Saxena and Kumar (1995), introduced the following basic
analogue of I-function in terms of the Mellin-Barnes type basic contour integral
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).
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 (2) we consider the expression:
On multiplying above equation by:
and making use of the following identity given by Askey
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.
Proof: Consider the L.H.S. of (3):
By definition of Iq- function, we get:
This completes the proof of theorem (3).
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 Foxs H-function.
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.