On a fractional differential operator involving multivariable Gimel-function
and product of general class of polynomials
Frédéric Ayant
Teacher in High School , France E-mail :[email protected]
ABSTRACT
In the present paper we use a fractional differential operator and product of general class of multivariable polynomials, Riemann Zeta function and multivariable function. On account of the kernel, due to the general class of multivariable polynomials and multivariable Gimel-function our findings provide interesting unification and extension of a number of results. Some new special cases of our main result are mentioned briefly.
KEYWORDS : Multivariable Gimel-function, multiple integral contours, fractional differential operator, general class of polynomials, Riemann Zeta function.
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
On a Fractional Differential Operator Involving
Multivariable Gimel-Function
and Product of General Class of Polynomials
Frédéric Ayant Teacher in High School , France
. . . .
. . . .
. . . .
(1.2)
and
(1.3)
1) stands for .
2) and verify :
.
3) .
4)
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. [7].
Remark 3.
If and
, then the generalized multivariable Gimel-function reduces in multivariable I-function defined by Prasad [6].
Remark 4.
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 ([9],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 ([8],p. 1, Eq. 1).
We shall note
(1.14)
Goyal and Laddha ([4], p. 99-108, Vol 11(2)) introduced an extension to the generalized Riemann Zeta function defined in the following slightly modified form
(1.15)
(1.16)
2. Required results.
We use the fractional derivative operator defined in the following manner :
Lemma 1. ([5], Mishra).
(2.1)
where and and are not necessarily integers.
We use the binomial expansion in the following manner :
Lemma 2.
(2.2)
where and
3. Main result.
Theorem.
(3.1)
provided that
where is defined by (1.4).
Proof
We first express the multivariable gimel-function by this multiple integrals contour with the help of (1.1), 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 extension to the generalized Riemann Zeta function by (1.15). Interchange the order of summations and integrations, which is justified under the conditions mentioned above, we have (say I).
(3.2)
Now we use the lemmae 1 and 2 and we interpret the resulting multiple integrals contour with the help of (1.1) about the gimel-function of r-variables, we obtain the desired theorem 1.
4. Special cases.
Our main results provides unifications and extensions of various results on fractional differential operators. For the sake of illustration, we mention the following special cases.
Taking and and in and , ,
and , then we obtain the following result
Corollary 1.
(4.1)
provided that
, , and
The generalized polynomials reduces to general class of polynomials study by Srivastava ([7],p. 1, Eq. 1) and taking , we obtain
(4.2)
provided that
, , and
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 ([12], p. 158-161).
Take , the class of polynomials defined above reduces in Jacobi polynomials
([12], p. 158-161).
Take , the class of polynomials defined above reduces in Konhauser polynomials
([12], p. 158-161).
We get interesting results involving Jacobi polynomials and Konhauser biorthogonal polynomials form our main results.
Remarks :
We obtain easily the same relations with the functions defined in the section 1. O.P. Garg et al. [3] have obtained the same relation with the multivariable H-function.
5. Conclusion.
The importance of our all the results lies in their manifold generality. By specialising the various parameters as well as variables in the multivariable Gimel-function, we get several fractional differential operators 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 Aleph-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. Again the class of multivariable polynomials involved in these integrals and multiplication formulae reduces to a large number of polynomials listed by Srivastava and Singh ([11], p.158-161), therefore, from the integrals we can further obtain various integrals involving a number of simpler polynomials.
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.
[3] O.P. Garg and V. Kumar and V. Sharma, On a fractional differential operator involving multivariable H-function Riemann Zeta function and product of general polynomials, Acta Ciencia Indica. 33(4) (2007), 1323-1326.
[4] S.P. Goyal and R.K. Laddha, On the generalized Riemann Zeta function and the generalized Lambert transform, Ganita Sandesh, 11(2) (1997), 99-108.
[5] A.P. Misra, On a fractional differential operator, Ganita, 26(1975), 1-18.
[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, A contour integral involving Fox’s H-function, Indian. J. Math. 14(1972), 1-6.
[9] 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.
[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),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), 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.
