A finite double integral involving a general multivariable
polynomial and multivariable Gimel-function
1 Teacher in High School , France E-mail : [email protected]
Abstract
In this paper we evaluate a general finite double integral involving the product of algebric and exponenetial functions, a general multivariable polynomial and the multivariable Gimel-function. Some new and interesting special cases og our main integral formula have been considered briefly.
Keywords :Multivariable Gimel-function, multiple integrals contour, double integrals, multivariable polynomial.
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 . We define a generalized transcendental function of several complex variables.
= (1.1)
with
A Finite Double Integral Involving a General
Multivariable Polynomial and Multivariable
Gimel-Function
F.Y.Ayant
. . . .
. . . .
. . . .
(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 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. [5].
Remark 3.
If and
, then the generalized multivariable Gimel-function reduces in multivariable I-function defined by Prasad [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 [9,10].
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)
The generalized polynomials defined by Srivastava ([8],p. 251, Eq. (C.1)), is given in the following manner :
(1.13)
Where are arbitrary positive integers and the coefficients are arbitrary constants, real or complex. I we take in the (1.13) and denote thus obtained by , we arrive at general class of polynomials study by Srivastava ([7],p. 1, Eq. 1).
We shall note
(1.14)
2. Required results.
Lemma 1. ([3], MacRobert p. 450)
(2.1)
provided
Lemma 2. ( Soni and Rathie, [6], p. 255 ).
(2.1)
provided are different at zero, .
3. Main integral.
Theorem.
(3.1)
provided
,
where is defined by (1.4).
The series on the right-hand side of (3.1) is absolutely convergent.
Proof
We first express the general class of mutivariable polynomials occurring on the left-hand side of (3.1) in the series forms with the help of the equation (1.13) and the multivariable Gimel-function by this multiple integrals contour with the help of (1.1), then interchanging the order of integrations and summations (which is justified due to absolute convergence of the integrals involved in the process under the conditions stated with result and evaluating the x and y-integrals with the help of lemmae 1 and 2, respectively, we find that (say) I
(2.2)
On expressing as a hypergeometric serie and changing the order of integrations and summation, interpreting the resulting expression in multiple integrals contour with the help of (1.1), we obtain the desired theorem 1.
3. Special cases.
In this sestion, we shall give several particular cases.
Taking in (2.1) and evaluating the y-integral, we obtain the following integral.
(3.1)
provided
are no zero integers, , ,
where is defined by (1.4).
The series on the right-hand side of (3.1) is absolutely converge.
Similarly, if we take and in (2.1) , we get the following resulting
Corollary 2.
(3.2)
provided
,
where is defined by (1.4).
The series on the right-hand side of (3.2) is absolutely converge.
We consider a class of polynomial of one variable.
Srivastava ([7],p. 1, Eq. 1). have introduced the general class of polynomials :
(3.3)
where is an arbitrary positive integer and the coeficients are arbitrary constants real or complex. On specializing these coefficients , yields a number of known polynomials as special cases. These include, among others, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, bessel polynomials and several others, see ([11], p. 158-161).
Taking by
We shall note
(3.4)
and we obtain
Corollary 3.
provided
are no zero integers , ,
,
where is defined by (1.4).
The series on the right-hand side of (3.5) is absolutely converge.
Remark :
We obtain the same double finite integrals with the functions defined in section I.
4. Conclusion.
The importance of our all the results lies in their manifold generality. Firstly, in view of general arguments utilized in these double integrals, we can obtain a large simpler double or single finite integrals. Secondly by specialising the various parameters as well as variables in the generalized multivariable polynomials, we obtain a large number of formulae involving simpler special functions ( ultraspherical -Gegenbauer, Legendre, Tchebyshev, Bateman’s, Hermite, Laguerre polynomials and others). Thirdy by specialising the various parameters as well as variables in the multivariable Gimel-function, we get a several formulae involving remarkably wide variety of useful functions ( or product of such functions) which are expressible 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 intersting cases appearing in literature of Pure and Applied Mathematics and Mathematical Physics.
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] T.M. MacRobert, Beta function formulae and integrals involving E-function, Math. Ann., 142(5) (1961), 450-452.
[4] Y.N. Prasad, Multivariable I-function , Vijnana Parishad Anusandhan Patrika 29 (1986) , 231-237.
[5] 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.
[6] S.S. Soni and A.K. Rathie, Integrals involving H-function, Vijnana Parishad Anusandhan Patrika, 26(1983); 293-296.
[7] H.M. Srivastava, A contour integral involving Fox’s H-function, Indian. J. Math. 14(1972), 1-6.
[8] H.M. Srivastava, A multilinear generating function for the Konhauser set of biorthogonal polynomials suggested by Laguerre polynomial, Pacific. J. Math. 177(1985), page 183-191.
[9] 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.
[11] 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.
