A Study of Unified Finite Integrals Involving Generalized Modified Bessel Function of Third Kind, (λŋµ,ʋ) General Class of Polynomials and the Multivariable Gimel-Function

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A study of unified finite integrals involving generalized modified Bessel function of

third kind

, general class of polynomials and the multivariable Gimel-function

1 Teacher in High School , France E-mail : [email protected]

ABSTRACT

In this paper,we first evaluate a unified and general finite integral whose integrand involves the product of the function , the class of polynomials and the multivariable Gimel function. The arguments of the function occurring in the integrand involve the product of factors of the form . On account of the most general nature og our main findings, a large number of new and known integrals can easily be obtained from it merely by specializing the functions and parameters involved therein. At the end of this paper, we shall give two particular cases.

Keywords: Generalized Bessel function of third kind, multivariable Gimel-function, class of polynomials, H-function.

2010 Mathematics Subject Classification. 33C60, 82C31

1. Introduction

We throughout this paper, let and be set of complex numbers, real numbers and positive integers respectively. Also . We define a generalized transcendental function of several complex variables.

= (1.1)

A Study of Unified Finite Integrals Involving

Generalized Modified Bessel Function of

Third Kind, (

λ

ŋ

_µ,

ʋ

) General Class of Polynomials

and the Multivariable Gimel-Function

F.Y.Ayant

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. . . .

(1.2)

and

(1.3)

The contour is in the - plane and run from to where if is a real number with loop, if necessary to ensure that the poles of

, to

the right of the contour and the poles of lie to the left of the

contour . The condition for absolute convergence of multiple Mellin-Barnes type contour (1.1) can be obtained of the corresponding conditions for multivariable H-function given by as :

where

(1.4)

Following the lines of Braaksma ([3] p. 278), we may establish the the asymptotic expansion in the following convenient form :

,

, where :

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Remark 1.

If and

, then the multivariable Gimel-function reduces in the multivariable Aleph- function defined by Ayant [1].

Remark 2.

If and

, then the multivariable Gimel-function reduces in a multivariable I-function defined by Prathima et al. [7].

Remark 3.

If and

, then the generalized multivariable Gimel-function reduces in multivariable I-function defined by Prasad [6].

Remark 4.

If the three above conditions are satisfied at the same time, then the generalized multivariable Gimel-function reduces in the multivariable H-function defined by Srivastava and Panda [10,11].

For more details about The Gimel function, see Ayant [2].

The generalization of the modified integral Bessel function of the third kind or Macdonald function will be represented by the following form [4,p.152, eq.(1.2)]

(1.5)

Srivastava [8] introduced the general class of polynomials :

(1.6)

Where is an arbtrary positive integer and the coefficient are arbitrary constants, real or complex.

By suitably specialized the coefficient the polynomials can be reduced to the classical orthogonal polynomials such as Jacobi, Hermite, Legendre and Laguerre polynomials etc, see Srivastava and Singh [12].

2. Main integral

In your investigation, we shall use the following notations.

(2.1)

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(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

We have the follwing resulting

Theorem

(2.10)

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, is given in (1.13).

Proof

First we express the modified integral Bessel function of the third kind in terms of the H-function of one variable [4, p. 155, eq. (2.6)] and the geberalized polynomial with the help of (1.5). Now, we express the multivariable Gimel-function and H-function of one variable in term of their respectives Mellin-Barnes type integrals contouf. Then we change the order of the series and integrals contour with the x-integral (which is permissible under the conditions stated above ). The left-hand side of (2.10) writes (say I)

(2.11)

Finally, on evaluating the x-integral obtained above with thr help of a special case of [9, p. 61, eq. (5.2.1)] and expressing the H-function thus obtained in terms of its Mellin-Barnes integral contour. Now interpreting the result in the result in Mellin-Barnes integrals contour in multivariable Gimel-function of (r+2)-variables, we obtain the equation (2.10)

Mellin-Barnes integrals contour in multivariable Gimel-function of (r+2)-variables, we obtain the equation (2.10)

3. Particular cases.

In this section, the multivariable Gimel-function reduce in multivariable H-function.

If we reduce

If we reduce into into [3, p.152, eq(1.3)], [3, p.152, eq(1.3)], into Laguerre polynomials and the multivariable H-function into into Laguerre polynomials and the multivariable H-function into generalized hypergeometric function by taking

generalized hypergeometric function by taking [9, p.18, eq.(2.6.3)] in the integral (2.10), we arrive at the [9, p.18, eq.(2.6.3)] in the integral (2.10), we arrive at the following relation, see Harjule and Jain [5].

following relation, see Harjule and Jain [5].

Corollary 1.

(3.1)(3.1)

where

(3.2)

(3.3)

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I we take

I we take in the main integral and further reduce in the main integral and further reduce to to

Meijer’s function

Meijer’s function by taking by taking , , into Jacobi polynomial into Jacobi polynomial see Srivastava and Singh [12] and the see Srivastava and Singh [12] and the

multivariable Gimel -function into Appell’s function F_3 , we obtain the following integral after algebraic

manipulations : manipulations : Corollary 2. Corollary 2.

(3.4)(3.4) where where (3.5) (3.5)

the existence conditions of (2.10) are satisfied.

4. Conclusion

The integral formulae involving in this paper are double fold generality in term of variables and parameters. By specializing the various parameters and variables involved, these formulae can suitably be applied to derive the corresponding results involving wide variety of useful functions (or product of several such functions) which can be expressed in terms of E, F, G, H, I, Aleph-function of one and several variables and simpler special functions of one and several variables. Hence the formulae derived in this paper are most general in character and may prove to be useful in several interesting cases appearing in literature of Pure and Applied Mathematics and Mathematical Physics.

REFERENCES

[1] F.Y. Ayant, A Fractional Integration of the Product of Two Multivariable Aleph-Functions and a General Class of Polynomials I, Int. Jr. of Mathematical Sciences & Applications, 7(2) (2017), 181-198.

[2] F.Y.Ayant, Some transformations and idendities form multivariable Gimel-function, International Journal of

Matematics and Technology (IJMTT), 59(4) (2018), 248-255.

[3] B.L.J. Braaksma, Asymptotics expansions and analytic continuations for a class of Barnes-integrals, Compositio Math. 15 (1962-1964), 239-341

[4] H.J.Glaeske,A.A. Kilbas[5] and M. Saigo, A modified Bessel type integral transform and its compositions wth fractional calculus operators on spaces and , Journal of comp.and Appl.Math.188(2000), 151-168.

[5] P Harjule and R. Jain, A study of unified finite integrals involving generalized modified Bessel function of third kind , general class of polynomials and the multivariable H-function, Journal of Indian Acad. Math. 38(2) (2016), 225-231

[6]Y.N. Prasad, Multivariable I-function , Vijnana Parisha Anusandhan Patrika 29 (1986), 231-237.

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[8] H.M. Srivastava, A contour integral involving Fox’s H-function, Indian. J. Math. 14(1972), 1-6.

[9] H.M. Srivastava, K.C. Gupta and S.P. Goyal, The H-function of one and two variables with application, South Asian publishers , New Delhi, (1982)

[10] H.M. Srivastava and R. Panda, Some expansion theorems and generating relations for the H-function of several complex variables, Comment. Math. Univ. St. Paul. 24(1975), page.119-137.

[11] H.M. Srivastava and R. Panda, Some expansion theorems and generating relations for the H-function of several complex variables II. Comment. Math. Univ. St. Paul. 25(1976), page.167-197

[12] H.M. Srivastava and N.P. Singh, The integration of certains products of the multivariable H-function with a general

class of polynomials, Rend. Circ. Mat. Palermo. 32(2)(1983), 157-187.