1. INTRODUCTION

A large number of integral formulae involving different types of special functions have been developed by many authors. Garg and Mittal [1]obtained an interesting unified integral involving Fox H-function. Considering the work of Garg and Mittal [1], Ali [2]gave three interesting unified integrals involving the hypergeometric function 1F2.By using Ali’s method [2] Choi and Agarwal [3] presented two generalized integral formulas involving the Bessel function of the first kind , which are expressed in terms of the generalized (Wright) hypergeometric functions. Agarwal[4] study some new unified integral formulae associated with the H -function. Each of these formulae involves a product of the H-function and Srivastavapolynomials with essentially arbitrary coefficients. They evaluated the formulae in terms of z[logarithmic derivative of z]. Recently Chouhan and Khan [5] presents two new unified integral formulae involving the Fox H-function and M-Series. These results were expressed in terms of the H function.

The Riemann-Liouville fractional integral operator of order [6], [7] and [8]is defined by where m is a positive integer and the integral exists.

A more general function known as H-function was introduced by Inayat-Hussain [9] in the following form Where and 1.i Here aj (j = 1,…,P) and bj (j = 1,…,Q) are complex parameters, 0( 1,...,P) and 0( 1,...,Q)jjjj and the exponents Aj (j = 1,…,N) and Bj (j = N+1,…,Q) can take any non-integer values . When allthe exponents Aj and Bjtakes the value unity, the H-function reduces to the well-known Fox’s H-function [10](see also [11]). Buschman and Srivastava [12]has proved that the integral represented by Eq.(2.2) is absolutely convergent when > 0 and | arg z | < 1/2 , where The following functions are represented in terms of H function by choosing parameters specifically. (i) The function connected with certain class of Feynman integrals Where (ii) The polylogarithm function of order s introducedby Erdelyi et.al. [14] is

The generalized polynomial set ,;Snx

is defined by the following Rodrigues typeformula [15] with the differential operator being defined as Where Raizada [15] presented ,;Snx in the following series form where

Kuipers and Meulenbeld [16] introduced generalized associated Legendre functions ,,,mnmnkkPzQz This function can be presented in terms of hypergeometric function 21,;;Fabcz as

In this section, two integrals will be evaluated. The integrals are associated with the product of the generalized polynomial set, the H-function and generalized associated Legendre functions. The integrals are as follow:

where The conditions of validity of Eq (3.1) are (a) 0),( Re (b) min {,,,,,,, }0uvpq (not all zero simultaneously) (c) 1 M1+Re()ReminRe (/)0jjjkub (d)

()()max ,1, .f



(e) If

1,2,...;0,1,2,...;22

mnmnkk220,1,2,...k12z,

PROOF. Let L.H.S. of Eq(3.1) is 2.To evaluate the integral, the generalized polynomial set ,,Snz

is replaced by its series representation from using Eq(2.8), H-function is replaced by its Mellin-Barnes contour integral form using Eq(2.2)and ,mnkQz is replaced by its hypergeometric function form using Eq(2.11)in the left hand side ofEq(3.1). Then the powers of (x - a), (b - x), (cx+d) and (gx+f) are collected. In the resulting expression the order of integration and summation is interchanged (which is permissible under the conditions stated with (3.1)) and integral is expressed as follows

Taking##311,,kRtuRtv#1Rtp

#1,Rtq and simplifying the powers of (cx+d) and (gx+f) by applying binomial expansions for x [a, b] The innermost integral is simplified with the help of the Eulerian type integral given by Eq(3.6). The beta function is simplified in terms of gamma function and resulting Mellin-Barnes contour integral is interpreted as H-function. After little simplification the right hand side of Eq(3.1) is obtained.

where and 12121212,,,(,,,), and bbaabbaaR are as given in Eqs (2.9), (2.10) and (2.11) respectively. The conditions of validity for (3.4) are (a) 0),( Re (b) min {,,,,,,,}0uvpq (not all zero simultaneously)

(c) 1 N

aku



1 N

av



(d)



(e) If

1,2,...;0,1,2,...;22

mnmnkk220,1,2,...k 12z, PROOF. The integral (3.4) can be evaluated in a similar way as that of the first integral.

Each of our integral formulae (3.1) and (3.4) are unified in nature and possesses manifold generality. On suitably specializing the parameters of theH-function, the generalized polynomial set ,,Sn[x] in our main integrals, a large number of new integrals can be obtained as their special cases. one of them are discussed below. In the third integral reducing H function to F(-z, s) function as given by Eq(2.6) and the generalized associated Legendre polynomial ,mnkQx to the associated Legendre polynomial nkQx as given by Eq(2.12), following result is obtained. Where L3 and L4 are same as given by Eqs(3.5) and (3.6).

The results obtained from these integrals can be applied to obtain Riemann-Liouville fractional calculus operator of unified functions. One of the examples is shown below. Taking b = z,  = v = 0 in Eq(3.1), the Riemann-Liouville fractional calculus operator of order of a unified function is obtained as where and other symbols are same as given in Eqs(2.9), (2.10) and (2.11) respectively. The conditions of validity of Eqs(5.1)can be obtained from those stated with (3.1) and (3.4).

The author is grateful to Vinita Agrawal, Department of Humanities & Sciences, Thakur College of Engineering & Technology, Mumbai-400101, Maharashtra, India for her useful suggestions and constant help during the preparation of this paper.

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Department of Mathematics, M A J Govt. P G College, Deeg, Bharatpur, Rajasthan