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Some results for terminating series
Journal of Inequalities and Applications volume 2013, Article number: 365 (2013)
The aim of this research paper is to find explicit expressions of
each for . Known results earlier obtained by Kim et al. and Chu follow special cases of our main findings. The results are derived with the help of generalizations of Gauss second, Kummer and Bailey summation theorems for the series obtained earlier by Lavoie et al.
The generalized hypergeometric functions with p numeratorial and q denominatorial parameters are defined by 
where denotes the Pochhammer symbol (or the shifted factorial, since ) defined for any complex number α by
Using the fundamental functional relation , can be written as
where Γ is the well-known gamma function.
It is well known that the series converges for all z if and for if , and the series diverges for all if . The convergence of the series for the case when is of much interest.
The series with converges absolutely if .
The series converges conditionally if and . The series diverges if .
It should be remarked here that whenever the hypergeometric function and the generalized hypergeometric functions can be summed in terms of gamma functions, the results are very important from the application point of view. This function has been extensively studied by many authors (see, e.g., Slater  and Exton ). We begin by recalling the well-known and classical Gauss second summation theorem (see )
the Kummer summation theorem (see )
provided for convergence, and the Bailey summation theorem (see )
In 1996, Lavoie, Grondin and Rathie  generalized the above mentioned classical summation theorems in the following form.
Generalization of the Gauss second summation theorem:
As usual, denotes the greatest integer less than or equal to x and its modulus is denoted by . For , we get Gauss second summation theorem (1.2). The coefficients and are given in Table 1.
The generalization of the Kummer summation theorem:
provided for . For , we get Kummer summation theorem (1.3). The coefficients and are given in Table 2.
The generalization of the Bailey summation theorem:
for . For , we get Kummer summation theorem (1.3). The coefficients and are given in Table 3.
On the other hand, in the investigation of a model of biological junction with quantum-like characteristic based upon Toeplitz operators, Samoletov  obtained, by using the principle of mathematical induction, the following sum containing factorials
where, as usual
In terms of the familiar gamma function and the Gauss hypergeometric function, Samoletov  also rewrote the sum (1.8) in its equivalent form
by setting . In the same paper, Srivastava  obtained for , equivalent expressions of (1.9) by using Plaff-Kummer transformation and Euler transformation.
The aim of this paper is to find explicit expressions of
each for . The results are derived with the help of the generalized Gauss second summation theorem, the generalized Kummer summation theorem and the generalized Bailey summation theorem for the series obtained earlier by Lavoie et al. Several known as well as new results have also been obtained from our main findings.
3 Proofs of (2.1) and (2.2)
In order to derive (2.1), we proceed as follows. Expressing as a series, we have
On reversing the series, we have
Using the well-known identities
we have, after a little algebra,
Summing up the series, we finally have
Now it can be easily seen that the on the right-hand side can be evaluated with the help of generalized Gauss second summation theorem (1.5) and after a little simplification, we easily arrive at the right-hand side of (2.1). Further, in (3.1), if we use the Plaff-transformation 
it takes the following form
The on the right-hand side can now be evaluated with the help of generalized Kummer summation theorem (1.6) and after a little algebra, we obtain the following equivalent form of our first result (2.1)
it takes the following form
The on the right-hand side can now be evaluated with the help of generalized Bailey summation theorem (1.7) and after a little simplification, we get the following equivalent form of our first result (2.1)
The coefficients and can be obtained from the table of and by changing a by and c by , respectively. It is not out of place to mention here that in (2.1) or its equivalent forms (3.4) or (3.7), if we take , we get the following results
We remark in passing that the results (3.8), (3.9) and (3.10) are recorded in  in another form which can be easily seen to be equivalent by using the reflection property of the gamma function. Other results (3.11) to (3.16) are believed to be new.
In (2.1) or its equivalent forms (3.3) or (3.7), it is not difficult to obtain the following very interesting results:
In 2010, the results (3.17) to (3.34) were again obtained by Chu .
The results (3.17) and (3.18) are also recorded in .
It is interesting to compare the results (3.17) and (3.19).
In order to derive (2.2), we proceed as follows. Expressing as a series, we have
On reversing the series, we have
Using appropriate identities, after a little algebra, we have
The on the right-hand side can now be evaluated with the help of generalized Bailey summation theorem (1.7), and after some simplification, we easily arrive at the right-hand side of (2.2). This completes the proof of (2.2).
Remark As mentioned in (2.1), we can also get two more equivalent forms of our second main result (2.2) by employing Plaff’s transformation (3.2) and Euler’s transformation (3.5), but the details are left as an exercise to the interested reader.
In (2.2), if we take , we get the following interesting results:
Remark The results (3.36) to (3.44) have also been recently obtained by Chu  by following an entirely different method.
4 Concluding remark
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The first-name author was supported by Wonkwang University in 2013.
The authors declare that they have no competing interests.
The authors have equal contributions to each part of this paper. All the authors have read and approved the final manuscript.
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Kim, Y.S., Rathie, A.K. Some results for terminating series. J Inequal Appl 2013, 365 (2013). https://doi.org/10.1186/1029-242X-2013-365
- generalized hypergeometric series
- Gauss summation theorem
- Gauss second theorem
- Kummer and Bailey theorems