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New Finite Integrals Involving Product of -function and Srivastava Polynomial



Praveen Agarwal and Mehar Chand
 
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ABSTRACT

The aim of the present research is to study some new finite integrals. Here, first we obtain six new finite double integrals involving the product of the -function and Srivastava polynomials. The values of the integral are obtained in terms of Ψ(z) (the logarithmic derivative of Γ(z)). Next for all findings, we establish an interesting integral relation in terms of -function. Present findings are the most general in nature and act as the key formulas from which we can obtain their special cases.

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  How to cite this article:

Praveen Agarwal and Mehar Chand, 2012. New Finite Integrals Involving Product of -function and Srivastava Polynomial. Asian Journal of Mathematics & Statistics, 5: 142-149.

DOI: 10.3923/ajms.2012.142.149

URL: https://scialert.net/abstract/?doi=ajms.2012.142.149
 
Received: January 20, 2012; Accepted: February 29, 2012; Published: May 18, 2012



INTRODUCTION

Inayat-Hussain (1987a) introduced generalization form of Fox's H-function which is popularly known as Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function. Now Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function stands on fairly firm footing through the research contributions of various authors like Inayat-Hussain (1987b), Rathie (1997), Gupta and Soni (2006), Gupta et al. (2007), Agarwal and Jain (2009), Agarwal (2011) and Buschman and Srivastava (1990).

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function is defined and represented in the following manner by Gupta et al. (2007):

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(1)

Where:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(2)

It may be noted that the Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial contains fractional powers of some of the gamma function and m, n, p, q are integers such that 1≤m≤q, 1≤n≤p, (αj)1,p, (βj)1,q are positive real numbers and (Aj)1,n, (Bj)m+1,q may take non-integer values which we assume to be positive for standardization purpose. (αj)1,p and (βj)1,q are complex numbers.

The nature of contour L, sufficient conditions of convergence of defining integral (1) and other details about the Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function can be seen in the papers of Gupta and Soni (2006) and Gupta et al. (2007).

The behavior of the Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function for small values of |z| follows easily from a result given by Rathie (1997):

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial

Where:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(3)

The following series representation for the Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function given by Saxena et al. (2002) will be required later on:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(4)

Where:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(5)

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(6)

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(7)

The following function which follows as special cases of the Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function will be required in the sequel (Gupta et al., 2007):

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(8)

The Srivastava polynomials Snm[x] will be defined and represented as follows (Srivastava, 1972, Eq. 1):

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(9)

where, n = 0, 1, 2,…, m is an arbitrary positive integer, the coefficients An,l (n, l≥0) are arbitrary constants, real or complex. Snm[x] yields number of known polynomials as its special cases. These include, among other, the Jacobi polynomials, the Bessel Polynomials, the Lagurre Polynomials, the Brafman Polynomials and several others (Srivastava and Singh, 1983).

The following well known Euler Integral Formula is required to establish the main integral (Srivastava and Karlsson, 1985, Eq. 2):

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(10)

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial

MAIN INTEGRALS

Let ψ (z) denote the logarithmic derivative of gamma function Γ (z) i.e.:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial

We have:

First integral:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(11)

Second integral:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(12)

Third integral:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(13)

Fourth integral:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(14)

Fifth integral:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(15)

Sixth integral:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(16)

The following interesting integral will be required to establish the results from Eq. 11-16:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(17)

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(18)

Where:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(19)

The above result Eq. 17 will be valid under the following conditions:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial

where, Ω is given by Eq. 7.

To evaluate the above integral we express Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial in its series form with the help of Eq. 9 and Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function in terms of Mellin-Barnes type of contour integral by Eq.1 and then interchanging the order of integration and summation, we get:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(20)

Further using the result Eq. 10 the above integral becomes:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(21)

Then interpret with the help of Eq. 1 and 21, we have the required result (Eq. 17) and if we express Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function in series form with the help of Eq. 4 we easily arrive at Eq. 18.

DERIVATION OF THE MAIN INTEGRALS

The result in Eq. 11 is established by taking the partial derivative on both sides of Eq. 18 with respect to a. Equation 12 and 13 are similarly established by taking the partial derivative of Eq. 18 with respect to b and c, respectively. Equation 14 is established by adding Eq. 11, 12 and 13; Eq. 15 is established by subtracting Eq. 11 and 12 from 13; Eq. 16 is established by subtracting Eq. 12 and Eq. 13 from Eq. 11.

SPECIAL CASES

If we put Aj = Bj = 1, Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function reduces to Fox’s H-function (Srivastava et al., 1982), then the Eq. 17 takes the following form:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(22)

Where:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial

If we put Aj = Bj = 1; αj = βj = 1, then the Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function reduces to general type of G-function (Meijer, 1946) which is also believe to be new.

The conditions of convergence of Eq. 22 can be easily obtained from those of Eq. 17.

By applying the our result given in Eq. 17 to the case of Hermite polynomials (Srivastava and Singh, 1983) by setting:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial

in which mi = 2; ni = n1, r = 1; Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial, we have the following interesting results:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(23)

The conditions of convergence of Eq. 23 can be easily obtained from those of Eq. 17.

By applying the our results given in Eq. 17 to the case of Lagurre polynomials (Srivastava and Singh, 1983) by setting Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial in which:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial

We have the following interesting results:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(24)

The conditions of convergence of Eq. 24 can be easily obtained from those of Eq. 17.

If we put n = p, m = 1, q = q+1, b1 = 0, β1 = 1, aj = 1-aj, bj = 1-bj, in Eq. 17 then the Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial-function reduces to generalized wright hypergeometric function (Wright, 1935) i.e.:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial

the Eq. 17 takes the following form:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial
(25)

Where:

Image for - New Finite Integrals Involving Product of -function and Srivastava Polynomial

The conditions of convergence of Eq. 25 can be easily obtained from those of Eq. 17.

ACKNOWLEDGMENTS

The authors express their sincerest thanks to the referees for some valuable suggestions.

REFERENCES

1:  Inayat-Hussain, A.A., 1987. New properties of hypergeometric series derivable from Feynman integrals: I. Transformation and reeducation formulae. J. Phys. A: Math. Gen., 20: 4109-4117.

2:  Inayat-Hussain, A.A., 1987. New properties of hypergeometric series derivable from Feynman integrals II. A generalisation of the H function. J. Phys. A: Math. Gen., 20: 4119-4128.
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3:  Rathie, A.K., 1997. A new generalization of generalized hypergeometric functions. Le Mathematiche, 52: 297-310.
Direct Link  |  

4:  Srivastava, H.M., 1972. A contour integral involving Fox's H-function. Indian J. Math., 14: 1-6.

5:  Srivastava, H.M., K.C. Gupta and S.P. Goyal, 1982. The H-Function of One and Two Variables with Applications. South Asian Publishers, New Dehli, Madras

6:  Srivastava, H.M. and N.P. Singh, 1983. The integration of certain products of the multivariable H-function with a general class of polynomials. Rend. Circ. Mat. Palermo, 2: 157-187.

7:  Srivastava, H.M. and P.W. Karlsson, 1985. Multiple Gaussian Hypergeometric Series. E. Horwood, Chichester, ISBN: 9780470201008, Pages: 425

8:  Gupta, K.C. and R.C. Soni, 2006. On a basic integral formula involving the product of the H-function and Fox H-function. J. Raj. Acad. Phys. Sci., 4: 157-164.

9:  Gupta, K.C., R. Jain and R. Agrawal, 2007. On existence conditions for a generalized Mellin-Barnes type integral. Natl. Acad. Sci. Lett., 30: 169-172.
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10:  Meijer, C.S., 1946. On the G-function. I-VIII. Nederl. Akad. Wetensch. Proc. Ser. A, 49: 227-237.

11:  Agarwal, P. and S. Jain, 2009. On unified finit integrals involving a multivariable polynomial and a generalized Mellin Barnes type of contour integral having general argument. Natl. Acad. Sci. Lett., 32: 281-286.

12:  Agarwal, P., 2011. On multiple integral relations involving generalized mellin-barnes type of contour integral. Tamsui Oxford J. Inform. Math. Sci., 27: 449-462.
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13:  Buschman, R.G. and H.M. Srivastava, 1990. The H function associated with a certain class of Feynman integrals. J. Phys. A: Math. Gen., 23: 4707-4710.
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14:  Saxena, R.K., C. Ram and S.L. Kalla, 2002. Applications of generalized H-function in bivariate distributions. Rew. Acad. Canar. Cienc., 14: 111-120.

15:  Wright, E.M., 1935. The asymptotic expansion of the generalized hypergeometric function. J. London Math. Soc., 10: 286-293.
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