Abstract
The aim of this paper is to establish an extension of Kummer's second theorem in the form = + , where + , . For , we recover Kummer's second theorem. The result is derived with the help of Kummer's second theorem and its contiguous results available in the literature. As an application, we obtain two general results for the terminating series. The results derived in this paper are simple, interesting, and easily established and may be useful in physics, engineering, and applied mathematics.
1. Introduction
The generalized hypergeometric function with numerator and denominator parameters is defined by [1] where denotes Pochhammer’s symbol (or the shifted or raised factorial, since ) defined by
Using the fundamental properties of Gamma function , can be written in the form where is the familiar Gamma function.
It is not out of place to mention here that whenever a generalized hypergeometric or hypergeometric function reduces to Gamma function, the results are very important from the applicative point of view. Thus, the classical summation theorem for the series such as those of Gauss, Gauss second, Kummer, and Bailey plays an important role in the theory of hypergeometric series. For generalization and extensions of these classical summation theorems, we refer to [2, 3].
By employing the above mentioned classical summation theorems, Bailey [4] had obtained a large number of very interesting results (including results due to Ramanujan, Gauss, Kummer, and Whipple) involving products of generalized hypergeometric series.
On the other hand, from the theory of differential equations, Kummer [5] established the following very interesting and useful result known in the literature as Kummer's second theorem:
Bailey [4] established the result (4) by employing the Gauss second summation theorem, and Choi and Rathie [6] established the result (4) (of course, by changing to ) by employing the classical Gauss summation theorem. From (4), Rainville [7] deduced the following two useful and classical results:
Using (5) and (6), it is not difficult to establish the following transformation due to Kummer:
In 1995, Rathie and Nagar [8] obtained two results closely related to Kummer’s second theorem (4); one of those results is given below:
In 2010, Kim et al. [1] have generalized the Kummer's second theorem and obtained explicit expressions of for by employing the generalized Gauss second summation theorem obtained earlier by Lavoie et al. [9]. We, however, would like to mention one of their results which we will require in our present investigation:
In 2008, Rathie and Pogàny [10] established a new summation formula for and, as an application, obtained the following result which is known as an extension of Kummer’s second Theorem (4): for .
It is noted that if in (11) we set , we immediately recover Kummer’s second Theorem (4).
Very recently Rakha [11] rederived the result (11) in its equivalent form by employing the classical Gauss summation theorem, and Kim et al. [12] derived (11) in a very elementary way and, as an application, obtained the following two elegant results:
It is interesting to mention here that the right-hand side of (12) is independent of , where .
Remark 1. (a) In (12) and (13), if we set , we recover (5) and (6), respectively.
(b) Using (12) and (13), Kim et al. [12] have obtained the following extension of transformation (7) due to Kummer:
for .
The aim of this paper is to establish another extension of Kummer’s second Theorem (4) by employing the known results (4), (8), and (10). As an application, we mention two interesting results for the terminating series. The results established in this paper are simple, interesting, and easily established and may be useful in physics, engineering, and applied mathematics.
2. Main Result
The result to be established in this paper is as follows: where and is given by .
2.1. Derivation
In order to derive (15), we proceed as follows. Denoting the left-hand side of (15) by and expressing as a series with the help of (1), we have
Now, it is not difficult to see that we have
Separating (18) into three terms, we have
For the second and third terms on the right-hand side of (19), changing to and to , respectively, and making use of the following results: we have, after some simplification,
Now, summing up the series with the help of (1), we have
Finally, observing the right-hand side of (22), we see that the first, second, and third expressions can now be evaluated with the help of the results (10), (8), and (4), respectively, and, after some simplification, we arrive at the desired result (15). This completes the proof of (15).
3. New Results for Terminating
In this section, from our newly obtained result (15), we will establish two new results for the terminating series. These are where , and .
3.1. Derivations
In order to derive the results (23), we proceed as follows. Denoting the left-hand side of (15) by , then expressing both of the functions involved in the series, we have
Replacing by in (24) and using the known result [7, page 56, Lemma 10]: we have
Using the identity we have, after some simplification,
Expressing the inner series in the last result, we get
Now, separating the into even and odd powers of and making use of the results: we finally have
Also, it is not difficult to see that
Now, if we denote the right-hand side of (15) by , then with the help of (32), it can be expressed as
Thus, from (31) and (33), if we equate the coefficients of and on both sides, we at once arrive at the results (23). This completes the proof.
Remark 2. (a) Setting in (15), we immediately recover Kummer’s second Theorem (4). Thus, (15) can be regarded as the extension of (4).
(b) Also, if we take in (12) and (13), we again at once get the result (5) and (6), respectively. Thus, our results (12) and (13) can be regarded as extensions of (5) and (6).
4. Extension of a Transformation due to Kummer
In this section, we will establish a natural extension of Kummer’s transformation: for and is given by
4.1. Derivation
In order to establish the result (34), we proceed as follows. Denote the left-hand side of (34) by ; we have expressing as a series, we have
Applying the generalized Binomial theorem we have
Using , we have changing to and using (25), we have
Using we have
Expressing the inner series, as , we find
from which, we have
Using the the following identities: together with the result, we have with
This completes the proof of (34).
Remark 3. In (34), if we take , we get (7). Thus, (34) may be regarded as an extension of (7).
Authors’ Contribution
All authors contributed equally to this paper. They read and approved the final paper.
Acknowledgments
The authors would like to express their sincere gratitude to the referees for their valuable comments and suggestions. They are so much appreciated to the College of Science, Sultan Qaboos University, Muscat, Oman, for supporting the publication charge of this paper.