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Abstract

This paper deals with some new integral relation of I- function of one variable.

Key words

I- function, multivariable polynomial

Introduction

The I- function of one variable is defined by Saxena [1] and we shall represent here in the following manner:

                                (1.1)

where   is a complex variable and

                                                               (1.2)

In which log |z| represent the natural logarithm of |z| and arg |z| is not necessarily the principle value. An empty product is interpreted as unity, also,

                         (1.3)

m, n and pi  are non-negative integers satisfying 0 < n < p_i, 0 < m < q_i,  , α_ji, (j=1,….. p_i; i = 1,………r) and β_ji (j = 1,….q_i ; i = 1,….r) are assumed to be positive quantities for standardization purpose . Also a_ji ( j = 1,…., p_i; i = 1,……,r) and b_ji  ( j = 1,…….., q_i; i = 1,…..,r) are complex numbers such that none of the points.

             (1.4)

which are the poles of (b_n – β_nS), h = 1,……m and the points.

S =                                             (1.5)

which are the poles of  coincide with one another, i.e. with

                                                            (1.6)

for n, h = 0, 1,2,….; h = 1,…., m;  l  = 1,…..,n.

Further, the contour L runs from - to +. Such that the poles of , h = 1……, m; lie to the right of L and the poles , l = 1,….,n lie to the left of L. The integral (1.1) converges, if |arg z| < ½ B π   (B>0), A < 0, where

Gradshteyin and Ryzhik [2] given table of Integrals, series, Sharma [3] evaluated the integrals involving general class of polynomial with H-function, Srivastava and Garg [4] established some integrals involving a general class of polynomials and the multivariable H- function. Recently, Satyanarayana and Pragathi Kumar [5] has evaluated Some finite integrals involving multivariable polynomials, Agarwal [6] established  integral involving the product of Srivastava’s polynomials and generalized Mellin-Barnes type of contour integral, Bhattar [7] established some integral  formulas involving two - function and multivariable’s general class of polynomiyals. Satyanarayana and Pragathi Kumar [5] has evaluated some finite integrals involving multivariable polynomials. Following them, I evaluated some new integrals involving multivariable polynomials, and I-function of one variable.

Formula Required

The following formulas will be required in our investigation

(i)            The second class of multivariable polynomials given by Srivastava [8,9] is defined as follows:

            .                   (2.1)

(ii)           The first class of multivariable polynomials introduced by Srivastava and Garg [4] is defined as follows:

Some New Finite Integrals Formulae

In this section we prove two integral formulae, which involving multivariable polynomials, and I function of one variable.

where m_i>0  (i=1,…,t), n_i>0 (i = 1,….,t)  h ≥ 0 ,g ≥ 0   (not both are zero simultaneously).

Provided the conditions stated in results (3.1) are satisfied.

Proof : To establish integral in (3.1), we express I-function occurring in its left –hand side interms of Mellin-Barnes [10] contour integral given by (3.1), the second class of polynomial given by (2.1). Then interchange the order of integration of summations and integration, we arrive the following:

Now we evaluate the above integral with help of integral (2.2). Interpreting the resulting contour integral of H-function we can easily arrive at desired result (3.1).

To establish integral in (3.2) can be easily established on the same lines similar to the proof of (3.1).

Special Cases of (3.1) and (3.2)

Take A (V_1,k₁;…;V_t,k_t) = A₁(V₁,k₁)… A_t (V_t,k_t) in (3.1) the multivariable polynomial  reduced to the product of well-known general class of polynomials  and the result (3.1) reduced to following form

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1. Substituting r=1 in (3.1), we obtain :

2. Substituting α_j=β_j=1 in (4.2) we obtain

           

1. Substituting r=1 in (3.2), we obtain :

2. Substituting α_j=β_j=1 in (4.4), we obtain

References

1. Saxena S (1982) Formal solution of certain new pair of dual integral equations involving H-function. Proc Nat Acad Sci India 52, A. III, 366-375.
2. Gradshteyin IS, Ryzhik IM (2001) Table of Integrals, series and products, 6/e. Academic press, New Delhi.
3. Sharma RP (2006) On finite integrals involving Jacobi polynomials and the H-function. Kyungpook. Math J 46: 307-313. 
4. Srivastava HM, Garg M (1987) Some integrals involving a general class of polynomials and the multivariable H-function. Rev Roamaine Phys 32: 685-692.
5. Satyanarayana B, Pragathi Kumar Y (2011) Some finite integrals involving multivariable polynomials, H-function of one variable and H-function of ‘r’ variables. African Journal of Mathematics and Computer Science Research 4: 281-285.
6. Agarwal P (2012) On a unified integral involving the product of Srivastava’s polynomials and generalized Mellin-Barnes type of contour integral. Advances in Mechanical Engineering and its Applications (AMEA) 158: 2.
7. Bhattar B (2014) Integral formulae’s involving two - function and multivariable’s general class of polynomiyals. ISCA Bushma.
8. Srivastava A (2010) The integration of certain products pertaining to the H-function with general polynomials. Ganita Sandesh 31/32: 51-58.
9. Srivatsava HM, Singh NP (1983) The integration of certain products of the multivariable H-function with a general class of polynomials. Rend Circ Mat Palermo 32.
10. Agarwal P, Chand M (2012) New theorems involving the generalized Mellin-Barnes type of contour integrals and general class of polynomials. GJSFRM 12.