"
### $
TRUNCATED BILATERAL HYPERGEOMETRIC SERIES ASSOCIATED WITH
NEGATIVE UNIT ARGUMENT
M. I. Qureshi and Kaleem A. Quraishi*
Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University), New Delhi-110025(India)
*Mathematics Section, Mewat Engineering College (Wakf), Palla, Nuh, Mewat-122107, Haryana (India)
E-mails: [email protected];[email protected] (Dedicated to Prof. R. P. Agarwal, June 15, 1925-February 9, 2008)
(Received on: 21-11-10; Accepted: 08-12-10)
---
ABSTRACTI
n this paper we obtain some summation theorems for truncated bilateral generalized hypergeometric series involving negative unit argument given byθ δ
− −
−
+
RP B
B B
B
H
g
h
2 2
]
1
;
)
(
1
;
)
[(
,B+1H
B+1[(
g
B),
1
−
ε
;
1
+
(
h
B),
−
ε
;
−
1
]
22PR−−δθ,θ δ
ρ
ϖ
ρ
ϖ
−− +
+
−
−
+
−
−
−
R P B
B B
B
H
g
h
2 2 2
2
[(
),
1
,
1
;
1
(
),
,
;
1
]
θ δ
υ
η
ϕ
η
ϕ
υ
−− +
+
−
−
−
+
−
−
−
−
R P B
B B
B
H
g
h
2 2 3
3
[(
),
1
,
1
,
1
;
1
(
),
,
,
;
1
]
, and B+EH
B+Eg
B−
Ξ
E+
h
B−
Ξ
E−
PR−−δθ2 2
]
1
;
)
(
),
(
1
;
)
(
1
),
[(
using series iteration techniques; where
ε
,
ϖ
,
ρ
,
ϕ
,
υ
,
η
andΞ
E are the functions of parameters.
,
,
,
,
,
,
,
2 1 21
g
g
Bh
h
h
Bg
Applying Rainville's limit formula for certain infinite products, some non terminating bilateral hypergeometric summation theorems with negative unit argument are also deduced, in terms of Gamma functions subject to certain conditions. The results presented here are presumably new.Keywords and Phrases: Pochhammer Symbol; Gamma function; Rainville's limitformula; Unilateral and bilateral series; Truncated and non terminating series.
2010 AMS Subject Classifications: 33-Special Functions, Primary 33C99; Secondary 33C20.
1. INTRODUCTION: Rainville's Limit Formula
If
)
(
)
)(
(
)
(
)
)(
(
2 1
2 1
b
n
b
n
b
n
a
n
a
n
a
n
U
n k+
+
+
+
+
+
=
(1.1)then product
∏
U
n can only converge ifk
=
anda
i=
b
i.
When these conditions are satisfied, we can express the infinite product in terms of Gamma functions [3,p.115(Q.No.11)]. Now limit formula for certain infinite products can be written in the following form, if
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
+
a
1+
+
a
2+
+
a
3+
+
+
a
s=
+
b
1+
+
b
2+
+
b
3+
+
+
b
s (1.2) !" # $ %Kaleem A. Quraishi** $ %
+ , $ ! , ! #
and no
a
sorb
s is a negative integer, then without any loss of absolute convergence, we have the following theorem [3,p.128(Q.1);5,pp.6-7(1.3.8);7,pp.14-15(Th.5)].
+
+
+
+
+
+
=
+
+
+
+
+
+
∞ → ∞
=
∏
k s k
k
k s k
k k
n s
s
b
b
b
a
a
a
b
n
b
n
b
n
a
n
a
n
a
n
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
lim
)
(
)
)(
(
)
(
)
)(
(
2 1
2 1
1 1 2
2 1
(1.3)
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
2 1
2 1
s s
a
a
a
b
b
b
+
Γ
+
Γ
+
Γ
+
Γ
+
Γ
+
Γ
=
(1.4)If condition (1.2) is not true, then product in (1.3) diverges.
Non Terminating Bilateral Generalized Hypergeometric Series:
(Dirichlet series or Laurent series) [9,pp.180-182(6.1.1.2,6.1.1.3,6.1.2.3); see also 2;4]
The values of parameters
a
1,
a
2,
,
a
A,
b
1,
b
2,
,
b
A are adjusted in such a manner that each term in the expansion of (1.5) is well defined then
∏
∏
∞
−∞ =
= = ∞
−∞ =
=
=
=
r A
j
r j
r A
j
r j
r r r A r
r r A r r
A A A
A A
A A A
b
z
a
b
b
b
z
a
a
a
z
b
b
b
a
a
a
H
z
b
a
H
1 1 2
1 2 1 2
1 2 1
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
;
,
,
,
;
,
,
,
;
)
(
;
)
(
(1.5)
where Pochhammer's symbol
(
c
)
k is given by∏
−
=
+
=
1 0
).
(
)
(
k
i
k
c
i
c
Truncated Bilateral Generalized Hypergeometric Series:
In 1967, Verma [10,pp.233-234(3.4)] gave the following definition of truncated bilateral generalized hypergeometric series
∏
∏
− − =
= =
=
=
N
M r
A
j r j
r A
j
r j M
N A
A A
A M
N A
A A A
b
z
a
z
b
b
b
a
a
a
H
z
b
a
H
1 1 1 2
1 2 1
)
(
)
(
;
,
,
,
;
,
,
,
;
)
(
;
)
(
where
A
,
N
∈
{
1
,
2
,
3
,
},
M
∈
{
0
,
1
,
2
,
3
,
}.
In 1996, R. P. Agarwal[1,p.19(13)] gave the following definition, with slightly modification in the above definition
∏
∏
− =
= =
=
=
N
M r
A
j r j
r A
j
r j M
N A
A A
A M
N A
A A A
b
z
a
z
b
b
b
a
a
a
H
z
b
a
H
1 1 2
1 2 1
)
(
)
(
;
,
,
,
;
,
,
,
;
)
(
;
)
(
(1.6)
where
A
,
M
,
N
∈
{
1
,
2
,
3
,
}.
When
M
,
N
→
∞
in (1.6), we get non terminating bilateral generalized hypergeometric series (1.5).* $ %
+ , $ ! , ! #
-In next sections we shall discuss the applications of summation theorems of Slater, Verma, Qureshi and Quraishi with positive unit argument, for truncated bilateral hypergeometric series involving negative unit argument.
Since Pochhammer's symbol is associated with Gamma function and Gamma function is undefined for zero and negative integers therefore numerator and denominator parameters are adjusted in such a way that each term of following results is completely well defined and meaningful then without any loss of convergence, we have the following theorems.
2. COMPANION OF FIRST THEOREM OF VERMA:
4
1
2 1
2 3
1 1 2
1 1 1 2
2
(
1
)
)
(
)
1
(
)
(
)
1
(
)
(
1
;
)
(
1
;
)
(
W
g
h
W
g
h
W
h
g
W
h
g
H
Bj
j B
j j
B
j
j B
j j
B
j
j B
j j R
P B
B B B
∏
∏
∏
∏
∏
∏
= =
= =
= = −
−
−
−
+
+
−
−
+
−
=
−
+
θ
δ
(2.1)
where
∏
∏
= − −
= − −
−
+
+
−
+
+
=
B
j P
j
P j B
j P
j
P j P
h
h
P
g
g
W
1 1 1
2
2
2
1
)!
(
2
3
2
2
)
2
(
δ δ
δ δ
δ
δ
(2.2)
∏
∏
= − −
= − −
−
+
+
−
+
+
=
B
j P
j
P j B
j P
j
P j P
h
h
P
g
g
W
1 1 1
1 1 1
1 2
2
3
2
2
)!
1
(
2
4
2
3
)
2
(
(2.3)
∏
∏
= − −
= − −
−
−
−
−
−
−
=
B
j R
j
R j B
j R
j
R j R
g
g
R
h
h
W
1 1 1
1 1 1
1 3
2
3
2
2
)!
1
(
2
4
2
3
)
2
(
(2.4)
∏
∏
= −− −−
= −− −−
− −
−
−
−
−
−
−
=
B
j R
j
R j B
j R
j
R j R
g
g
R
h
h
W
1 1 1
1 1 1
1 4
2
4
2
3
)!
1
(
2
5
2
4
)
2
(
θ θ
θ θ
θ
θ
(2.5)
subject to the following conditions given by
)
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 2 B B r 1 1 2 2 B Br
g
g
g
g
g
g
S
h
h
h
h
h
h
S
+
+
+
=
−
+
−
+
−
+
(2.6)0
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 21
2B+
g
+
g
g
+
g
g
B+
g
B≠
S
(2.7)when
}
,
4
,
3
,
2
{
},
,
3
,
2
,
1
{
},
1
,
0
{
,
);
2
(
,
,
3
,
2
,
1
∈
∈
∈
=
B
B
P
r
θ
δ
andR
∈
{
3
,
4
,
5
,
}.
(2.8)Proof of (2.1)
* $ % + , $ ! , ! # . − − = − = − − =
+
Φ
+
Φ
=
Φ
θ θ 1 0 1 0 1 2 0)
1
2
(
)
2
(
)
(
R i R i R ii
i
i
(2.9) which is the unification of− = − = − =
+
Φ
+
Φ
=
Φ
1 0 1 0 1 2 0)
1
2
(
)
2
(
)
(
R i R i R ii
i
i
and − = − = − =+
Φ
+
Φ
=
Φ
2 0 1 0 2 2 0)
1
2
(
)
2
(
)
(
R i R i R ii
i
i
(2.10) whereθ
∈
{
0
,
1
}
andR
∈
{
3
,
4
,
5
,
}.
− = − = − =
+
Φ
+
Φ
=
Φ
1 0 0 2 0)
1
2
(
)
2
(
)
(
P i P i P ii
i
i
δ δ (2.11) which is the unification of− = = =
+
Φ
+
Φ
=
Φ
1 0 0 2 0)
1
2
(
)
2
(
)
(
P i P i P ii
i
i
and − = − = − =+
Φ
+
Φ
=
Φ
1 0 1 0 1 2 0)
1
2
(
)
2
(
)
(
P i P i P ii
i
i
(2.12)where
δ
∈
{
0
,
1
}
andP
∈
{
2
,
3
,
4
,
}.
Consider the left hand side of (2.1)− + − =
+
+
+
−
=
Ξ
δ θ P Ri i i B i
i i B i i
h
h
h
g
g
g
22 1 2
2 1
)
1
(
)
1
(
)
1
(
)
1
(
)
(
)
(
)
(
− = − + − =+
+
+
−
+
+
+
+
−
=
δ θ Pi i i B i
i i B i i R
i i i B i
i i B i i
h
h
h
g
g
g
h
h
h
g
g
g
20 1 2
2 1 1
2 1 2
2 1
)
1
(
)
1
(
)
1
(
)
1
(
)
(
)
(
)
(
)
1
(
)
1
(
)
1
(
)
1
(
)
(
)
(
)
(
(2.13)∏
∏
− = − − = = =+
+
+
−
+
−
−
−
−
−
−
−
+
−
−
=
δ θ Pi i i B i
i i B i i R
i i i B i
i i B i i B j j B j j
h
h
h
g
g
g
g
g
g
h
h
h
g
h
20 1 2
2 1 1
2
0 1 2
2 1 1 1
)
1
(
)
1
(
)
1
(
)
1
(
)
(
)
(
)
(
)
2
(
)
2
(
)
2
(
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
)
(
(2.14)Now applying the series identities (2.9), (2.11) and properties of Pochhammer's symbol, after simplification we get
∏
∏
− = = =+
−
−
−
−
−
−
−
−
−
−
−
−
+
−
−
=
Ξ
10 1 1 2 2
2 2 1 1 1 1
2
3
2
2
2
3
2
2
2
3
2
2
2
2
2
1
2
2
2
1
2
2
2
1
)
1
(
)
(
R i i B i B i i i i i B i B i i i i B j j B j jg
g
g
g
g
g
h
h
h
h
h
h
g
h
∏
∏
−− = = =+
−
−
−
−
−
−
−
−
−
−
−
−
−
−
+
θ 10 1 1 2 2
2 2 1 1 1 2 1 2
2
4
2
3
2
4
2
3
2
4
2
3
2
3
2
2
2
3
2
2
2
3
2
2
)
1
(
)
(
R i i B i B i i i i i B i B i i i i B j j B j jg
g
g
g
g
g
h
h
h
h
h
h
g
h
− =−
+
+
+
+
+
+
+
+
+
+
δ P i i B i B i i i i i B i B i i i ih
h
h
h
h
h
g
g
g
g
g
g
0 1 1 2 2
* $ %
+ , $ ! , ! # /
∏
∏
−=
= =
+
+
+
+
+
+
+
+
+
+
+
+
+
−
1
0 1 1 2 2
2 2
1 1
1 1
2
3
2
2
2
3
2
2
2
3
2
2
2
2
2
1
2
2
2
1
2
2
2
1
)
1
(
)
(
P
i
i B
i B
i i
i i
i B
i B
i i
i i
B
j
j B
j j
h
h
h
h
h
h
g
g
g
g
g
g
h
g
(2.15)
Now write all finite series of (2.15) in truncated unilateral generalized hypergeometric notation, we get
−
+
+
+
+
+
+
+
+
+
=
Ξ
− +
δ
P B
B B B
B
B
h
h
h
h
h
h
g
g
g
g
g
g
F
1
;
2
2
,
2
1
,
,
2
2
,
2
1
,
2
2
,
2
1
;
2
1
,
2
,
,
2
1
,
2
,
2
1
,
2
,
1
2 2
1 1
2 2
1 1
2 1 2
−
+
+
+
+
+
+
+
+
+
+
+
+
+
−
− +
= =
∏
∏
1 2
2 1
1
2 2
1 1
2 1 2 1
1
1
;
2
3
,
2
2
,
,
2
3
,
2
2
,
2
3
,
2
2
;
2
2
,
2
1
,
,
2
2
,
2
1
,
2
2
,
2
1
,
1
)
1
(
)
(
P B
B
B B
B B B
j
j B
j j
h
h
h
h
h
h
g
g
g
g
g
g
F
h
g
+
−
−
−
−
−
−
−
−
−
−
−
−
+
−
−
− +
= =
∏
∏
1 2
2 1
1
2 2
1 1
2 1 2 1
1
1
;
2
3
,
2
2
,
,
2
3
,
2
2
,
2
3
,
2
2
;
2
2
,
2
1
,
,
2
2
,
2
1
,
2
2
,
2
1
,
1
)
1
(
)
(
R B
B
B B
B B B
j
j B
j j
g
g
g
g
g
g
h
h
h
h
h
h
F
g
h
θ
− − +
= =
−
−
−
−
−
−
−
−
−
−
−
−
−
−
+
∏
∏
1 2
2 1
1
2 2
1 1
2 1 2 1
2 1
2
1
;
2
4
,
2
3
,
,
2
4
,
2
3
,
2
4
,
2
3
;
2
3
,
2
2
,
,
2
3
,
2
2
,
2
3
,
2
2
,
1
)
1
(
)
(
R B
B
B B
B B B
j
j B
j j
g
g
g
g
g
g
h
h
h
h
h
h
F
g
h
(2.16)
By the applications of Slater's theorem[8;9,pp.83-84(2.6.1.1, 2.6.1.7); see also 6,equations (3.5.1)-(3.5.3)] in the right hand side of (2.16), we obtain the right hand side of (2.1).
Deduction of (2.1)
In
W
1,
W
2,
W
3,
W
4 of (2.1), (1.2) type conditions associated with (1.3) type products are satisfied, therefore we can take the limitR
,
P
→
∞
8 1
2 1
2 7
1 1 6
1 1 5
)
1
(
)
(
)
1
(
)
(
)
1
(
)
(
1
;
)
(
1
;
)
(
W
g
h
W
g
h
W
h
g
W
h
g
H
Bj
j B
j j
B
j
j B
j j
B
j
j B
j j
B B B B
∏
∏
∏
∏
∏
∏
= =
= =
= =
−
−
+
+
−
−
+
−
=
−
+
(2.17)where
∏
∏
= =
+
Γ
+
Γ
+
Γ
+
Γ
=
B
j
j j
B
j
j j
g
g
h
h
W
1 1 5
2
3
2
2
2
2
2
1
* $ %
+ , $ ! , ! # 0
∏
∏
= =
+
Γ
+
Γ
+
Γ
+
Γ
=
B
j
j j
B
j
j j
g
g
h
h
W
1 1 6
2
4
2
3
2
3
2
2
(2.19)
∏
∏
= =
−
Γ
−
Γ
−
Γ
−
Γ
=
B
j
j j
B
j
j j
h
h
g
g
W
1 1 7
2
4
2
3
2
3
2
2
(2.20)
∏
∏
= =
−
Γ
−
Γ
−
Γ
−
Γ
=
B
j
j j
B
j
j j
h
h
g
g
W
1 1 8
2
5
2
4
2
4
2
3
(2.21)
subject to the conditions (2.6)-(2.8).
3. COMPANION OF SECOND THEOREM OF VERMA:
If we proceed on the same parallel lines of preceding section and apply Verma theorem [10,p.233(3.3); 1,p.19(4.12); see also 6,equations (3.2.1)-(3.2.4)], we obtain
4 1
2 1
2 3
1 1 2
1 1 1
2 2 1
1
)
1
(
)
(
)
2
(
)
1
(
)
(
)
1
(
)
1
(
)
(
)
1
(
1
;
),
(
1
;
1
),
(
W
g
h
W
g
h
W
h
g
W
h
g
H
Bj
j B
j j
B
j
j B
j j
B
j
j B
j j R
P B
B B B
∏
∏
∏
∏
∏
∏
= =
= =
= = −
− +
+
−
−
+
+
+
−
+
−
+
−
−
=
−
−
+
−
ε
ε
ε
ε
ε
ε
ε
ε
θδ
(3.1) subject to the following conditions given by
)
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 2 B B r 1 1 2 2 B Br
g
g
g
g
g
g
S
h
h
h
h
h
h
S
+
+
+
=
−
+
−
+
−
+
(3.2))
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 2 2 1 1 2 22B
g
g
g
g
g
Bg
BS
Bh
h
h
h
h
Bh
BS
+
+
+
≠
−
+
−
+
−
+
(3.3)0
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 21
2B+
g
+
g
g
+
g
g
B+
g
B≠
S
(3.4))}
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
{
)
1
,
,
,
1
,
,
1
,
,
2
(
2 2 1
1 2
2 2 1 1 2
2 2 1 1 1 2
B B B
B B
B
B B
B
h
h
h
h
h
h
S
g
g
g
g
g
g
S
g
g
g
g
g
g
S
+
−
+
−
+
−
−
+
+
+
+
+
+
−
=
+ε
(3.5) when
}
,
4
,
3
,
2
{
},
,
3
,
2
,
1
{
},
1
,
0
{
,
);
1
2
(
,
,
3
,
2
,
1
−
∈
∈
∈
=
B
B
P
r
θ
δ
andR
∈
{
3
,
4
,
5
,
}.
(3.6)* $ %
+ , $ ! , ! # 1
Deduction of (3.1)
Similarly in
W
1,
W
2,
W
3,
W
4of (3.1), (1.2) type conditions associated with (1.3) type products are satisfied, therefore we can take the limitR
,
P
→
∞
8
1
2 1
2
7
1 1 6
1 1 5
1 1
)
1
(
)
(
)
2
(
)
1
(
)
(
)
1
(
)
1
(
)
(
)
1
(
1
;
),
(
1
;
1
),
(
W
g
h
W
g
h
W
h
g
W
h
g
H
Bj
j B
j j
B
j
j B
j j
B
j
j B
j j
B B B B
∏
∏
∏
∏
∏
∏
= =
= =
= = +
+
−
−
+
+
+
−
+
−
+
−
−
=
−
−
+
−
ε
ε
ε
ε
ε
ε
ε
ε
(3.7) subject to the conditions (3.2)-(3.6) and
W
5,
W
6,
W
7,
W
8are given by the equations (2.18), (2.19), (2.20), (2.21)respectively.
4. COMPANION OF FIRST THEOREM OF QURESHI AND QURAISHI:
If we proceed on the same parallel lines of preceding sections and apply first theorem of authors[6,equations (3.3.1)-(3.3.6)], we obtain
θ
δ
ρ
ϖ
ρ
ϖ
−− +
+
−
−
−
+
−
−
RP B
B B B
h
g
H
2
2 2
2
1
;
,
),
(
1
;
1
,
1
),
(
4 1
2 1
2 3
1
1 2
1 1 1
)
1
(
)
(
)
2
(
)
2
(
)
1
(
)
(
)
1
(
)
1
(
)
1
(
)
(
)
1
(
)
1
(
W
g
h
W
g
h
W
h
g
W
Bj
j B
j j
B
j
j B
j j
B
j
j B
j j
∏
∏
∏
∏
∏
∏
= =
=
=
= =
−
−
+
+
+
+
−
+
+
−
+
−
−
−
=
ρ
ϖ
ρ
ϖ
ρ
ϖ
ρ
ϖ
ϖρ
ρ
ϖ
(4.1) subject to the following conditions given by
)
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 2 B B r 1 1 2 2 B Br
g
g
g
g
g
g
S
h
h
h
h
h
h
S
+
+
+
=
−
+
−
+
−
+
(4.2))
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 2 2 1 1 1 2 21
2B
g
g
g
g
g
Bg
BS
Bh
h
h
h
h
Bh
BS
−+
+
+
≠
−−
+
−
+
−
+
(4.3)0
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 21
2B+
g
+
g
g
+
g
g
B+
g
B≠
S
(4.4))}
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
{
2
)
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
2 2 1
1 1
2 2
2 1 1 1 2
2 2 1
1 2
2 2 1 1 2
B B B
B B
B
B B B
B B
B
h
h
h
h
h
h
S
g
g
g
g
g
g
S
D
h
h
h
h
h
h
S
g
g
g
g
g
g
S
+
−
+
−
+
−
−
+
+
+
+
+
−
+
−
+
−
+
+
+
+
−
=
− −
ϖ
(4.5))}
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
{
2
)
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
2 2 1
1 1
2 2
2 1 1 1 2
2 2 1
1 2
2 2 1 1 2
B B B
B B B
B B B
B B B
h
h
h
h
h
h
S
g
g
g
g
g
g
S
D
h
h
h
h
h
h
S
g
g
g
g
g
g
S
+
−
+
−
+
−
−
+
+
+
−
+
−
+
−
+
−
+
+
+
+
−
=
− −
ρ
(4.6)
−
+
−
+
−
+
−
−
+
+
+
=
22 2 1
1 2
2 2 1 1
2
(
2
,
,
1
,
,
1
,
,
,
1
)
(
1
,
,
1
,
,
,
1
,
)}
{
S
Bg
g
g
g
g
Bg
BS
Bh
h
h
h
h
Bh
BD
×
+
+
+
−
4
{
S
2B+1(
2
,
g
1,
1
g
1,
g
2,
1
g
2,
,
g
B,
1
g
B)}
)}
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
{
S
2B 1g
1+
g
1g
2+
g
2g
B+
g
B−
S
2B 1−
+
h
1h
1−
+
h
2h
2−
+
h
Bh
B×
− −* $ %
+ , $ ! , ! #
when
r
=
1
,
2
,
3
,
,
(
2
B
−
2
);
θ
,
δ
∈
{
0
,
1
},
B
,
P
∈
{
2
,
3
,
4
,
}
andR
∈
{
3
,
4
,
5
,
}.
(4.8) andW
1,
W
2,
W
3,
W
4are given by the equations (2.2), (2.3), (2.4), (2.5) respectively.Deduction of (4.1)
When
R
,
P
→
∞
in (4.1), then−
−
−
+
−
−
+
+
1
;
,
),
(
1
;
1
,
1
),
(
2 2
ρ
ϖ
ρ
ϖ
B B B B
h
g
H
8 1
2 1
2 7
1
1 6
1 1 5
)
1
(
)
(
)
2
(
)
2
(
)
1
(
)
(
)
1
(
)
1
(
)
1
(
)
(
)
1
(
)
1
(
W
g
h
W
g
h
W
h
g
W
Bj
j B
j j
B
j
j B
j j
B
j
j B
j j
∏
∏
∏
∏
∏
∏
= =
=
=
= =
−
−
+
+
+
+
−
+
+
−
+
−
−
−
=
ρ
ϖ
ρ
ϖ
ρ
ϖ
ρ
ϖ
ϖρ
ρ
ϖ
(4.9) subject to the conditions (4.2)-(4.8) and
W
5,
W
6,
W
7,
W
8 are given by the equations (2.18), (2.19), (2.20), (2.21)respectively.
5. COMPANION OF SECOND THEOREM OF QURESHI AND QURAISHI:
If we proceed on the same parallel lines of preceding sections and apply second theorem of authors[6,equations (3.4.1)-(3.4.8)], we can obtain
−
+
−
−
−
−
=
−
−
−
−
+
−
−
−
∏
∏
=
= −
− +
+ 2
1
1 1
2
2 3
3
)
1
(
)
(
)
1
)(
1
(
)
1
(
1
;
,
,
),
(
1
;
1
,
1
,
1
),
(
W
h
g
W
h
g
H
Bj
j B
j j R
P B
B B B
υ
η
ϕ
υ
η
ϕ
υ
η
ϕ
υ
η
ϕ
θδ
4 1
2 1
2 3
1
1
)
1
(
)
(
)
2
)(
2
(
)
2
(
)
1
(
)
(
)
1
)(
1
(
)
1
(
W
g
h
W
g
h
B
j
j B
j j
B
j
j B
j j
∏
∏
∏
∏
=
=
=
=
−
−
+
+
+
+
+
−
+
+
+
−
ηυ
ϕ
υ
η
ϕ
ηυ
ϕ
υ
η
ϕ
(5.1)
subject to the following conditions given by
)
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 2 B B r 1 1 2 2 B Br
g
g
g
g
g
g
S
h
h
h
h
h
h
S
+
+
+
=
−
+
−
+
−
+
(5.2))
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 2 2 2 1 1 2 22
2B
g
g
g
g
g
Bg
BS
Bh
h
h
h
h
Bh
BS
−+
+
+
≠
−−
+
−
+
−
+
(5.3)0
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 21
2B+
g
+
g
g
+
g
g
B+
g
B≠
S
(5.4)where
2
,
2
,
2
υ
η
ϕ
* $ %
+ , $ ! , ! #
+
+
−
+
−
+
−
−
+
+
+
−−
3 2
2 1
1 2
2 2
2 1 1 2
2
(
2
,
,
1
,
,
1
,
,
,
1
)
(
1
,
,
1
,
,
,
1
,
)}(
2
)
[{
S
Bg
g
g
g
g
Bg
BS
Bh
h
h
h
h
Bh
Bm
+
+
−
+
−
+
−
−
+
+
+
+
− − 22 2 1
1 1
2 2
2 1 1 1
2
(
2
,
,
1
,
,
1
,
,
,
1
)
(
1
,
,
1
,
,
,
1
,
)}(
2
)
{
S
Bg
g
g
g
g
Bg
BS
Bh
h
h
h
h
Bh
Bm
+
+
−
+
−
+
−
−
+
+
+
+
{
S
2B(
2
,
g
1,
1
g
1,
g
2,
1
g
2,
,
g
B,
1
g
B)
S
2B(
1
h
1,
h
1,
1
h
2,
h
2,
,
1
h
B,
h
B)}(
2
m
)
0
)}]
1
,
,
,
1
,
,
1
,
,
2
(
{
2 1 1+
1 2+
2+
=
+
S
B+g
g
g
g
g
Bg
B (5.5) whenr
=
1
,
2
,
3
,
,
(
2
B
−
3
);
θ
,
δ
∈
{
0
,
1
},
B
,
P
∈
{
2
,
3
,
4
,
}
andR
∈
{
3
,
4
,
5
,
}.
(5.6) andW
1,
W
2,
W
3,
W
4are given by the equations (2.2), (2.3), (2.4), (2.5) respectively.Deduction of (5.1)
When
R
,
P
→
∞
in (5.1), then−
+
−
−
−
−
=
−
−
−
−
+
−
−
−
∏
∏
=
= +
+ 6
1
1 5
3 3
)
1
(
)
(
)
1
)(
1
(
)
1
(
1
;
,
,
),
(
1
;
1
,
1
,
1
),
(
W
h
g
W
h
g
H
Bj
j B
j j
B B B B
υ
η
ϕ
υ
η
ϕ
υ
η
ϕ
υ
η
ϕ
8 1
2 1
2 7
1
1
)
1
(
)
(
)
2
)(
2
(
)
2
(
)
1
(
)
(
)
1
)(
1
(
)
1
(
W
g
h
W
g
h
B
j
j B
j j
B
j
j B
j j
∏
∏
∏
∏
=
=
=
=
−
−
+
+
+
+
+
−
+
+
+
−
ηυ
ϕ
υ
η
ϕ
ηυ
ϕ
υ
η
ϕ
(5.7)
subject to the conditions (5.2)-(5.6) and
W
5,
W
6,
W
7,
W
8 are given by the equations (2.18), (2.19), (2.20), (2.21) respectively.6. COMPANION OF THIRD THEOREM OF QURESHI AND QURAISHI:
If we apply third theorem of authors[6,equations (3.6.1)-(3.6.4)] and proceed on the same parallel lines of preceding sections, we can obtain
θ
δ
−
− +
+
−
Ξ
−
Ξ
−
Ξ
−
+
Ξ
−
Ξ
−
Ξ
−
RP E
B
E B
E B E B
h
g
H
2
2 2
1 2 1
1
;
,
,
,
),
(
1
;
1
,
,
1
,
1
),
(
4 1
2 1
1
2 1
3 1
1
1 1
2 1
1
1 1
1
)
1
(
)
(
)
(
)
2
(
)
1
(
)
(
)
(
)
1
(
)
1
(
)
(
)
(
)
1
(
W
g
h
W
g
h
W
h
g
W
Bj
j E
j j
B
j j E
j
j
B
j
j E
j j
B
j j E
j
j
B
j
j E
j j
B
j j E
j j
∏
∏
∏
∏
∏
∏
∏
∏
∏
∏
∏
∏
= =
= =
= =
= =
= =
= =
−
Ξ
−
Ξ
+
+
+
−
Ξ
Ξ
+
−
+
Ξ
−
Ξ
−
=
(6.1)subject to the following conditions given by
)
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 2 B B r 1 1 2 2 B Br
g
g
g
g
g
g
S
h
h
h
h
h
h
S
+
+
+
=
−
+
−
+
−
+
(6.2))
,
1
,
,
,
1
,
,
1
(
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 2 2 1 1 1 2 21
2B E
g
g
g
g
g
Bg
BS
B Eh
h
h
h
h
Bh
BS
− ++
+
+
≠
− +−
+
−
+
−
+
(6.3)0
)
1
,
,
,
1
,
,
1
,
,
2
(
1 1 2 21
2B+
g
+
g
g
+
g
g
B+
g
B≠
S
(6.4)where
2
,
,
2
,
2
2
1
Ξ
Ξ
EΞ
* $ %
+ , $ ! , ! #
+
+
−
+
−
+
−
−
+
+
+
− ++ −
E B
B E
B B B E
B
g
g
g
g
g
g
S
h
h
h
h
h
h
m
S
(
2
,
,
1
,
,
1
,
,
,
1
)
(
1
,
,
1
,
,
,
1
,
)}(
2
)
[{
2 1 1 1 2 2 2 1 1 1 2 2+ +
− +
− +
− −
+ +
+ − + −
+ −
1 2
2 1
1 2
2 2
2 1 1 2
2 (2, ,1 , ,1 , , ,1 ) ( 1 , , 1 , , , 1 , )}(2 )
{S B E g g g g gB gB S B E h h h h hB hB m E
+
+
−
+
−
+
−
−
+
+
+
+
+
− − 22 2 1
1 1
2 2
2 1 1 1
2
(
2
,
,
1
,
,
1
,
,
,
1
)
(
1
,
,
1
,
,
,
1
,
)}(
2
)
{
S
Bg
g
g
g
g
Bg
BS
Bh
h
h
h
h
Bh
Bm
+
+
−
+
−
+
−
−
+
+
+
+
{
S
2B(
2
,
g
1,
1
g
1,
g
2,
1
g
2,
,
g
B,
1
g
B)
S
2B(
1
h
1,
h
1,
1
h
2,
h
2,
,
1
h
B,
h
B)}(
2
m
)
0
)}]
1
,
,
,
1
,
,
1
,
,
2
(
{
2 1 1+
1 2+
2+
=
+
S
B+g
g
g
g
g
Bg
B (6.5) whenr
=
1
,
2
,
3
,
,
(
2
B
−
E
);
θ
,
δ
∈
{
0
,
1
},
B
∈
{
1
,
2
,
3
,
},
P
∈
{
2
,
3
,
4
,
},
R
∈
{
3
,
4
,
5
,
},
E
<
2
B
(6.6) andW
1,
W
2,
W
3,
W
4are given by the equations (2.2), (2.3), (2.4), (2.5) respectively.Deduction of (6.1)
When
R
,
P
→
∞
in (6.1), then−
Ξ
−
Ξ
−
Ξ
−
+
Ξ
−
Ξ
−
Ξ
−
+
+
1
;
,
,
,
),
(
1
;
1
,
,
1
,
1
),
(
2 1
2 1
E B
E B
E B E B
h
g
H
8 1
2 1
1
2 1
7 1
1
1 1
6 1
1
1 1
5
)
1
(
)
(
)
(
)
2
(
)
1
(
)
(
)
(
)
1
(
)
1
(
)
(
)
(
)
1
(
W
g
h
W
g
h
W
h
g
W
Bj
j E
j j
B
j j E
j
j
B
j
j E
j j
B
j j E
j
j
B
j
j E
j j
B
j j E
j j
∏
∏
∏
∏
∏
∏
∏
∏
∏
∏
∏
∏
= =
= =
= =
= =
= =
= =
−
Ξ
−
Ξ
+
+
+
−
Ξ
Ξ
+
−
+
Ξ
−
Ξ
−
=
(6.7)subject to the conditions (6.2)-(6.6) and
W
5,
W
6,
W
7,
W
8are given by the equations (2.18), (2.19), (2.20), (2.21) respectively.REFERENCES:
1. Agarwal, R. P.; Resonance of Ramanujan's Mathematics, Vol. I, New Age International Pvt. Ltd., New Delhi, 1996. 2. Bailey, W. N.; Series of Hypergeometric Type which are Infinite in Both Directions, Quart. J. Math. (Oxford), 7(1936),
105-115.
3. Bromwich, T. J. I.’A; An Introduction to the Theory of Infinite Series, Third Edition, Chelsea Publishing Company, New York, 1991.
4. Dougall, J.; On Vandermonde's Theorem and Some More General Expansions, Proc. Edin. Math. Soc., 25(1907), 114-132. 5. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.; Higher Transcendental Functions, Vol. I(Bateman
Manuscript Project), McGraw-Hill Book Co. Inc., New York, Toronto and London, 1953.
6. Qureshi, M. I. and Quraishi, Kaleem A.; Unification and Generalization of Certain Results on Truncated Unilateral Hypergeometric Series, Accepted for Publication in International Transactions in Applied Sciences.
7. Rainville, E. D.; Special Functions, The Macmillan Co. Inc., New York 1960; Reprinted by Chelsea Publ. Co. Bronx, New York, 1971.
8. Slater, L. J.; A Note on the Partial Sum of a Certain Hypergeometric Series, Math. Gaz., 39 (1955), 217-218.
