1. Introduction
The generalized hypergeometric function with
p numerator and
q denominator parameters is defined by [
1,
2,
3,
4]:
where
denotes the Pochhammer symbol (or the raised factorial, since
) defined, for any complex number
a, by
Using the property of the gamma function relation
,
can be written in the form
Here, as usual,
p and
q are non-negative integers and parameters
and
can have arbitrary complex values with zero or negative integer values of
excluded. The generalized hypergeometric function
converges for
and
and
, where
s is the parametric excess defined by
For
and
, we recover Gauss’s well-known hypergeometric function, viz.,
It is not out of place to mention here that whenever hypergeometric functions or the generalized hypergeometric functions reduce to gamma functions, the results are very important from the application point of view. Thus, the classical summation theorems, such as the two of Gauss and those of Kummer and Bailey for the series ; those of Watson, Dixon, Whipple, and Saalschütz for the series ; and others, play an important role in the theory of generalized hypergeometric series. Here, we would like to mention the following summation theorems for ready reference.
Gauss’s summation theorem [1,2,3,4]:
provided
.
Gauss’s second summation theorem [1,2,3,4]:
Bailey’s summation theorem [1,2,3,4]:
Kummer’s summation theorem [1,2,3,4]:
The above-mentioned classical summation theorems have wide applications. By employing Gauss’s second summation theorem we can easily obtain the following two elegant results recorded in [
2,
3]:
and
each for
.
Also, by employing the Bailey summation theorem (
7), we may obtain the following interesting result recorded in [
5]:
In addition to this, we mention the following very interesting summation formulas (including Ramanujan’s summations):
and
As pointed out by Berndt [
6], the above-mentioned Ramanujan summations (excluding (
13) and (
16)) can be obtained very quickly by employing hypergeometric identities. For this, the summation (
12) can be obtained by Gauss’s second summation theorem (
6) by letting
or by employing Bailey’s summation theorem (
7) by letting
and
. The summation (
13) is the well-known Leibniz–Gregory pi series, and can be obtained by employing Kummer’s summation theorem (
8) by letting
and
. The summations (
14) and (
15) can be obtained from Kummer’s summation theorem (
8) by letting
and
and
, respectively. Summation (
16), which is due to Brychkov [
5], can also be obtained by employing Kummer’s summation theorem by letting
.
Further, in terms of hypergeometric function, the summations (
12)–(
16) can be written in the following manner:
and
Moreover, in a very interesting, popular, and useful research paper (in addition to other known and new results), Bailey [
7] obtained the following results by employing the classical Kummer summation theorem (
8):
Recently, a good deal of progress has been made in the direction of generalizing the above-mentioned classical summation theorems. For this, we refer to a paper by Rakha and Rathie [
8]. Also, the above-mentioned classical summation theorems have been extended by Kim et al. [
9]. In our present investigation, we use the following extension of Kummer’s summation theorem [
9]. For
, we have
When
, we recover the Kummer summation theorem (
8).
The paper is organized as follows. In
Section 2, extensions of Gauss’s second summation theorem and Bailey’s summation theorem are given by a different method to that given by Kim et al. [
9]. In
Section 3, extensions of the results (
17)–(
21) are given by employing extensions of Gauss’s second summation theorem (
29), Bailey’s summation theorem (
30), and Kummer’s summation theorem (
28). A few interesting special cases (known as well as new) and the limiting cases are also provided. In
Section 4, extensions of the results (
22) and (23), involving products of generalized hypergeometric series, are established. In
Section 5, extensions of Bailey’s results (
24) and (25) are given. In
Section 6, extensions of Bailey’s results (
26) and (27) are given. Finally, in
Section 7, we derive an extension of the well-known classical Dixon theorem obtained earlier by Kim et al. [
9] by following a different method. The results established in this paper are simple, interesting, easily established, and may be useful.
7. Extension of Classical Dixon’s Summation Theorem
In this section, we establish the following extension of the classical Dixon’s summation theorem for the series asserted in the following theorem.
Theorem 6. The following results hold true for .provided with Proof. In order to derive (
54), we proceed as follows. Denoting the left-hand side of (
54) by
, and then, expressing
as a series, we have
Using a known result [
3] (Exercise 5, p. 69),
we have
Expressing
as a series, we have
Using the known identities
we have, after some simplification,
Now, replacing
k by
and using the result [
3] (Equation (2), p. 57)
we have, after some algebra,
Summing up the inner series, we have
Finally,
can now be evaluated with the help of the extended Kummer’s summation theorem (
28) and after much simplification, summing up the result and series in terms of
, which can be further evaluated with the help of the classical Gauss summation theorem (
5), we arrive at the right-hand side of (
54). This completes the proof of (
54). □
Corollary 10. In result (54), if we take , we obtainprovided , which is the classical Dixon’s summation theorem [3,11]. Thus, result (54) is regarded as an extension of the result (55).
Remark: For some interesting summation formulas involving hypergeometric functions in two variables (i.e., in Kampé de Fériet), we refer the reader to research papers by Verma et al. [
12] and Wang and Chen [
13].
Concluding remarks: In this research paper, we have provided applications of the extension of Kummer’s summation theorem due to Kim et al. in establishing (i) extensions of Gauss’s second summation theorem and Bailey’s summation theorem; (ii) extensions of several summations (including Ramanujan’s summations); (iii) extensions of several results involving products of generalized hypergeometric series; and (iv) extension of classical Dixon’s summation theorem. As special cases, we recover several known summations (including several Ramanujan summations and the Gregory–Lebniz pi summation) and various results involving products of generalized hypergeometric series due to Bailey. We believe that the results established in this research paper are simple, interesting, easily established, have not appeared in the literature previously, and represent a definite contribution in the theory of generalized hypergeometric series of one and several variables. It is hoped that the results could be of potential use in the area of applied mathematics, statistics, engineering, and mathematical physics. We conclude this research paper by remarking that the new results (
29), (
30), (
36)–(
40), (
41), (
42), (
48), (
49), (
52), (
53), and (
54), in their most general form, for example, generalization of result (29) in the form
for
and
, is under investigation and will form a part of the subsequent paper in this direction.