An expansion formula for the multivariable Aleph-function F.Y. 1 Teacher in High School, France E-mail : fredericayant@gmail.com ABSTRACT The aim of the paper is to establish a general expansion formula for the Aleph-function of several variables. It is significant to observe that a large number of finite infinite series for this function can be easily summed up by using the summation theorem for ordinary hypergeometric series in the main result. Also, by appropriately specializing the parameters of the multivariable Aleph-function, one can obtain expansion formulas for simpler special functions of one several variables. Keywords:Multivariable Aleph-function,Mellin-Barnes contour,generalized hypergeometric function,expansion formula, I-function of several variables, Aleph-function of two variables. 2010 Mathematics Subject Classification. 33C99, 33C60, 44A20 1. Introduction preliminaries. We begin by recalling the definition of the Aleph-function of several variables. he Aleph-function of several variables generalize the multivariable I-function defined by Sharma Ahmad [3], itself is an a generalisation of G 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 will be defined represented as follows. We have : = (1.1) with For more details, see Ayant [2]. The reals numbers are positives for, are positives for The condition for absolute convergence of multiple Mellin-Barnes type contour (1.1) can be obtained by extension of the corresponding conditions for multivariable H-function given by as :,, with,, (1.2)
The complex numbers are not zero.throughout this document, we assume the existence absolute convergence conditions of the multivariable Aleph-function. We may establish the the asymptotic expansion in the following convenient form :...,......,..., with : We will use these following notations in this paper ; (1.3) (1.4) (1.5) (1.6), (1.7) The multivariable Aleph-function write : (1.8) (1.9) Let. We note for exempt: 2. General expansion formula We have the following general formula
= ; (2.1) are defined by (1.5), (1.6), (1.7) (1.8) respectively. sts for (2.2) sts for (2.3) The formula (2.1) holds if any one of the following sets of conditions is satisfied. a) b) c) d) (2.4) (2.5) Proof of (2.1) To evaluate (2.1), we first replace the multivariable Aleph-function occurring on the left-h side of (2.1) by the Mellin-Barnes contour integral (1.1), change the order of summation integration ( (which is permissible under the conditions stated), then we get the desired result (2.1) on using the definition of the generalized hypergeometric function ( see [2],page 40]. 3. Particular cases
a) If in (2.1), using the Gauss theorem ( [6], page 243 (III.3)), we get the following summation formula : (3.1) b) If in (2.1), use the Kummer's theorem ( [6], page 243 (III.5)), we have the following result : (3.2) 4. Aleph-function of two variables If, we obtain the Aleph-function of two variables defined by K.Sharma [5], we have the following general expansion formula. =
; (4.1) with the same notations validity conditions. We have the two particular cases a) (4.2) b) (4.3) Remark : If, the Aleph-function of two variables degenere to the I-function of two variables defined by Sharma et al [4]. For more details, see the paper of Agrawal et al [1]. 6. Conclusion In this paper, we have established a general expansion formula involving the multivariable Aleph-function by using the generalized hypergeometric function. Due to general nature of the multivariable aleph-function expansion formula involving here, our formula is capable to be reduced into many known news expansions involving the special functions of one several variables. REFERENCES [1] Agrawal M.K Jain S.S.L. An expansion formula for the I-function of two variables. Vijnana Parishad Anushan Patrika 1997, vol 40 (2), page 121-129. [2] Ayant F.Y. An integral associated with the Aleph-functions of several variables. International Journal of Mathematics Trends Technology (IJMTT).. 2016 Vol 31 (3), page 142-154. [3] Sharma C.K. Ahmad S.S.: On the multivariable I-function. Acta ciencia Indica Math, 1994 vol 20,no2, p 113-
116. [4] C.K. Sharma P.L. mishra : On the I-function of two variables its properties. Acta Ciencia Indica Math, 1991 Vol 17 page 667-672. [5] Sharma K. On the integral representation applications of the generalized function of two variables, International Journal of Mathematical Engineering Sciences, Vol 3, issue1 ( 2014 ), page1-13. [6] Slater L.J. Generalized hypergeometric function. Cambridge University Press, Cambridge London (1966). Personal adress : 411 Avenue Joseph Raynaud Le parc Fleuri, Bat B 83140, Six-Fours les plages Tel : 06-83-12-49-68 Department : VAR Country : FRANCE