CERTAIN NEW CONTINUED FRACTIONS FOR THE RATIO OF TWO
3ψ
3SERIES
Maheshwar Pathak* & Pankaj Srivastava**
*Department of Mathematics, University of Petroleum & Energy Studies, Dehradun-248007, India **Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad-211004, India
(Received on: 18-04-12; Accepted on: 09-05-12)
_______________________________________________________________________________________________
ABSTRACT
I
n the present paper we have developed certain new continued fractions representations for the ratios of two basic bilateral hypergeometric series3ψ
3.Keywords: Basic hypergeometric series, Basic bilateral hypergeometric series, Continued fraction.
AMS Subject Classification: 33D15, 11A55.
________________________________________________________________________________________________
1. INTRODUCTION
Now a days continued fractions have been become center of attraction for pure and applied mathematicians. The Indian mathematician Aryabhata(d. 550 AD) [4] used a continued fraction to solve a linear indeterminate equation. The development of continued fractions theory was stopped for more than a thousand years due to the specific applications. Major impacts of continued fractions in the different field of mathematics have been noticed from the beginning of 20th century. The Indian genius Ramanujan [16, 17] had shown new facet of continued fractions in the field of 𝑞𝑞-series.
Ramanujan and his works on continued fractions had inspired a number of mathematicians working in the field of 𝑞𝑞
-series to carry out researches in this direction. There are a number of mathematicians namely R. P. Agarwal [1, 2], G. E. Andrews and D. Bowman [3], B. C. Berndt [5], N. A. Bhagirathi [6,7], S. Bhargava, C. Adiga and D. D. Somashekara [8], R. Y. Denis [9, 10, 11], R. Y. Denis and S. N. Singh [12], Maheshwar Pathak and Pankaj Srivastava [14], P. Rai [15], S. N. Singh [18, 19, 20], Pankaj Srivastava [21], A. Verma, R. Y. Denis and K. Srinivasa Rao [22] etc. have established several results for basic hypergeometric series 2
φ
1, 3φ
2 and basic bilateral hypergeometric series 2ψ
2, 3ψ
3 in terms of continued fractions. In the present paper certain new results for the quotients of two3
ψ
3 series in terms of continued fractions have been established following the techniques developed by PankajSrivastava [21] and also certain special cases have been deduced.
2. DEFINITIONS AND NOTATIONS
We shall use the following 𝑞𝑞-symbols: For
1 and
r1
q
<
q
<
1
0
( ; )
(1
),
1
n
s n
s
a q
aq
n
−
=
=
∏
−
≥
1
0
( ;
)
(1
),
1
n
r rs
n s
a q
aq
n
−
=
=
∏
−
≥
0 0
( ; )
a q
=
1, ( ;
a q
r)
=
1
0
( ;
r)
(1
rs)
s
a q
aq
∞ ∞
=
=
∏
−
( )
a
n=
( ; ) .
a q
nCorresponding author:
Maheshwar Pathak*
© 2012, IJMA. All Rights Reserved 1803
A generalized basic hypergeometric series with base 𝑞𝑞 is defined as:
1 2 1
1
0
1 2 1 1 1
,
, ...,
( ; ) ...(
; )
; ;
,
,
, ...,
( ; ) ...(
; ) ( ; )
n
A n A n
A A
n
A n A n n
a
a
a
a q
a q z
q
z
b
b
b
b q
b
q
q q
φ
− ∞=
− −
=
∑
where |𝑧𝑧| < 1, |𝑞𝑞| < 1.
A generalized basic bilateral hypergeometric series with base 𝑞𝑞 is defined as:
1 2 1
1 2 1
,
, ...,
( ; ) ...(
; )
; ;
,
,
, ...,
( ; ) ...(
; )
n
A n A n
A A
n
A n A n
a
a
a
a q
a q z
q
z
b
b
b
b q
b
q
ψ
∞=−∞
=
∑
where |𝑏𝑏1𝑏𝑏2… 𝑏𝑏𝐴𝐴/𝑎𝑎1𝑎𝑎2… 𝑎𝑎𝐴𝐴| < |𝑧𝑧| < 1, |𝑞𝑞| < 1.
A finite continued fraction is an expression of the type:
3 5 7 9 1
1 11
2 4 6 8 10 12 ...
.
n n
a
a
a
a
a
a
a
a
a
a
a
a
a
a
−
+
+
+
+
+
+
where 𝑎𝑎1, 𝑎𝑎2, 𝑎𝑎3, 𝑎𝑎4, … are real or complex numbers.
It is an infinite continued fraction when 𝑛𝑛 → ∞:
In order to establish main results, we shall make use of the following known results:
S. N. Singh [20] has established the following general transformation formula between basic and basic bilateral hypergeometric series:
1
1 1
1
1
3 3
1
1 1
3 2 1
,
,
,
;
,
,
,
, ...,
; ;
,
,
,
;
,
,
,
, ...,
(1
)(1
)( ; ) ...( ; )
,
,
,
(1
)(1
)(
/ ; ) ...(
/ ; )
r
r r
m r m
r r
r
m
m r m
r r
m r m
bq az q
b
b
a
q
q
cq
dq
q
q
az
b a
b
b
b
q
z
q q
bq
b
b
q
az
q
c
d
b az
a
b
b
b
bc
bd b q
b q
a
bcq
bdq
b q
c
d b b q
b
b q
ψ
φ
∞ + +
∞
+ +
=
−
−
×
×
−
−
1, ...,
; ;
,
,
, ...,
r m r r
b q
q
z
bc
bd
b
b
(1)
Bhargava, Adiga and Somashekara [8] have established the following results for the quotients of two 3
φ
2 series,which are as follows:
3 2
0
3 2
0
2 2 1 2 2
,
,
; ;
/
,
(1
)
,
,
(1
)
(1
)(1
/
)
; ;
/
,
...
...
(1
)
(1
)(1
/
)
(1
)
n n
n n
a
b
cq
q
de abc
dq
e
d
E
a
b
cq
d
dq
e cq
q
de abc
d
e
F
E
F
dq
dq
e cq
dq
φ
φ
+ +
−
=
−
+ −
−
+
−
+
−
−
+ −
+
(2)where
(
n/
)(1
n)(1
n)(
n),
0,1, 2,...
n
E
=
deq
abc
−
aq
−
bq
dq
−
c
n
=
and
1 1 1
( /
)[1 (
n/ )][1 (
n/ )](1
n),
0,1, 2,...
n
© 2012, IJMA. All Rights Reserved 1804
3 2
0
3 2
0
2 2 1 2 2
,
,
; ;
/
,
(1
)[1 (
/
)]
,
,
(1
)[1 (
/
)]
(1
)
; ;
/
,
...
...
(1
)[1 (
/
)]
(1
)
(1
)[1 (
/
)]
n n
n n
a
b
c
q
deq abc
dq
e
d
de abc
D
a
b
c
d
de abc
dq
q
de abc
d
e
C
D
C
dq
de abc
dq
dq
de abc
φ
φ
+ +
−
−
=
−
−
+ −
+
−
−
+
−
+ −
−
+
(3)where
1 1 1
[1 (
/ )][1 (
/ )][1 (
/ )],
0,1, 2,...
n n n n
n
C
= −
eq
−
dq
+a
−
dq
+b
−
dq
+c
n
=
and
(
/
)(1
)(1
)(1
),
0,1, 2,...
n n n
n
D
=
de abc
−
aq
−
bq
−
cq
n
=
3 2
3 2
,
,
; ;
/
,
,
,
; ;
/
,
aq
bq
c
q
de abcq
d
eq
a
b
c
q
de abc
d
e
φ
φ
(1
) / [1 (
/
)] [(
/
) ( / )][1 ( / )] (
/
)(1
)(1
)(1
)
1
(1
)[1 (
/
)]
(1
)
e
de abcq
c de abcq
e c
d c
de abcq
aq
bq
c
d
de abcq
aq
−
−
−
−
−
−
−
=
+
−
−
+
−
+
1 1
...
...
[1 (
/
)]
(1
)
[1 (
/
)]
(1
)
n n
B
A
B
A
de abcq
aq
de abcq
aq
−
+ −
+
−
+ −
+
(4) where
1
(
/
)(1
n)(1
n),
1, 2, 3,...
n
A
=
de abcq
−
bq
+−
cq
n
=
and
1 1
[1 (
n/ )][1 (
n/ )],
1, 2, 3,...
n
B
=
aq
−
dq
−a
−
eq
−a
n
=
The identity due to Gasper and Rahman [13, III.9, pp. 241] is:
3 2 3 2
,
,
( / ,
/
; )
,
/ ,
/
; ;
/
; ;
/
,
( ,
/
; )
,
/
a
b
c
e a de bc q
a
d b
d c
q
de abc
q
e a
d
e
e de abc q
d
de bc
φ
∞φ
∞
=
(5)3. MAIN RESULTS
We have established the following results which are as follows:
1 2
1 2
2 2
1 2
2 2
1 2
2
1
1 2
3 3
2 2
1 2 2
2 2 2
1 2
3 3
1 2
0 0
1 2
2 1 2 2
/ ,
/ ,
/
1
; ;
(
; ) (
/ ; )
/ ,
/ ,
/
(1
)
(
/ ; ) ( ; )
(1
)
/ ,
/ ,
/
1
; ;
/ ,
/ ,
/
...
(1
)(1
/
)
(1
)
m m
m m
m m
m m
m m
m m
m
a b
b q
b
b q
b
q
b q q
b
b q
aq
q b
b b
b
b
b
b q b q
b q
b
a b
b q
b
b q
b
q
aq
q b
b b
b
b
E
F
b q
b b q
b q
ψ
ψ
+
+
+
+
−
=
×
−
+
−
−
+ −
+
2 1 2 1 2 22 1 2 2
...
(1
)(1
/
)
(1
)
n n
m
n n
E
F
b q
+b b q
+b q
+
−
−
+ −
+
(6)
where
1 2 1 2
1 2 2
(
n/
m m)(1
n)(1
m n)(
n m),
0,1, 2,...
n
E
=
q
aq
+−
aq
−
b q
+b q
−
b q
n
=
and
2 1 1 1 1 2 1
1 2 2 2 1 2
( /
m)(1
n/ )(1
n/
m)(1
m n),
0,1, 2,...
n
© 2012, IJMA. All Rights Reserved 1805
1 2
1 2 1 2
1 2
1 2
1 2
1 1
1 2
3 3
1 2 1 2 1 2 2
1 2 1 2 2
1 2
3 3
1 2
1 2
/ ,
/ ,
/
1
; ;
/ ,
/ ,
/
(
)(
)(1
)
(1
) (1
)
(1
)(1
)(
)
(
)(
)
/ ,
/ ,
/
1
; ;
/ ,
/ ,
/
(1
)(
)
(1
m m
m m m m
m m
m m
m m
a b
b q
b
b q
b
q
aq
q b
b q b
b q b
b b b b
b q
b q
a
b
b
b
b b q
b b q
a b
a b
b q
b
b q
b
q
aq
q b
b b
b
b
b
a b
ψ
ψ
+ +
+
+
−
−
−
−
−
=
×
−
−
−
−
−
−
−
−
0 0
2 2 1 2 2
1 2 1 1 2 1 1 2
...
...
)(
)
(1
)
(1
)(1
/ )
(1
)
(1
)(1
/ )
n n
n n
D
C
D
C
b
a b
b q
b q
b
a
b q
+b q
+b
a
−
+ −
+ −
−
+
−
+
−
−
+
(7)where
1 2 1 2
( ) 1 1 1
1 1 2
(1
/ )(1
)(1
),
0,1, 2,...
n m m n n m n m
n
C
= −
q
− +−
b q
+a
−
b q
+ +−
b q
+ +n
=
and
1 2
2 1 2
(
/ )(1
n)(1
n m)(1
n m/
),
0,1, 2,...
n
D
=
b
a
−
aq
−
q
−−
b q
−b
n
=
1 2
1 2
1 2
1 2
1
1 2
3 3 1
1 2
1 2
3 3
1 2
,
/ ,
/
1
; ;
/ ,
/ ,
/
/ ,
/ ,
/
1
; ;
/ ,
/ ,
/
/
m mm m
m m
m m
b q
b
b q
b
q
aq
q b
b q b
b
b
a b
b q
b
b q
b
q
aq
q b
b b
b
b
aq b
ψ
ψ
+
+ +
+
1
1
1 1
1 1
(
1)(1
)(
)
(
)(
)(1
)
m m
a
b q
b b
a b b b q
b
−
−
−
=
−
−
−
1 2 2 1 2 2 2 1 2 1 2
1 2
1 1 1 1
1 2 1 2 1 2
1 2
(1
)(1 1 /
)
(1 /
/
)(1
) (1 /
)(1
)(1
)(1
)
1
(1
)(1 1 /
)
(1
)
m m m m m m m m m m m
m m
b
aq
b q
aq
b q
b
q
aq
aq
b q
b q
b
aq
aq
+ + + + − − + + +
+ +
−
−
−
−
−
−
−
+
−
−
+
−
+
1 2 1 2
1 1
2
1
...
1... ,
0
(1 1 /
)
(1
)
(1 1 /
)
(1
)
n n
m m m m
B
A
B
A
m
aq
+ +aq
aq
+ +aq
≠
−
+
−
+
−
+
−
+
(8)
where
1 2 1 1 1 2
1 2
(1/
m m)(1
m n)(1
m n),
1, 2, 3,...
n
A
=
aq
+ +−
b q
+ +−
b q
+n
=
and
1 1
1
(1
/ ),
1, 2, 3,...
2
(1
/ )
nn
n
B
=
aq
−
b q
−
a
−
b q
−a
n
=
4. PROOF OF MAIN RESULTS
The proof of main results 6, 7 and 8 are given below:
Proof of (6): Taking 𝑟𝑟 = 2, 𝑐𝑐 = 𝑑𝑑 = 0 in transformation formula (1), we get
1 2
1 2
3 3
1 2
/ ,
/ ,
/
; ;
/ ,
/ ,
/
m m
a b
b q
b
b q
b
q
z
q b
b b
b
b
ψ
1 2
1 2
1 2
1 2 1 2
3 2
1 2 1 2
,
,
, ;
( ; ) ( ; )
,
,
; ;
(
/ ; ) (
/ ; )
,
,
,
,
;
m m
m m
m m
bq az q
q
q
b q
b q
a
b q
b q
az
b a
q
z
q q
bq
b b q
b
b q
b
b
az
q
b az
a
φ
∞∞
=
×
×
© 2012, IJMA. All Rights Reserved 1806 Proof of (7): Taking 𝑧𝑧 = 1/𝑎𝑎𝑞𝑞𝑚𝑚1+𝑚𝑚2 in (9), we get
(
)
(
)
1 2 1 2
1 2
1 2 1 2 1 2
1 2
1 2
1 2
1 2
1
1 2
3 3 1
1 2
1 2 1 2
3 2
1 2 1 2
,
,1 /
, / ;
/ ,
/ ,
/
1
; ;
/ ,
/ ,
/
/ ,
,1 /
,
/ ;
( ; ) ( ; )
,
,
1
; ;
(
/ ; ) (
/ ; )
,
m m m m
m m
m m m m m m
m m
m m
m m
m m
q bq
bq
q a q
a b
b q
b
b q
b
q
aq
q b
b b
b
b
q b q
q
bq a q
b q
b q
a
b q
b q
q
b b q
b
b q
b
b
a q
ψ
φ
+ + +
∞
+ + + +
∞
+
=
×
×
(10)making use of result (5) in (10) and then replace 𝑏𝑏1, 𝑏𝑏2 by 𝑏𝑏1𝑞𝑞, 𝑏𝑏2𝑞𝑞 in that result. Again we use the result (5) in (10) and take the ratio of these two results, finally making use of the result (3) and after simplification; we get the result (7).
Proof of (8): Replacing 𝑎𝑎 by 𝑎𝑎𝑞𝑞 and 𝑏𝑏1 by 𝑏𝑏1𝑞𝑞 in (10), we get
(
)
(
)
1 2 1 2
1 2
1 2 1 2 1 2
1 1
1 2
3 3 1 1
1 2
,
,1 /
,1 / ;
,
/ ,
/
1
; ;
/ ,
,1 /
, / ;
/ ,
/ ,
/
/
m m m m m mm m m m m m
q bq
bq
a q
b q
b
b q
b
q
aq
q b q
q
b a q
q b
b q b
b
b
aq b
ψ
+ + +
+
∞
+ + + + +
∞
=
1 2
1 2
1 2
1 2
1
1 2 1 2
3 2 1
1 2 1 2
(
; ) ( ; )
,
,
1
; ;
(
/ ; ) (
/ ; )
,
m m
m m
m m
m m
b q q
b q
aq
b q
b q
q
b q b q
b
b q
φ
b q
b
a q
+
+ +
×
×
(11)taking the ratio of (11) and (10) and making use of result (4) and after simplification, we get the result (8).
5. SPECIAL CASES:
In this section, we shall consider certain applications of the main results obtained in the §3.
(1) Putting
m
1=
0
in result (6), we get(
) (
)
(
) (
)
(
(
) (
)
)
(
)
(
) (
)(
) (
)
2
2
2 2
2
2 2 2
2
2
1 2 2 2
2
2 2 2
1
2 2 2 2 1 2
2 2 2
2
0
2 2 1 1 2 2
2 2 1 2 2
/ ,
/
1
; ;
/ ,
/
;
/ ;
1
/ ;
;
1
1
1
/
/ ,
/
1
; ;
/ ,
/
...
.
1
1
1
/
1
m
m
m m o
m
m m m
m
n n
n m
n
a b
b q
b
q
aq
q b
b
b
b q q
b
b q
b
E
b q b q
b q
b
b q
b b q
a b
b q
b
q
aq
q b
b
b
F
E
F
b q
b q
b b q
b q
ψ
ψ
+
+
+ +
+
−
=
×
+
−
+
−
−
−
+
−
−
+
−
+
..
(12)where
(
)(
)(
)
(
)
2 2
1 2 2
/
m1
1
m,
0,1, 2,...
n n n n
n
E
=
q
aq
−
aq
−
b q
b q
−
b q
n
=
and
(
2 1)(
1)(
1)
(
2 1)
1
/
21
2/
1
2/
11
2,
0,1, 2,...
m n n m n
n
F
=
b b q
+−
b q
+a
−
b q
+b
−
b q
+ +n
=
© 2012, IJMA. All Rights Reserved 1807
2 2 2
2
/ ,
/
1
; ;
/ ,
/
a b
b q b
q
a
q b
b
b
ψ
(
)
(
)
(
(
)
)
(
)(
)
(
)
(
)
(
)
(
)
2 0
2
2 2 1 2
2 2 1 2 2
2 2 1 2 2
, / ,1 / ,
1
1
1
1
/
/ ,
/ ,1 / ,
;
...
...
1
1
1
/
1
;
o n n
n n
q q a
b
q
b
G
b
b q
b b q
q b qb a
a b q
H
G
H
b q
b q
b b q
b q
qb
∞
∞
+ +
−
=
×
−
+
−
−
+
−
+
−
−
+
−
+
(14) where
(
/
)(
1
)(
1
1)(
2 2)
,
0,1, 2,...
n n n n
n
G
=
q
a
−
aq
−
b q
b q
−
b
n
=
and
(
)
(
1)(
1)(
1)
1
/
21
2/
1
2/
1
2,
0,1, 2,...
n n n
n
H
=
b b q
−
b q
+a
−
b q
+b
−
b q
+n
=
(3) Putting
b
1=
a
andb
=
bq
in result (7), we get(
)(
)
(
)
(
)(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)(
)
(
)(
) (
)
(
)
(
)
1 2
1 2
2 1
2
1 2
1 2
2 2 2
2
2 2
2
1 1 2 2
2 2
2 2
2
2 0 0
2
2 2
1
/ ,
/
1
; ;
1 / ,
/
1
1
1
1
1
/ ,
/
1
; ;
1 / ,
/
1
...
1
1
1
1
/
m m
m m
m m
m
m m
m m
n
aq
b
b q
b
q
aq
b
b
b
b q
a
b
bq
a bq b
aq
b qb q
a b
a
b
bq
aq
b
b q
b
q
aq
b
b
bq
a
a b
D
C
D
a
a b
aq
aq
b
a
m
aq
ψ
ψ
+
− −
+
−
−
−
−
−
=
×
−
−
−
−
−
−
−
−
−
+
−
+
−
−
+
(
2 1) (
2 2)
(
)
2
...
1
1
1
/
n
n n
C
aq
+aq
+b
a
−
+
−
−
+
(14) where
(
)
(
)(
)
1 2 1 2
( ) 1 1 1
2
1
1
1
,
0,1, 2,...
n m m n n m n m
n
C
= −
q
− +−
q
+−
aq
+ +−
b q
+ +n
=
and
(4) Putting 𝑏𝑏 = 1 in result (6), we get
(
)
(
) (
)
(
)
1 2
1 2
2
1 2
1 2
1
1 2
3 2
1 2
2
1
2 2 1 2
1 2
3 2
1 2
,
,
1
; ;
,
1
1
1
1
/
,
,
1
; ;
,
m m
m m
o m m m
m m
a
b q
b q
q
aq
b
b
b
E
b
b q
b b q
a
b q
b q
q
aq
b
b
φ
φ
+
+
+
+
−
=
−
+
−
−
+
(
)
(
)(
1)(
2)
2
/
1
1
1
/
,
0,1, 2,...
n m n m
n n
© 2012, IJMA. All Rights Reserved 1808
(
) (
)
(
) (
)
2
0
2 2 1 1 2 2
2 2 1 2 2
...
...
1
1
1
/
1
n n
n m
n
F
E
F
b q
b q
+b b q
+b q
+
−
+
−
−
+
−
+
(15) where
(
1 2)(
)
(
1)(
2)
1 2 2
/
m m1
1
m n m,
0,1, 2,...
n n n
n
E
=
q
aq
+−
aq
−
b q
+b q
−
b q
n
=
and
(
2 1)(
1)(
1 1)(
2 1)
1
/
21
2/
1
2/
11
2,
0,1, 2,...
m n n m m n
n
F
=
b b q
+−
b q
+a
−
b q
+b q
−
b q
+ +n
=
(5) Putting
a
=
α
q
andb
=
1
in result (7), we get(
)
(
)
(
(
)(
)(
)
)
(
)
(
)
(
)
(
) (
)
(
)
1 2
1 2
1 2
1 2
1 1
1 2
3 2 1
1 2
2 1 2
2 1 2
1 2
3 2 1
1 2
0
2 2 1 2 2
1 1 2 1 1 2
,
,
1
; ;
,
1
1
1
,
,
1
; ;
,
...
...
1
1
1
/
1
1
1
/
m m
m m
m m
m m
o n n
n n
q
b q
b q
q
q
b q
b q
q
b
b
q b
q b
b
q b
q
b q
b q
q
q
b
b
D
C
D
C
b q
b q
b
q
b q
b q
b
q
α
φ
α
α
α
α
α
α
φ
α
α
α
+ +
+ +
+ +
+ +
−
−
−
=
−
−
−
+
−
+
−
−
+
−
+
−
−
+
(16)
where
α
>
1
(
)
(
)(
)
1 2 1 2
( ) 1 1
1 1 2
1
/
1
1
,
0,1, 2,...
n m m n n m n m
n
C
= −
q
− +−
b q
α
−
b q
+ +−
b q
+ +n
=
and
(
)
(
)(
)(
)
1 2
1
2
/
1
1
1
1/
2,
0,1, 2,...
n m n m
n n
D
=
b
α
q
−
α
q
+−
q
−−
b q
−b
n
=
(6) Putting
b
1=
0
in result (8), we get(
)
2
1 2 1 2 1 2
1 2 2
1 2
2
2 2 1 1 ( 1)
2 2
1 2
2 2 2
2
/ ,
/
1
; ;
/ ,
/
(
1)
1/(1 1/
)
(
/ )(1
)
(
)
1
1
(1 1/
)
/ ,
/
1
; ;
/ ,
/
m
m m m m m m
m m m
m m
aq b
b q
b
q
aq
q b
b
b
a
aq
b q
a
q
a b
b
aq
a b
b q
b
q
aq
q b
b
b
ψ
ψ
+ + + + − + −
+ + +
−
−
−
=
×
−
+
−
−
+
1 2
1 2 1 2
1
2 1 1
2
1 1
2
(1 1 /
)(1
)(1
)
...
... ,
0
(1
)
(1 1 /
)
(1
)
(1 1 /
)
(1
)
m m
n n
m m m m
aq
aq
B
A
B
A
m
aq
aq
aq
aq
aq
m
b q
+ ++ + + +
−
−
−
≠
−
+
−
+
−
+
−
+
−
+
(17) where
1 2 1 2
2
(1/
m m)(1
m n),
1, 2, 3,...
n
A
=
aq
+ +−
b q
+n
=
and
1 2
(1
n/ ),
1, 2, 3,...
n
© 2012, IJMA. All Rights Reserved 1809 (7) Putting 𝑏𝑏 = 1 and 𝑏𝑏1= 0 in result (8), we get
2
1 2 1 2 1 2 1 2 2
1 2 2
1 2
1 2
2
2 1 1 1 ( 1) 1
2 2 2
1 2
2 2 1
2
1 1
1
,
1
; ;
/ (1
)
(1/
)(1
)(1
)
1/ (1 1/
)
1
(1
)(1 1/
)
(1
)
,
1
; ;
...
(1 1/
)
(1
)
(1 1/
m
m m m m m m m m m
m m m
m m
n m m
aq
b q
q
aq
b
aq
b q
a
q
aq
aq
b q
b
aq
aq
a
b q
q
aq
b
B
B
A
aq
aq
aq
φ
φ
+ + + + − + − + +
+ + +
+ +
−
−
−
−
=
+
−
−
+
−
+
−
+ −
+
−
1 2 1)
(1
)
... ,
20
n m m
A
m
aq
+ +
≠
+ −
+
(18) where
1 2 1 2
2
(1/
m m)(1
m n),
1, 2, 3,...
n
A
=
aq
+ +−
b q
+n
=
and
1 2
(1
n/ ),
1, 2, 3,...
n
B
=
aq
−
b q
−a
n
=
REFERENCES
[1] Agarwal, R. P., “Pade approximants continued fractions and Heine's 𝑞𝑞-series”, J. Math. Phys. Sci. 3 26 (1992),
281-290.
[2] Agarwal, R. P., “Resonance of Ramanujan's mathematics Vol.III”, New Age International Pvt. Ltd. Publishers, New
Delhi, 1998.
[3] Andrews, G.E. and Bowman, D., “A full extension of the Rogers-Ramanujan continued fraction”, Proc. Amer.
Math. Soc. (11) 123 (1995), 3343-3350.
[4] Brezinski, Claude, “History of continued fractions and Pade approximants”, Springer-Verlag, New York, 1980.
[5] Berndt, B. C., “Ramanujan's Notebook part III”, Springer-Verlag, 1991.
[6] Bhagirathi, N. A., “On certain investigations in 𝑞𝑞-series and continued fractions”, Math. Student 1-4 56 (1988), 158-170.
[7] Bhagirathi, N. A., “On basic bilateral hypergeometric series and continued fractions”, Math. Student 1-4 56 (1988), 135-141.
[8] Bhargava, S., Adiga, C., Somashekara, D. D., “On certain continued fractions related to 3
φ
2 basic hypergeometric functions”, J. Math. Phys. Sci. 21 (1987), 613-629.[9] Denis, R. Y., “On basic hypergeometric function and continued fractions”, Math. Student 1-4 52 (1984), 129-136.
[10] Denis, R. Y., “On certain 𝑞𝑞-series and continued fractions”, Math. Student 44 (1983), 70-76.
[11] Denis, R. Y., “On generalization of continued fraction of Gauss”, Internat. J. Math. & Math. Sci. (4) 13 (1990), 741-746.
[12] Denis, R. Y. and Singh, S. N., “Certain result involving 𝑞𝑞-series and continued fractions”, Far East J. Math. Sci. (FJMS) (5) 3 (2001), 731-737.
[13] Gasper, G. and Rahman, M., “Basic Hypergeometric Series”, Cambridge University Press, New York, 1991.
[14] Pathak, Maheshwar and Srivastava, Pankaj, “A note on continued fractions and 3
ψ
3 series”, Ital. J. Pure Appl. Math. 27 (2010), 191-200.[15] Rai, P., “Certain bilateral extensions of the Rogers-Ramanujan continued fraction”, Ganita (1) 57 (2006), 73-87.
© 2012, IJMA. All Rights Reserved 1810
[17] Ramanujan, S., “The lost note book and other unpublished papers”, Narosa, New Delhi 1988.
[18] Singh, S. N., “On 𝑞𝑞-hypergeometric function and continued fractions”, Math. Student 1-4 56 (1988), 81-84. [19] Singh, S. N., “Basic hypergeometric series and continued fractions”, Math. Student 1-4 56 (1988), 91-96.
[20] Singh, S. N., “Certain transformation formulae for basic and bibasic hypergeometric series”, Proc. Nat. Acad. Sci. India (III) 65(A) (1995), 319-329.
[21] Srivastava, Pankaj, “Certain continued fractions for quotients of two 3
ψ
3 series”, Proc. Nat. Acad. Sci. India (IV) 78(A) (2008), 327-330.[22] Verma, A., Denis, R. Y. and Srinivasa Rao, K., “New continued fractions involving basic hypergeometric 3
φ
2function”, Jour. Math. Phy. Sci. 6 21 (1987), 585-592.
