EXTENSION OF A SUMMATION DUE TO RAMANUJAN
ARJUN K. RATHIE Department of Mathematics, School of Mathematical and Physical Sciences, Central University of Kerala, Riverside Transit Campus, Padennakad, P.O. Nileshwar, KASARAGOD, 671328, KERALA, INDIA e-mail address:
[email protected]
Abstract : In this short research note, we aim to establish an interesting extension of a summation due to Ramanujan. The result is derived with the help of extension of Gauss summation theorem available in the literature. Keywords : Generalized hypergeometric series, Gauss summation theorem, Dixon summation theorem, Ramanujan summations. 2010 Mathematics Subject Classification : 33C90 , 33C05
1. INTRODUCTION The generalized hypergeometric function
with p numeratorial and q
denominatorial parameters is defined by [7],
,…., ,…,
∞
; =
( ) …..
( ) …
!
(1.1)
where ( ) denotes the Pochhammer symbol (or the shifted or raising factorial, since (1) = !) defined for any complex number ( ) =
1 , = 0 (1.2) ( + 1) … . ( + − 1) , ∈
Using the fundamental function relation, ( ) =
by
Γ( + ) , ( ∈ Γ( )
Γ( + 1) = Γ( ),
( )
can be written as
∪ {0} ) (1.3)
where Γ is the well known Gamma function. For convergence of
[ ] and its other elementary properties, we refer [7].
It is interesting to mention here two special cases of (1.1) for (i) (ii)
= 1,
= 2, = 1 and for
= 1 respectively defined by
,
;
( ) ( ) (1.4) ( ) !
=
and =
( ) (1.5) ( ) !
The function
is popularly known as the Gauss’s hypergeometric function and
;
the function
is known as Kummer’s
function. The functions
and
function or the confluent hypergeometric
form the core of special functions and include as
their special cases almost all of the commonly used elementary functions. It is well known that in the theory of hypergeometric and generalized hypergeometric series, classical summation theorem such as those of Gauss, Gauss second, Kummer and Bailey for the series
; Watson, Dixon, Whipple and Saalschutz for the series
and others play an important role. Here we would like to mention two of them. These are: Gauss’s summation theorem [7] :
,
; 1 =
provided ℜ( − and
Γ( ) Γ( − − ) (1.6) Γ( − ) Γ( − )
− ) > 0 .
Dixon’s summation theorem [7] :
=
1+
, − ,
,
1+
Γ 1 + 2 Γ(1 +
−
; 1
− )Γ(1 +
Γ(1 + )Γ 1 + 2 −
− )Γ 1 + 2 −
Γ 1+2−
Γ(1 +
− − )
−
(1.7)
provided ℜ( − 2 − 2 ) > −2 Recently good deal of progress has been done in generalizing and extending [2,4] the above mentioned classical summation theorems [2,4] for the series
,
and others.
Here we will mention the extensions of (1.6) and (1.7) given in [2,3]
,
=
, + 1 ; 1 + 1 ,
Γ( + 1) Γ( − − ) ( − Γ( − + 1) Γ( − + 1)
provided ℜ( −
− ) > 0 and
− ) +
(1.8)
≠ 0, −1, −2, ⋯
and , 2+
=
, , − , 1 + 2
( − 1) Γ 1 2
+1 ; 1 − ,
3 1 1 − ) Γ 2 + 2 − − Γ 2 1 1 1 3 Γ 2 − + 2 Γ 2 − + 2 Γ(2 + − − )
Γ(1 +
− ) Γ(2 +
1 1 Γ 2 Γ(1 + − ) Γ(1 + − ) Γ 1 + 2 − − + (1.9) ( − 1) Γ 1 + 1 Γ 1 − + 1 Γ 1 − + 1 Γ(1 + − − ) 2 2 2 2 2
provided ℜ( − 2 − 2 ) > −2 and = 1− =
1
(1 +
1+ − 1+ − −
and
are given by
− ) (1.10) (1 +
− − 2 ) − 2
2
−
− + 1 (1.11)
= , (1.8) reduces to (1.6) and for
Clearly for
=1+
−
, (1.9) reduces to (1.7)
respectively. Applications of the above mentioned classical summation theorems are well known now. It has been pointed out by Berndt [1] that several interesting summations due to Ramanujan can be obtained quite simply by employing above mentioned classical summation theorems. Here, we would like to mention the following interesting summations due to Ramanujan[1,5,6]. These are 1+
1+
1 1 5 2
+
1 1.3 9 2.4
+ ⋯ = 4 Γ
3 4
(1.12)
1 1 1 1.3 + + ⋯ = 5 2 9 2.4 8 √2 Γ
3 4
(1.13)
and
1+
1 1 1 1.3 + + ⋯ = 5 2 9 2.4 2 √2 Γ
3 4
(1.14)
As explained by Berndt[1] that the summations (1.12) and (1.13) due to Ramanujan can be obtained quite simply by employing classical Dixon summation theorem (1.7) by taking ( ) =
= ,
( ) = ,
=
and
= =
respectively. The summation (1.14) obtained by Ramanujan[6] using an integral representation.
Very recently Rathie and Paris [8] pointed out that the summation (1.14) can also be obtained in a very simple manner taking = ,
= and
by employing Gauss summation theorem (1.6) by
= .
In 2010, Kim, et al. [2] obtained the extensions of Ramanujan summations (1.12) and (1.13) by employing extension of Dixon summation theorem (1.9) in the following form {(1.15) written here with minor correction} 1+
1 1 5 2
+1 1 1.3 + 2 9 2.4
+1 4 + ⋯ = 3 3
1
−1 + 3Γ
3 4
1−
1 (1.15) 4
and
1+
5( + 1) 1 1.3 + 9 9 2.4
1 1 5 2 5
=
48 √2 Γ each for
3 4
5( + 2) + ⋯ 13
5 5 −1 − 4 32√2 Γ
3 4
1 − 1 (1.16) 4
≠ 0, −1, −2, ⋯
In this short research note, we aim to give an extension of Ramanujan summation (1.14). 2. Extension of Ramanujan summation (1.14) : The summation to be established in this note is
1+
for
1 2
1 9
+1
+
1.3 2.4
1.5 9.13
+2
+ ⋯ =
5 12 √2 Γ
3 4
1+
1 (2.1) 4
≠ 0, −1, −2, ⋯
Proof : In order to derive (2.1) , we proceed as follows. Setting extended Gauss summation theorem (1.8), we see that
= ,
=
and
=
in
1 , 2
. . . =
1 , 4
=
as a series with the help of the definition (1.1), we have
1 4 9 4
; 1
9 , 4 Expressing 1 2
+1
( + 1) ( ) !
Using Pochhammer symbol and expanding it , we have after some simplification = 1 +
1 2
1 9
+1
+
1.3 2.4
1.5 9.13
+2
+ ⋯ (2.2)
Also with the same substitution in the right-hand side, we have 9 1 1 Γ 4 Γ 2 1 . . .= 1+ 7 2 4 Γ 4 Using the results Γ( + 1) = Γ( ) , Γ( )Γ(1 − ) =
, and Γ
= √ , we
have after some simplification,
=
5 12 √2 Γ
3 4
1+
1 (2.3) 4
Finally equating (2.2) and (2.3), we get (2.1). This completes the proof of (2.1).
We conclude this note with the remark that our extended summation (2.1) reduces to Ramanujan summation by taking
= . Moreover, by giving different values for d in our
general summation (2.1), we can obtain a large number of new summations.
REFERENCES
[1]
Berndt,B.C. , Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, (1987).
[2]
Kim, Y. S., Rakha, M. A. and Rathie, A. K. Extensions of Certain Classical Summation Theorems for the Series
,
, and
with Applications in
Ramanujan's Summations, Int. J. Math. Math. Sci., 309503, 26 pages , (2010). [3]
Prudnikov, A. P., Brychkov, Yu. A., Marichev, O. I. Integrals and series, Vol.3., More special functions, Gordon and Breach Science Publishers, New York, (1990).
[4]
Rakha, M. A. and Rathie, A. K. Generalization of classical summation theorems for the series
and
with applications, Integral Transforms and Special
Functions, 22(11), 823-840 (2011). [5]
Ramanujan, S. , Notebooks(2 Volumes), Tata Institute of Fundamental Research, Bombay (1957).
[6]
Ramanujan, S., On question 330 of Professor Sanjana, J., Indian Math. Soc., 4, 59-61, (1912), Reprinted in collected papers of Srinivasa Ramanujan (Eds. G.H.Hardy, P.V. Seshu Aiyar & B.M. Wilson), Chelsea, New York,(1962).
[7]
Rainville,E.D., Special Functions , Chelsea Publishing Company, New York, (1960).
[8]
Rathie,A.K., Paris, R.B., A note on some summations due to Ramanujan, their generalizations and some allied series, arxiv: 13014359v1[math CV], (2013).
