A double expansion formula for generalized multivariable Gimel-function
involving Jacobi polynomials and Bessel functions
Frédéric Ayant
Teacher in High School , France E-mail :[email protected]
ABSTRACT
In this paper, we present a double expansion formula for the generalized multivariable Gimel-function involving Jacobi polynomials and Bessel functions.
KEYWORDS : Multivariable Gimel-function, multiple integral contours, Jacobi polynomial, Bessel function, Double expansion serie.
2010 Mathematics Subject Classification. 33C99, 33C60, 44A20
1. Introduction and preliminaries.
Throughout this paper, let and be set of complex numbers, real numbers and positive integers respectively.
Also .
The subject of expansion formulae and Fourier series of special functions occupies a large place in the literature of special functions. Certain double expansion formulae and double Fourier series of generalized hypergeometric functions play an important rôle in the development of the theories of special functions and two- dimensional boundary value problems. In this paper, we establish a double expansion formula for generalized multivariable Gimel-function.
We define a generalized transcendental function of several complex variables.
= (1.1)
with
A Double Expansion Formula for Generalized
Multivariable Gimel-Function Involving Jacobi
Polynomials and Bessel Functions
. . . .
. . . .
. . . .
(1.2)
and
(1.3)
1) stands for .
2)
and verify :
.
3) .
5)
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 ([2] p. 278), we may establish the the asymptotic expansion in the following convenient form :
,
, where :
and
Remark 1.
If and
, then the generalized multivariable Gimel-function reduces in the generalized multivariable Aleph- function ( extension of multivariable Aleph-function defined by Ayant [1]).
If and
, then the generalized multivariable Gimel-function reduces in a generalized multivariable I-function (extension of multivariable I-function defined by Prathima et al. [7]).
Remark 3.
If and
, then the generalized multivariable Gimel-function reduces in generalized of multivariable I-function (extension of 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 generalized multivariable H-function (extension of multivariable H-function defined by Srivastava and panda [8,9]).
In your investigation, we shall use the following notations.
(1.5)
(1.6)
(1.7)
(1.8)
(1.9)
(1.10)
(1.11)
(1.12)
2. Required results.
Lemma 1. ([4], p. 1240, Eq. 4)
(2.1)
provided
Lemma 2. (see Luke,[5])
(2.2)
provided .
The orthogonality property of the Jacobi polynomials [3].
Lemma 3.
(2.3)
provided
Orthogonality property for Bessel’s function [5]
Lemma 3.
(2.4)
provided
3. Main integrals.
In this section, we establish two integrals.
Theorem 1.
(3.1)
provided
,
where is defined by (1.4).
To prove the theorem 1, we replace the generalized multivariable Gimel-function by this multiple integrals contour with the help of (1.1), change the order of integrations which is justified under the conditions mentioned above. We get
(3.2)
Evaluate the inner integral with the help of lemma 1 and interpreting the Mellin-Barnes multiple integrals contour in terms of the multivariable Gimel-function, we get the desired result (3.1).
Theorem 2.
(3.3)
provided
,
where is defined by (1.4).
Proof
To prove the theorem 2, we replace the generalized multivariable Gimel-function by this multiple integrals contour with the help of (1.1), change the order of integrations which is justified under the conditions mentioned above. We get
(3.4)
Evaluate the inner integral with the help of lemma 2 and interpreting the Mellin-Barnes multiple integrals contour in terms of the multivariable Gimel-function, we get the desired result (3.3).
4. Double expansion formula.
The double expansion formula to be establish is
Theorem 3.
where
(4.2)
(4.3)
(4.4)
valid under the existence conditions mentioned in (3.1) and (3.3).
Proof
To establish (4.1), let
(4.5)
The above equation is valid, since is continuous and bounded variation in the region .
Multiplying both sides of (4.5) by , integrating with respect to from 1 to and using the (2.4) and (3.3), we obtain
(4.6)
where
(4.7)
(4.8)
Multiplying both sides of (4.6) by , integrating with respect to from -1 to 1 and using
(2.3) and (3.1), we obtain
(4.9)
where
(4.10)
(4.12)
Finally substituting the value of in (4.5), we obtain the desired result (4.1).
5. Conclusion.
Since on specializing the parameters of generalized Gimel-function of several variables yield almost all special functions appearing in Applied Mathematics and Physical Sciences. Therefore the result presented in this study is of a general character and hence may encompass several cases of interest.
REFERENCES.
[1] F. Ayant, An integral associated with the Aleph-functions of several variables. International Journal of Mathematics Trends and Technology (IJMTT), 31(3) (2016), 142-154.
[2] B.L.J. Braaksma, Asymptotics expansions and analytic continuations for a class of Barnes-integrals, Compositio Math. 15 (1962-1964), 239-341.
[3] A. Erdelyi, W. Magnus, F. Oberhettinger transforms and F.G. Tricomi, Tables of integrals, Vol II, McGraw-Hill, New York (1954).
[4] I.S. Gradsteyn and I.M. Ryzhik, Tables of integrals, Series and Product, Academic press, New York (1980).
[5] Y.L. Luke, Integrals of Bessel functions, McGraw Hill, New York (1962).
[6] Y.N. Prasad, Multivariable I-function , Vijnana Parishad Anusandhan Patrika 29 (1986) , 231-237.
[7] J. Prathima, V. Nambisan and S.K. Kurumujji, A Study of I-function of Several Complex Variables, International Journal of Engineering Mathematics, Vol (2014), 1-12.
[8] 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),119-137.
[9] 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), 167-197.
