- Research Article
- Open Access
On a new class of summation formulas involving the generalized hypergeometric polynomial
© Rathie and Kılıçman; licensee Springer. 2014
- Received: 24 September 2013
- Accepted: 9 January 2014
- Published: 28 January 2014
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 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.
- generalized hypergeometric series
- extension of Gauss and Kummer summation theorems
For details as regards convergence etc. of , we refer to .
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 ; Watson, Dixon, Whipple and Saalschütz for the series 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 [4, 5] and the references therein.
provided that .
provided and .
We remark in passing that the result (1.10) is a presumably new result.
Now we observe that the first on the right-hand side of (1.11) can be evaluated with the help of the known result (1.6) while the second on the right-hand side of (1.11) can be evaluated with the help of Kummer’s summation theorem (1.5), after some simplification, we easily arrive at the right-hand side of (1.10). This completes the proof of (1.10). □
Remark In (1.10), if we set we recover (1.7).
which is a terminating form of the confluent hypergeometric function , has frequently occurred [, p.268].
For several new and interesting results closely related to (1.15) and (1.16), see Kim et al. .
The aim of this research paper is to first establish a general transformation formula which generalizes (1.13). Then by employing extensions of Gauss’ summation theorem, Kummer’s summation theorem, and its contiguous results, we establish three new and interesting summation formulas for the generalized hypergeometric polynomial which generalize the results (1.15), (1.16), and (1.17).
We conclude this section by remarking that the results established in this research paper are simple, interesting, easily established, and (potentially) useful.
Finally, expressing the inner series as a hypergeometric function, we then easily arrive at the right-hand side of (2.1). This completes the proof of (2.1). □
The summation formulas to be established in this section are given by the following theorems.
which is valid for .
Finally, summing up the two series on the right-hand side, we easily arrive at the desired result (2.1). This completes the proof of Theorem 1.
which is valid for .
Finally, summing up the four series on the right-hand side, we arrive at the desired result (3.2). This completes the proof of Theorem 2.
Now, if we make use of extension of Gauss summation theorem (1.8), after some simplification, we easily arrive at the desired results (3.3). This completes the proof of Theorem 3. □
We also note that if one puts , one obtains Theorem 2, and if one obtains Theorem 1.
- (a)In (3.1), if we set , we have(4.1)
- (b)In (3.2), if we set , we have(4.2)
- (c)In (3.3), if we set so that , we have(4.3)
which upon using the result (1.12) reduces to the well-known result (1.17).
Similarly, other results can also be obtained.
The authors are very grateful to the referees for their valuable suggestions and comments that improved the paper.
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