i = A, B_i = B, α_j = β_i = 1, j = 1, 2, 3, ---- A, i = 1, 2, 3, --- B, Eq. 2 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:

Proof: To prove Eq. 3 we consider the expression:

On multiplying Eq. 4 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, A_i = A, B_i = B, we get the following well know basic analogue of Fox's H-function (Saxena and Kumar, 1995):

3 q-integral of Basic analogue of I-function: In this section we establish few number of results based on q-integral defined by Jackson (1904).

Theorem 1:

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

Since, It is well known that Jackson (1904):

Hence, makes the form:

Since, by Jackson (1904):

This completes the proof of the theorem.

Similarly we can easily prove the following theorems.

Theorem 2:

Theorem 3: Recently a definition for q-analogue of Euler’s definition for Gamma function has been given by Kac and Chebing (2002) as:

Special cases: If we take r = 1, A_i = A, B_i = B in theorems, we get the well-known results of basic analogue of Fox’s H-function (Saxena and Kumar, 1995).

CONCLUSION

In this study, we have obtained some results for basic analogues of 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