On a new class of summation formulas involving the generalized hypergeometric F 2
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EARCH ARTICLE
Open Access
On a new class of summation formulas involving the generalized hypergeometric F polynomial Arjun K Rathie1 and Adem Kılıçman2* *
Correspondence: [email protected] 2 Department of Mathematics, Institute of Mathematical Research, University Putra Malaysia (UPM), Serdang, Selangor 43400, Malaysia Full list of author information is available at the end of the article
Abstract The aim of this research paper is to establish a quite general transformation involving the generalized hypergeometric function. Extensions of Kummer’s first transformation, Gauss and Kummer summation, and its contiguous results are then applied to obtain a new class of summation formulas involving the generalized hypergeometric 2 F2 polynomial, which have not previously appeared in the literature. A few well-known results obtained earlier by Kim et al. (Int. J. Math. Math. Sci. 2010:309503, 2010; Integral Transforms Spec. Funct. 23(6):435-444, 2012) and Exton have been obtained as special cases of our main findings. MSC: 33C15; 33C20 Keywords: generalized hypergeometric series; extension of Gauss and Kummer summation theorems; polynomial
1 Introduction and results required The generalized hypergeometric function with p numeratorial and q denominatorial parameters is defined by [–] p Fq
α , . . . , αp ; z = p F q α , . . . , α p ; β , . . . , βq ; z β , . . . , βq ; =
∞ (α )n · · · (αp )n zn n=
(β )n · · · (βq )n n!
,
(.)
where (α)n denotes the well-known Pochhammer symbol defined for any α ∈ C by ⎧ ⎨α(α + )(α + ) · · · (α + n – ) (n ∈ N = {, , , . . .}), (α)n = ⎩ (n = ),
(.)
and is the well-known Gamma function defined by
∞
(s) =
e–x xs– dx
(s) > .
(.)
For details as regards convergence etc. of p Fq , we refer to []. © 2014 Rathie and Kılıçman; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Rathie and Kılıçman Advances in Difference Equations 2014, 2014:43 http://www.advancesindifferenceequations.com/content/2014/1/43
Page 2 of 10
It is interesting to mention here that whenever a generalized hypergeometric function reduces to gamma functions, the results are very important from the application point of view. Thus well-known classical summation theorems such as those of Gauss, Gauss second, Kummer, and Bailey for the series F ; Watson, Dixon, Whipple and Saalschütz for the series F and others play an important role in the theory of hypergeometric, generalized hypergeometric series and other branches of applied mathematics. Recently a good deal of progress has been made in the direction of generalizations and extensions of the above mentioned classical summation theorems. For this, we refer to the research papers [, ] and the references therein. This function has been extensively studied in great de
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