Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 128458, 6 pages
http://dx.doi.org/10.1155/2013/128458
Research Article
On an Extension of Kummer’s Second Theorem
Medhat A. Rakha,
1,2
Mohamed M. Awad,
2
and Arjun K. Rathie
3
1
Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36 (123), Alkhoud, Muscat, Oman
2
Department of Mathematics, Faculty of Science, Suez Canal University, Ismailia 41511, Egypt
3
Department of Mathematics, School of Mathematical and Physical Sciences, Central University of Kerala,
Riverside Transit Campus, Padennakkad P.O. Nileshwar, Kasaragod, Kerala 671 328, India
Correspondence should be addressed to Medhat A. Rakha; medhat@squ.edu.om
Received 4 December 2012; Accepted 26 February 2013
Academic Editor: Adem Kilic¸man
Copyright © 2013 Medhat A. Rakha et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
e aim of this paper is to establish an extension of Kummer’s second theorem in the form
−/2
2
2
[
, 2+;
2+2, ;
] =
0
1
[
−;
2
/16
+3/2;
] + ((/ − 1/2)/( + 1))
0
1
[
−;
2
/16
+3/2;
] + (
2
/2(2 + 3))
0
1
[
−;
2
/16
+5/2;
], where = (1/( + 1))(1/2 − /)+
/( + 1), ̸ = 0, −1, −2, . . .. For = 2, we recover Kummer’s second theorem. e 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 termi-
nating
3
2
(2) series. e results derived in this paper are simple, interesting, and easily established and may be useful in physics,
engineering, and applied mathematics.
1. Introduction
e generalized hypergeometric function
with numer-
ator and denominator parameters is defined by [1]
[
[
1
,...,
;
1
,...,
;
]
]
=
[
1
,...,
;
1
,...,
; ]
=
∞
∑
=0
(
1
)
⋅ ⋅ ⋅ (
)
(
1
)
⋅ ⋅ ⋅ (
)
!
,
(1)
where ()
denotes Pochhammer’s symbol (or the shiſted or
raised factorial, since (1)
= !) defined by
()
={
( + 1) ⋅ ⋅ ⋅ ( + − 1) , ∈ N,
1, = 0.
(2)
Using the fundamental properties of Gamma function Γ( +
1) = Γ(), ()
can be written in the form
()
=
Γ ( + )
Γ ()
, (3)
where Γ is the familiar Gamma function.
It is not out of place to mention here that whenever a
generalized hypergeometric or hypergeometric function
2
1
reduces to Gamma function, the results are very important
from the applicative point of view. us, the classical sum-
mation theorem for the series
2
1
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 gener-
alized hypergeometric series.
On the other hand, from the theory of differential equa-
tions, Kummer [5] established the following very interesting
and useful result known in the literature as Kummer’s second
theorem:
−/2
1
1
[
[
;
2;
]
]
=
0
1
[
[
[
[
[
−;
2
16
+
1
2
;
]
]
]
]
]
. (4)