Certain finite finite integral involving sequence of functions, a general class of
polynomials and multivariable Aleph-functions IV
1 Teacher in High School , France E-mail : [email protected]
ABSTRACT
In the present paper we evaluate a finite integral with involving the product of sequence of functions, a logarithm function of general argument, elliptic integral of first species, product of two multivariable Aleph-functions and general class of polynomials of several variables. The importance of the result established in this paper lies in the fact they involve the Aleph-function of several variables which is sufficiently general in nature and capable to yielding a large of results merely by specializating the parameters their in.
Keywords:Multivariable Aleph-function, general class of polynomials, Aleph-function of two variables, I-function of several variables, sequence of functions, elliptic integral of first species.
2010 Mathematics Subject Classification. 33C99, 33C60, 44A20
1.Introduction and preliminaries.
The function Aleph of several variables generalize the multivariable I-function recently study by C.K. Sharma and Ahmad [6] , itself is an a generalisation of G and H-functions of multiple variables. The multiple Mellin-Barnes integral occuring in this paper will be referred to as the multivariables Aleph-function throughout our present study and will be defined and represented as follows.
We define :
= (1.1)
with
(1.2)
and (1.3)
Suppose , as usual , that the parameters
;
;
with , ,
are complex numbers , and the and are assumed to be positive real numbers for standardization purpose such that
(1.4)
The reals numbers are positives for to , are positives for to
The contour is in the -p lane and run from to where is a real number with loop , if necessary ,ensure that the poles of with to are separated from those of
with to and with to to the left of the
contour . The condition for absolute convergence of multiple Mellin-Barnes type contour (1.9) can be obtained by extension of the corresponding conditions for multivariable H-function given by as :
, where
, with , , (1.5)
The complex numbers are not zero.Throughout this document , we assume the existence and absolute convergence conditions of the multivariable Aleph-function.
We may establish the the asymptotic expansion in the following convenient form :
,
,
Serie representation of Aleph-function of several variables is given by
(1.6)
Where , are given respectively in (1.2), (1.3) and
which is valid under the conditions (1.7)
for (1.8)
Consider the Aleph-function of s variables
= (1.9)
with
(1.10)
Suppose , as usual , that the parameters
;
;
with , ,
are complex numbers , and the and are assumed to be positive real numbers for standardization purpose such that
(1.12)
The reals numbers are positives for , are positives for
The contour is in the -p lane and run from to where is a real number with loop , if necessary ,ensure that the poles of with to are separated from those of
with to and with to to the left of the
contour . The condition for absolute convergence of multiple Mellin-Barnes type contour (1.9) can be obtained by extension of the corresponding conditions for multivariable H-function given by as :
, where
, with , , (1.13)
The complex numbers are not zero.Throughout this document , we assume the existence and absolute convergence conditions of the multivariable Aleph-function.
We may establish the the asymptotic expansion in the following convenient form :
,
,
where, with : and
We will use these following notations in this paper
; (1.15)
W (1.16)
(1.17)
(1.18)
(1.19)
(1.20)
The multivariable Aleph-function write :
(1.21)
The generalized polynomials defined by Srivastava [10], is given in the following manner :
(1.22)
Where are arbitrary positive integers and the coefficients are arbitrary
constants, real or complex. In the present paper, we use the following notation
(1.23)
In the document , we note :
(1.24)
where , are given respectively in (1.2) and (1.3)
2. Sequence of function and elliptic integral of first species
(2.1)
where (2.2)
and the infinite series on the right side (2.1) is absolutely convergent,
and
(2.3)
where is a sequence of constants.
By suitably specializing the parameters involving in (2.1), a general sequence of function reduced to generalized polynomials set studied by Raizada [4], a class of polynomials introduced by Fujiwara [3] and several others authors.
The elliptic integrals of first species are defined by :
(2.4)
For more details, see Whittaker and Watson ([11], page 515).
3. Required integral
We have the following integral, see Brychkov ([2], 4.21.3 Eq.8 page 269)
(3.1)
where and . the function is the generalized hypergeometric function, see Slater
([9], page 40, Eq2.1). isthe the elliptic integrals of first species.
4. Main integral
(4.1)
where
Provided that
a) ,
b)
c) , where is defined by (1.5) ;
d) , where is defined by (1.13) ;
e) The series on the right-hand side of (3.1) is assumed to be absolutely convergent
Proof
Expressing the general sequence of functions in serie with the
help of equation (2.1), the Aleph-function of r variables in series with the help of equation (1.6), the general class of polynomial of several variables in serie with the help of equation (1.22) and the Aleph-function of s variables in Mellin-Barnes contour integral with the help of equation (1.9), changing the order of integration ans summation (which is easily seen to be justified due to the absolute convergence of the integral and the summations involved in the process) and then evaluating the resulting integral with the help of equation (3.1). Finally expressing the
hypergeometric function in serie, use the relation with and interpreting the result
simple integral have been derived in this section for multivariable I-functions defined by Sharma et al [6].
Corollary 1
(5.1)
under the same notationa and conditions that (4.1) with
6. Aleph-function of two variables
If , we obtain the Aleph-function of two variables defined by K.Sharma [8], and we have the following simple integral.
(6.1)
under the same notationa and conditions that (4.1) with
7. I-function of two variables
If , then the Aleph-function of two variables degenere in the I-function of two variables defined by sharma et al [7] and we obtain the same formula with the I-function of two variables
Corollary 3
(7.1)
under the same notationa and conditions that (4.1) with and
8. Conclusion
In this paper we have evaluated a finite integral involving the multivariable Aleph-functions, a class of polynomials of several variables and the general of sequence of functions.The integral established in this paper is of very general nature as it contains Multivariable Aleph-function, which is a general function of several variables studied so far. Thus, the integral established in this research work would serve as a key formula from which, upon specializing the parameters, as many as desired results involving the special functions of one and several variables can be obtained.
REFERENCES
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