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International Journal of Engineering Mathematics
Volume 2014 (2014), Article ID 931395, 11 pages
A Study of I-Function of Several Complex Variables
1Department of Mathematics, Manipal Institute of Technology, Manipal, Karnataka 576104, India
2SCSVMV, Sri Jayendra Saraswathi Street, Enathur, Kanchipuram, Tamil Nadu 631561, India
3Department of Mathematics, College of Engineering, Trikaripur, Kerala 670307, India
4Department of Mathematics, P.A. College of Engineering, Mangalore, Karnataka 574153, India
Received 27 June 2013; Revised 5 September 2013; Accepted 23 September 2013; Published 27 January 2014
Academic Editor: Alberto Cardona
Copyright © 2014 Prathima Jayarama et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The aim of this paper is to introduce a natural generalization of the well-known, interesting, and useful Fox H-function into generalized function of several variables, namely, the I-function of ‘‘’’ variables. For , we get the I-function introduced and studied by Arjun Rathie (1997) and, for , we get I-function of two variables introduced very recently by ShanthaKumari et al. (2012). Convergent conditions, elementary properties, and special cases have also been given. The results presented in this paper generalize the results of H-function of ‘‘’’ variables available in the literature.
In 1997, Rathie introduced the generalization of the well-known Fox’s H-function  which has very recently found interesting applications in wireless communication [2–4]. Motivated by the I-function, very recently Shantha Kumari, Nambisan, and Rathie introduced I-function of two variables  which is a natural generalization of the H-function of two variables introduced earlier by Mittal and Gupta  and discussed some of its important properties.
In the present paper, we aim to develop I-function of “” variables which may be regarded as the natural generalization of the H-function of “” variables introduced earlier by Srivastava and Panda . We also discussed some of the important properties.
The remainder of this paper is organized as follows.
In Section 2, we have defined the I-function of “” variables by means of multiple Mellin-Barnes type contour integrals. In Section 3, we have given the convergence conditions for this function. In Section 4, we obtained the series representation and behaviour of the function for small values of the variables. In Section 5, we have mentioned special cases of our function giving relations with other functions available in the literature. Finally, in Section 6, we have mentioned a few important properties.
2. The I-Function of Several Variables
The generalized Fox H-function, namely, I-function of “” variables, is defined and represented in the following manner:where , , are given by where .
Also,(i), for ;(ii);(iii)an empty product is interpreted as unity;(iv)the parameters , , , , , , and are nonnegative integers such that , , , and (not all zero simulataneously);(v), , , and are assumed to be positive quantities for standardisation purpose. However, the definition of I-function of “” variables will have a meaning even if some of the quantities are zero or negative numbers. For these, we may obtain corresponding transformation formulas which will be given in a later section;(vi), , , and are complex numbers;(vii)the exponents , , , and of various gamma functions involved in (2) and (3) may take noninteger values;(viii)the contour in the complex -plane is of Mellin-Barnes type which runs from to , ( real) with indentation, if necessary, in such a manner that all singularities of , lie to the right and , are to the left of .
Following the results of Braaksma  the I-function of “” variables is analytic if
3. Convergence Conditions
Integral (1) converges absolutely if where and if and , , then integral (1) converges absolutely under the following conditions:(i), , where is given by (4) and (ii) with , ( and are real, ), and are chosen so that for we have .
Outline of the Proof. The convergence of integral (1) depends on the asymptotic behaviour of the functions , , defined by (2) and (3), respectively. Such asymptotic behaviour is based on the following relation for the gamma function , , : Along the contour , if we put and take the limit as for , we obtain by virtue of (8) that
Similarly, we have Also, Hence, substituting (10)-(11) in (1) and using (12) we have, after much simplification, where is independent of and , , and are given by (6), (7), and (8), respectively, for each .
Hence, the result follows.
Remark 1. If in (1), then the function will be denoted bywhere where .
Remark 2. If , , where and if in (1), then the corresponding function will be denoted bywhere
4. Series Representation
if(i), and , where is given by (4);(ii) for , , , , , ,then This result can be proved on computing the residues at the poles as follows:
The behaviour of the function is given by where On the other hand, when , the associated function given by (16) has the behaviour where
5. Elementary Special Cases
In this section, we mention some interesting and useful special cases of the I-function of “” variables.(i)If all the exponents , , , and in (1) are equal to unity, we obtain H-function of “” variables defined by Srivastava and Panda .(ii)When , (1) degenerates into the product of mutually independent I- functions of one variable introduced by Rathie .(iii)When and , (1) reduces to the I-function defined by Rathie .(iv)When , , , , and and is replaced by , , (1) reduces to the generalized Lauricella function .
where the functions , are Wright’s generalized hypergeometric functions . where the functions are Wright’s generalized Bessel functions . where , are the generalized Riemann zeta functions [13, page 27, 1.11, (1)], which are the generalizations of Hurwitz zeta functions and Riemann zeta functions [13, page 24, 1.10, (1) and 1.12, (1)]. where are the polylogarithms of order . For , , the R.H.S. of (28) reduces to the product of Euler’s dilogarithm [13, page 31, 1.11.1, equation (2)].
6. Elementary Properties and Transformation Formulas
The properties given below are immediate consequence of the definition (1) and hence they are given here without proof:
for , , where , , where , where , where , where , , where , ,