On a general class of multiple Eulerian integrals II
Frederic Ayant E-mail : [email protected]
Vinod Gill
Department of Mathematics Amity University, Rajasthan, Jaipur-303002, India E-mail : [email protected]
ABSTRACT
Recently, Raina and Srivastava [5] and Srivastava and Hussain [11] have provided closed-form expressions for a number of a general Eulerian integrals involving multivariable H-functions. Motivated by these recent works, we aim at evaluating a general class of multiple eulerian integrals involving a multivariable I-function defined by Prathima et al [4] with general arguments. These integrals will serve as a key formula from which one can deduce numerous useful integrals.
Keywords :Multivariable I-function, multiple Eulerian integral ,class of polynomials, sequence of polynomials.
2010 Mathematics Subject Classification. 33C45, 33C60, 26D20
1. Introduction and preliminaries.
The well-known Eulerian Beta integral
(1.1)
is a basic result for evaluation of numerous other potentially useful integrals involving various special functions and polynomials. Raina and Srivastava [5], Saigo and Saxena [8], Srivastava and Hussain [11], Srivastava and Garg [10] etc have established a number of Eulerian integrals involving various general class of polynomials, Meijer's G-function and Fox's H-function of one and more variables with general arguments.
The explicit form og the generalized polynomial set [7, p.71, (2.3.4)] is
(1.2)
where (1.3)
where and
We recall here the following definition of the general class of polynomials introduced and studied by Srivastava [9]
(1.4)
where and is an arbitrary positive integer. The coefficients are arbitrary constants, real or complex.
=
(1.5)
(1.6)
where , , are given by :
(1.7)
(1.8)
For more details, see Prathima et al [4].
Following the result of Braaksma [1] the I-function of r variables is analytic if :
(1.9)
The integral (2.1) converges absolutely if
where
(1.10)
The complex numbers are not zero.Throughout this document , we assume the existence and absolute convergence conditions of the multivariable I-function.
We will note :
(1.11)
(1.12)
2. Main integral
the following notations (2.1) and (2.2).
(2.1)
(2.2)
for
Lemma ( [2] p.287)
(2.3)
with ,
Theorem
We have the following result
(2.4)
We obtain a I-function of variables
(2.5)
Provided that
(i)
for
(ii) ; for
(iii) When min
(a)
(b)
When max
(c)
(d)
When inequalities (a) and (d) are satisfied.
When inequalities (b) and (c) are satisfied.
Where is defined by (1.10)
The multiple series of R.H.S. of (2.4) converges absolutely.
Proof
To establish the multiple integral formula (2.4), we first use the series representations for the polynomials sets and respectively in its left hand side. Further, using contour integral representation for the multivariable I-function defined by Prathima et al [4] and then interchanging the order of integration and summation suitably, which is permissible under the conditions stated above, we find that
(2.6)
Now by writing in terms of contour integral and changing the order of integration therein, we
obtain
L.H.S =
(2.7)
Substituting the value of from (2.2) and after simplifications, we get
L.H.S =
(2.8)
If for
L.H.S =
(2.9)
Now using (2.1) and then evaluating the inner-most integral by using the lemma (2.3), we get
L.H.S
(2.10)
3. Particular cases
The multivariable I-function occurring in the main integral can be suitably specialized to a remarkably wide variety of special functions which are expressible in terms of E, G, H and I-function of one and several variables. Again by suitably specializing various parameters and coefficients, the general class of polynomials and the general sequence of functions can be reduced to a large number of orthogonal polynomials and hypergeometric polynomials. Thus using various special cases of these special functions, we can obtain a large number of others integrals involving simpler special functions and polynomials of one and several variables.
On taking and in (2.4), the general class of polynomials reduces to unity an we get
Corollary 1
(3.1)
where
Putting in (2.4), we arrive at the following integral form
Corollary 2
(3.3)
where
(3.4)
with the same notations and corresponding validity conditions that (2.4).
Putting in (2.4), we obtain the following result.
(3.5)
where
(3.6)
and
(3.7)
for
with the same notations and corresponding validity conditions that (2.4).
If , the multivariable I-function reduces to I-function of two variables defined by Kumari et al [3]. We obtain
Corollary 4
(3.8)
where are equal to respectively for and we have the same conditions that (2.4) with .
Corolary 5
If , the multivariable I-function reduces to I-function of one variable defined by Rathie [6]. We obtain
(3.9)
where
(3.10)
we have the same conditions that (2.4) with .
5. Conclusion
Our main integral formula is unified in nature and possesses manifold generality. It acts a key formula and using various special cases of the multivariable I-function, general class of polynomials and a general sequence of functions, one can obtain a large number of other integrals involving simpler special functions and polynomials of one and several variables.
References
[1] B. L. J. Braaksma, “Asymptotic expansions and analytic continuations for a class of Barnes integrals,”Compositio Mathematical, vol. 15 (1964), pp. 239–341.
[2] I.S. Gradsteyn and I.M. Ryxhik, Table of integrals, series and products: Academic press, New York , (1980). [2] I.S. Gradsteyn and I.M. Ryxhik, Table of integrals, series and products: Academic press, New York , (1980).
[3] K. Shantha Kumari, T. M. Vasudevan Nambisan and A. K. Rathie,A study of I-function of two variables, Le Matematiche, 69(1) (2014), 285-305.
[4] J. Prathima, V. Nambisan and S.K. Kurumujji, A Study of I-function of Several Complex Variables, International Journalof Engineering Mathematics Vol(2014) , 2014, 1-12.
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[6] A.K. Rathie, A new generalization of generalized hypergeometric function, Le Matematiche, 52(2) (1997), 297-310.
[7] S.K. Raizada, A study of unified representation of special functions of mathematical physics and their use in statistical and boundary value problems, Ph.D. Thesis, Bundelkhand University, Jhansi, India, 1991.
[8] M. Saigo, and R.K. Saxena, Unified fractional integral formulas forthe multivariable H-function. J.Fractional Calculus 15 (l999), 91-107.
[9] H.M. Srivastava, A contour integral involving Fox’s H-function, Indian J. Math, 14 (1972), 1-6.
[10] H.M. Srivastava and M. Garg, Some integrals involving general class of polynomials and the multivariable H-function. Rev. Roumaine. Phys. 32 (1987) 685-692.
[11] H.M. Srivastava and M.A. Hussain, Fractional integration of the H-function of several variables. Comput. Math. Appl. 30 (9) (1995),73-85.
[12] H.M.Srivastava, H.M.and R. Panda, Some bilateral generating functions for a class of generalized hypergeometric polynomials, J. Reine Angew. Math. (1976), 265-274.
