# A Generalized Inverse Binomial Summation Theorem and Some Hypergeometric Transformation Formulas

Authors: S. M. Ripon
Journal: International Journal of Combinatorics
 (2016)

---

## Abstract

A generalized binomial theorem is developed in terms of Bell polynomials and by applying this identity some sums involving inverse binomial coefficient are calculated. A technique is derived for calculating a class of hypergeometric transformation formulas and also some curiousq series identities.


---

## Body

## 1. Introduction

Some of the most recent developments are on the use of different techniques for obtaining sums of hypergeometric series. In this paper, we present a new method for calculating the following summations and also a generalized theorem related to these series is investigated. We investigated the following summations formula with some restrictions for the functionsf
(
k
,
y
):(1)
∑
k
=
1
∞
-
1
m
k
x
k
+
n
k
n
+
1
m
f
k
,
y
,
∑
k
=
1
∞
-
1
m
k
x
k
+
n
k
n
+
1
m
.We assume that the function f
(
k
-
n
,
y
) has no poles at (
n
-
k
+
l
), where l is an integer ranged from 1
≤
l
≤
k.As it turns out, the above summation formula for the constant functionf
(
k
,
y
)
=
1 gives us a new generalized hypergeometric transformation formula of the type(2)
F
m
+
1
m
1
1
1
⋯
1
︷
m
+
1
n
+
2
⋯
n
+
2
︸
m
;
-
1
m
x
⟶
d
m
d
r
m
F
m
m
-
1
-
r
⋯
-
r
︷
m
1
1
1
⋯
1
︸
m
-
1
;
-
x
r
=
n
.We further investigated some q series closely related to the famous Roger-Ramanujan identities.(3)
∑
v
=
1
∞
q
v
-
1
q
r
+
v
a
q
r
v
.In our present investigation dealing with the series identity, we shall also make use of other such higher transcendental functions, the Riemann Zeta function and Hurwitz Zeta function, which are defined by(4)
ζ
s
=
∑
k
=
1
∞
1
k
s
,
ζ
s
,
a
+
1
=
∑
k
=
1
∞
1
k
+
a
s
=
ζ
s
-
H
a
s
,where H
a
(
s
) is the generalized harmonic numbers defined by(5)
H
a
s
=
∑
k
=
1
a
1
k
s
=
-
1
s
Γ
s
ψ
s
-
1
1
-
ψ
s
-
1
a
+
1
;
for
s
≥
1
,
a
≥
0
.In the above identity we used the ψ notation for denoting the generalized polygamma function of order m which is given by the m
+
1th times logarithmic derivative of Gamma function(6)
ψ
m
z
=
d
m
d
z
m
ψ
1
z
=
d
m
+
1
d
z
m
+
1
ln
⁡
Γ
z
;
ψ
1
=
-
γ
.
γ denotes the famous Euler-Mascheroni constant. The derivatives of generalized harmonic number are given:(7)
d
m
d
n
m
H
n
p
=
-
1
m
+
1
Γ
m
+
p
Γ
p
ζ
m
+
p
,
n
+
1
. And we used (
λ
)
v for the Pochhammer symbol defined (for λ
,
v
∈
C and in terms of the Gamma function) by(8)
λ
v
=
Γ
λ
+
v
Γ
λ
=
λ
λ
+
1
⋯
λ
+
v
-
1
;
λ
0
=
1
. The generalized hypergeometric function is defined by(9)
p
F
q
a
1
a
2
⋯
a
p
b
1
b
2
⋯
b
q
;
z
=
∑
k
=
0
∞
a
1
k
a
2
k
⋯
a
p
k
b
1
k
b
2
k
⋯
b
q
k
z
k
k
!
.The notation for generalized hypergeometric functions was introduced by Pochhammer in 1870 and modified by Barnes [1] and later by Maier and Slater [2]. A number of notational variations are commonly used. Most common notation is introduced by Graham et al. [3] using square brackets and a semicolon.The complete Bell polynomials of ordern are defined as(10)
Y
n
x
1
,
x
2
,
…
,
x
n
=
∑
k
1
+
2
k
2
+
3
k
3
+
⋯
+
n
k
n
=
n
n
!
k
1
!
k
2
!
⋯
k
n
!
x
1
1
!
k
1
x
2
2
!
k
2
⋯
x
n
n
!
k
n
. First few terms of these polynomials can be derived by (11)
Y
1
x
1
=
x
1
,
Y
2
x
1
,
x
2
=
x
1
2
+
x
2
,
Y
3
x
1
,
x
2
,
x
3
=
x
1
3
+
3
x
1
x
2
+
x
3
.


## 2. Main Results

Theorem 1.
For integersm
>
0
,
n
⩾
0, (12)
∑
k
=
1
∞
-
1
m
k
x
k
+
n
k
n
+
1
m
f
k
,
y
=
-
1
m
m
!
n
!
m
F
m
m
n
,
x
,
y
-
∑
k
=
0
n
∑
j
=
0
m
-
1
j
x
k
m
!
n
!
m
n
k
m
m
j
f
m
-
j
k
-
n
,
y
Y
m
m
B
1
,
…
,
m
B
m
,where [
F
m
(
r
,
x
,
y
)
=
∑
k
=
0
∞
x
k
r
k
m
f
(
k
-
r
,
y
) and F
m
(
m
)
(
n
,
x
,
y
)
=
d
m
/
d
r
m
F
m
r
,
x
,
y
r
=
n.Proof.
Let us define a functionF
m
(
r
,
x
,
y
) where r
,
x
,
y
∈
R such that(13)
F
m
r
,
x
,
y
=
∑
k
=
0
∞
x
k
r
k
m
f
k
-
r
,
y
.We also assume the function f
(
k
-
r
,
y
) has no poles at (
n
-
k
+
l
) for each l and m such that(14)
1
⩽
l
⩽
k
,
m
∈
Z
+
.We introduce another notation for the mth derivative with respect to r of the defined function F
m
(
r
,
x
,
y
):(15)
F
m
m
n
,
x
,
y
=
d
m
d
r
m
F
m
r
,
x
,
y
r
=
n
.Furthermore, we see the series also defined for r. It can be extended to the interval [
0
,
∞
], because when k
→
(
r
+
l
), where r is a positive integer, the further terms of the series just vanish.(16)
l
i
m
k
→
r
+
l
r
k
m
f
k
-
r
,
y
=
0
.Making use of certain special properties of Bell polynomials we can evaluate successive derivative of a given function. Let us consider a function f
(
x
,
y
,
z
) which has a Taylor series expansion around x; the detailed procedure of these kinds is extensively discussed in the paper [4]. Following the same process discussed in [5, 6] we can write the following elegant identity by virtue of Bell polynomials:(17)
d
n
d
r
n
e
f
r
,
x
,
y
=
e
f
r
,
x
,
y
Y
n
f
1
r
,
x
,
y
,
f
2
r
,
x
,
y
,
…
,
f
n
r
,
x
,
y
. The following identity can be recovered from the same method used in [5, page 8]:(18)
d
j
d
r
j
r
k
m
=
r
k
m
Y
j
m
H
r
-
H
r
-
k
,
m
H
r
1
-
H
r
-
k
1
,
…
,
m
H
r
j
-
1
-
H
r
-
k
j
-
1
.
H
r
j represents the jth order derivative harmonic number H
r
(
1
) with respect to r. The derivative of harmonic numbers can be evaluated by using the formula given in(19)
H
r
j
-
1
-
H
r
-
k
j
-
1
=
d
j
-
1
d
r
j
-
1
H
r
1
-
H
r
-
k
1
r
=
n
=
-
1
j
-
1
Γ
j
H
n
j
-
H
n
-
k
j
=
B
j
. Hence, upon considering (18) and (19), we find(20)
d
j
d
r
j
r
k
m
r
=
n
=
n
k
m
Y
j
m
B
1
,
m
B
2
,
…
,
m
B
j
.The above identity was derived extensively in [5] and a modified version was calculated in [7]. Now differentiating (13) m times with respect to r and considering the case r
=
n, where r is a positive integer as well as using the definition from (15), we can derive(21)
F
m
m
n
,
x
,
y
=
d
m
d
r
m
F
m
r
,
x
,
y
r
=
n
=
∑
k
=
0
∞
∑
j
=
0
m
x
k
m
j
d
j
d
r
j
r
k
m
r
=
n
d
m
-
j
d
r
m
-
j
f
k
-
r
,
y
r
=
n
.Making use of the identity derived in (20) we find(22)
F
m
m
n
,
x
,
y
=
∑
k
=
0
n
∑
j
=
0
m
x
k
m
j
f
m
-
j
k
-
r
,
y
n
k
m
Y
j
m
B
1
,
m
B
2
,
…
,
m
B
j
+
∑
k
=
n
+
1
∞
∑
j
=
0
m
x
k
m
j
f
m
-
j
k
-
r
,
y
n
k
m
Y
j
m
B
1
,
m
B
2
,
…
,
m
B
j
.From the asymptotic properties of Bell polynomials we have(23)
Y
j
f
1
x
,
f
2
1
x
,
…
,
f
j
1
x
=
f
j
1
x
∑
k
1
+
2
k
2
+
⋯
+
j
k
j
=
j
G
k
1
,
k
2
,
…
,
k
j
.Considering the above property in (23), for m
≥
1 and n
,
l
∈
Z
+ are both positive integers, we can readily evaluate the following limits:(24)
l
i
m
k
→
n
+
l
n
k
m
Y
1
m
Γ
1
H
n
1
-
H
n
-
k
1
=
0
,
l
i
m
k
→
n
+
1
n
k
m
Y
2
m
Γ
1
H
n
1
-
H
n
-
k
1
,
-
m
Γ
2
H
n
2
-
H
n
-
k
2
=
0
. Considering next few cases, we can move to final relation(25)
l
i
m
k
→
n
+
1
n
k
m
Y
m
-
1
m
B
1
,
m
B
2
,
…
,
m
B
m
-
1
=
0
,where B
m is defined as in (19). Recalling the properties of Bell polynomials mentioned in (25) if we consider the limit k
→
(
n
+
l
) for l
=
1,2
,
…
,
∞ we observe that for every 0
≤
j
≤
m the limiting value vanishes but it surprisingly gives us limiting value for j
=
m. (26)
l
i
m
k
→
n
+
l
∑
j
=
0
m
x
k
m
j
f
m
-
j
k
-
r
,
y
n
k
m
Y
j
m
B
1
,
…
,
m
B
j
=
0
+
l
i
m
k
→
n
+
l
x
k
m
m
f
m
-
m
k
-
r
,
y
n
k
m
Y
m
m
B
1
,
m
B
2
,
…
,
m
B
m
=
x
n
+
l
f
l
,
y
l
i
m
k
→
n
+
l
n
k
m
Y
m
m
B
1
,
m
B
2
,
…
,
m
B
m
.Furthermore, expanding the product of generalized harmonic numbers in asymptotic form with Big O notation, we can further deduce(27)
H
n
1
-
H
n
-
k
1
m
1
H
n
2
-
H
n
-
k
2
m
2
⋯
H
n
i
-
H
n
-
k
i
m
i
=
1
n
-
k
+
1
u
+
O
1
n
-
k
+
1
v
+
⋯ with u
=
m
1
+
2
m
2
+
⋯
+
i
m
i and m
1
+
2
m
2
+
⋯
+
i
m
i
>
v. Substituting the above asymptotic identity in (26) we find(28)
l
i
m
k
→
n
+
l
n
k
m
Y
m
m
B
1
,
…
,
m
B
m
=
l
i
m
k
→
n
+
l
n
k
m
1
n
-
k
+
1
m
+
O
1
n
-
k
+
1
m
-
1
+
⋯
Y
m
m
Γ
1
,
…
,
-
1
m
-
1
m
Γ
m
. After some tedious calculations we find(29)
l
i
m
k
→
n
+
l
n
k
m
Y
m
m
B
1
,
…
,
m
B
m
=
-
1
l
-
1
Γ
l
Γ
n
+
1
Γ
n
+
l
+
1
m
Y
m
m
Γ
1
,
…
,
-
1
m
-
1
m
Γ
m
.In view of the known identity for Bell polynomials,(30)
Y
m
m
Γ
1
,
…
,
-
1
m
-
1
m
Γ
m
=
m
!
.We can finally obtain a closed form for the limit(31)
l
i
m
k
→
n
+
l
n
k
m
Y
m
m
B
1
,
m
B
2
,
…
,
m
B
m
=
-
1
l
-
1
n
!
l
n
+
1
m
m
!
.Substituting the limiting value of (26) in (31), we can readily obtain a closed form for the required limit(32)
l
i
m
k
→
n
+
l
∑
j
=
0
m
x
k
m
j
f
m
-
j
k
-
r
,
y
n
k
m
Y
j
m
B
1
,
m
B
2
,
…
,
m
B
j
=
x
n
+
l
f
l
,
y
-
1
l
-
1
n
!
l
n
+
1
m
m
!
.Applying the above limiting value, we have(33)
∑
k
=
n
+
1
∞
∑
j
=
0
m
x
k
m
j
f
m
-
j
k
-
r
,
y
n
k
m
Y
j
m
B
1
,
m
B
2
,
…
,
m
B
j
=
m
!
∑
k
=
1
∞
x
n
+
k
f
k
,
y
-
1
k
-
1
n
!
k
n
+
1
m
=
-
1
m
m
!
n
!
m
∑
k
=
1
∞
-
1
m
k
x
k
+
n
k
n
+
1
m
f
k
,
y
.Finally by combining (22) and (33), we finally obtain Theorem 1.Theorem 2.
For integersm
>
0
,
n
⩾
0,(34)
∑
k
=
1
∞
-
1
m
k
x
k
+
n
k
n
+
1
m
f
k
+
n
,
y
=
-
1
m
m
!
n
!
m
Q
m
m
n
,
x
,
y
-
∑
k
=
0
n
x
k
f
k
,
y
n
k
m
Y
m
m
B
1
,
m
B
2
,
…
,
m
B
m
,where Q
m
(
r
,
x
,
y
)
=
∑
k
=
0
∞
x
k
r
k
m
f
(
k
,
y
) and Q
m
(
m
)
(
n
,
x
,
y
)
=
d
m
/
d
r
m
Q
m
(
r
,
x
,
y
)
r
=
n.Proof.
The proof is similar to the previous one. Similarly as before we consider the function(35)
Q
m
r
,
x
,
y
=
∑
k
=
0
∞
x
k
r
k
m
f
k
,
y
.Then by the same process and with same restrictions we can easily obtain Theorem 2.Theorem 3.
For integersm
>
0
,
n
⩾
0, and |
x
|
<
1, (36)
F
m
m
n
,
x
=
d
m
d
r
m
F
m
m
-
1
-
r
⋯
-
r
︷
m
1
1
1
⋯
1
︸
m
-
1
;
-
x
r
=
n
=
m
!
n
!
m
-
1
m
∑
k
=
1
∞
-
1
m
k
x
k
+
n
k
n
+
1
m
+
∑
k
=
0
n
x
k
n
k
m
Y
m
m
B
1
,
m
B
2
,
…
,
m
B
m
=
x
n
+
1
m
!
n
+
1
m
F
m
+
1
m
1
1
1
⋯
1
︷
m
+
1
n
+
2
⋯
n
+
2
︸
m
;
-
1
m
x
+
∑
k
=
0
n
x
k
n
k
m
Y
m
m
B
1
,
m
B
2
,
…
,
m
B
m
.Proof.
Let us takef
(
k
,
y
)
=
1 as a constant function in Theorem 1:(37)
∑
k
=
1
∞
-
1
m
k
x
k
+
n
k
n
+
1
m
=
-
1
m
m
!
n
!
m
F
m
m
n
,
x
-
∑
k
=
0
n
x
k
n
k
m
Y
m
m
B
1
,
m
B
2
,
…
,
m
B
m
.It proves the first part of Theorem 3. Writing the function F
m
(
m
)
(
n
,
x
) and F
m
(
n
,
x
) as Hyper geometric functions defined in first section we can derive(38)
m
!
n
!
m
-
1
m
∑
k
=
1
∞
-
1
m
k
x
k
+
n
k
n
+
1
m
=
x
n
+
1
m
!
n
+
1
m
F
m
+
1
m
1
1
1
⋯
1
︷
m
+
1
n
+
2
⋯
n
+
2
︸
m
;
-
1
m
x
,
F
m
r
,
x
=
∑
k
=
0
∞
x
k
r
k
m
=
F
m
m
-
1
-
r
⋯
-
r
︷
m
1
1
1
⋯
1
︸
m
-
1
;
-
x
.Finally combining (37) and (38) we can easily obtain Theorem 3.Theorem 4.
For integersn
⩾
0 and |
x
|
<
1, (39)
∑
k
=
1
∞
x
k
+
n
k
n
+
1
H
n
+
k
1
=
∑
k
=
1
n
x
k
n
!
n
k
H
k
1
H
n
1
-
H
n
-
k
1
+
1
1
+
x
n
!
Φ
1
1
+
x
,
2
,
n
+
1
-
1
+
x
n
n
!
H
n
1
ln
⁡
1
+
x
+
ln
⁡
1
+
x
ln
⁡
x
1
+
x
+
ζ
2
,
n
+
1
.Proof.
Considering Theorem2 for m
=
1
,
y
=
1
,
f
(
k
,
y
)
=
H
k
(
1
) and using the property of bell polynomials,(40)
∑
k
=
1
∞
x
k
+
n
k
n
+
1
H
n
+
k
1
=
-
1
n
!
d
d
r
Q
1
r
,
x
,
1
r
=
n
-
∑
k
=
0
n
x
k
n
k
H
k
1
H
n
1
-
H
n
-
k
1
.Taking into account an identity derived in Boyadzhiev’s paper [8] titled “Harmonic Number Identities via Euler’s Transform,” (41)
Q
1
r
,
x
,
1
=
H
r
1
1
+
x
r
-
∑
k
=
1
r
1
+
x
r
-
k
k
=
H
r
1
1
+
x
r
+
1
+
x
r
ln
⁡
x
1
+
x
-
∑
k
=
1
∞
1
+
x
-
k
k
+
r
.Differentiating the identity in (41) with respect to r and using the known formula d
/
d
r
H
r
(
1
)
r
=
n
=
ζ
(
2
,
n
+
1
), (42)
d
d
r
Q
1
r
,
x
,
1
r
=
n
=
1
+
x
n
H
n
1
ln
⁡
1
+
x
+
ln
⁡
1
+
x
ln
⁡
x
1
+
x
+
ζ
2
,
n
+
1
-
∑
k
=
0
∞
1
+
x
-
k
-
1
k
+
n
+
1
2
=
1
+
x
n
H
n
1
ln
⁡
1
+
x
+
ln
⁡
1
+
x
ln
⁡
x
1
+
x
+
ζ
2
,
n
+
1
-
1
1
+
x
Φ
1
1
+
x
,
2
,
n
+
1
.Considering the above expression together with (40) we can conclude Theorem 2. Special case of Theorem 2 for x
=
1 can be found in [5]. Theorem 2 also generalizes many identities derived in [9]. The finite summation in the right-hand side of Theorem 2 can be obtained from [7, 10] for different values of x. Interested readers can also find some computer assisted proofs of these identities in [11].Theorem 5.
For integersn
≥
0,(43)
∑
k
=
1
∞
-
1
k
k
n
+
1
2
H
n
+
k
1
=
1
n
!
2
∑
k
=
1
n
n
k
2
H
k
1
H
n
2
-
H
n
-
k
2
-
2
H
n
1
-
H
n
-
k
1
2
+
1
n
!
2
2
n
n
4
H
2
n
1
-
H
n
1
H
2
n
2
-
H
n
2
+
2
H
2
n
3
-
3
H
2
n
3
+
ζ
3
+
1
n
!
2
2
n
n
2
H
n
1
-
H
2
n
1
2
H
2
n
1
-
H
n
1
2
+
H
n
1
-
H
2
n
1
+
ζ
2
.Proof.
Settingm
=
2
,
y
=
2
,
x
=
1, andf
(
k
,
y
)
=
H
k
(
2
) in Theorem 2 and by virtue of Bell polynomials(44)
∑
k
=
1
∞
-
1
k
k
n
+
1
2
H
n
+
k
1
=
1
2
n
!
2
d
2
d
r
2
Q
2
r
,
1,1
r
=
n
+
∑
k
=
1
n
n
k
2
H
k
1
2
H
n
2
-
H
n
-
k
2
-
4
H
n
1
-
H
n
-
k
1
2
.The following harmonic number identity can be found in many texts of mathematical literatures. Mainly Chu and De Donno [12] and Paule and Schneider [11] discussed these types of summation formulas in great detail.(45)
Q
2
r
,
1,1
=
∑
k
=
1
r
r
k
2
H
k
1
=
2
r
r
2
H
r
1
-
H
2
r
1
.Differentiating (45) two times with respect to r and using the formula involving differentiation of generalized harmonic numbers stated in the first section, (46)
d
2
d
r
2
Q
2
r
,
1,1
r
=
n
=
d
2
d
r
2
2
r
r
2
H
r
1
-
H
2
r
1
r
=
n
=
2
n
n
8
H
2
n
1
-
H
n
1
H
2
n
2
-
H
n
2
+
4
H
2
n
3
-
6
H
2
n
3
+
2
ζ
3
+
2
n
n
2
2
H
n
1
-
H
2
n
1
2
H
2
n
1
-
H
n
1
2
+
H
n
1
-
H
2
n
1
+
ζ
2
.Finally substituting the expressions of (45) and (46) in (44), we easily obtain Theorem 5.Some Special CorollariesCorollary 6.
For integersn
≥
0
,
p
≤
n, (47)
x
n
+
1
n
+
1
n
+
1
p
F
2
1
1
1
n
-
p
+
2
;
-
x
=
∑
k
=
1
n
x
k
n
k
k
p
H
n
-
k
1
-
H
n
1
+
x
p
1
+
x
n
-
p
n
p
ln
⁡
1
+
x
+
H
n
1
-
H
n
-
p
1
.Using the following identity (48) given in Volume 2 of Gould’s book [13] and considering Theorem 2 for m
=
1; f
(
k
,
p
)
=
k
p and(48)
∑
k
=
0
n
x
k
n
k
k
p
=
x
p
1
+
x
n
-
p
n
p
.Some special cases of this formula can be found in the published literature [14–16].Corollary 7.
For integersn
≥
0
,
p
≥
1,(49)
∑
i
=
0
p
-
1
-
z
i
n
+
p
+
1
i
F
2
1
1
1
n
+
p
-
i
+
2
;
1
=
1
-
z
n
-
p
+
1
F
2
1
1
1
n
+
p
+
2
;
z
+
n
+
p
+
1
-
z
n
+
p
ln
⁡
1
+
x
+
H
n
1
-
H
n
+
p
1
-
∑
k
=
0
n
-
1
k
n
k
n
+
p
+
1
p
k
+
p
p
H
n
1
-
H
n
-
k
1
1
-
z
k
+
p
-
∑
i
=
0
p
-
1
-
z
i
k
+
p
i
.Considering the following identity (50) illustrated in Volume 2 of Gould’s book [13] and using Theorem 2 for p
≥
1
x
=
-
1
,
m
=
1
;
f
(
k
,
p
,
z
)
=
1
/
k
+
p
k
(
1
-
z
)
k
+
p
-
∑
i
=
0
p
-
1
(
-
z
)
i
k
+
p
i,(50)
∑
k
=
1
n
-
1
k
n
k
f
k
,
p
,
z
=
-
1
p
n
+
p
p
z
n
+
p
.Corollary 8.
For integersn
≥
0 and p
≥
0,(51)
-
1
n
n
+
1
p
+
1
∏
i
=
1
α
Y
i
α
+
p
+
1
F
α
+
p
1
1
n
+
2
⋯
n
+
2
︷
p
-
1
Y
1
⋯
Y
α
︷
α
n
+
1
⋯
n
+
1
︸
p
Y
1
+
1
⋯
Y
α
+
1
︸
α
;
1
=
∑
j
=
1
α
X
j
p
-
1
H
n
1
-
H
n
-
X
j
1
n
!
n
-
X
j
n
∏
i
=
1
,
i
≠
j
α
X
j
-
X
i
+
∑
k
=
1
n
-
1
k
k
p
n
k
n
!
∏
i
=
1
α
k
-
X
i
H
n
1
-
H
n
-
k
1
.Proof.
The following identity is given in Volume 5 of Gould’s book [13]:(52)
∑
k
=
1
n
-
1
k
k
p
n
k
n
!
∏
i
=
1
α
k
-
X
i
=
-
∑
j
=
1
α
X
j
p
-
1
n
-
X
j
n
∏
i
=
1
,
i
≠
j
α
X
j
-
X
i
.Considering Theorem 2 for x
=
-
1
,
m
=
1; f
(
k
,
X
i
)
=
k
p
/
∏
i
=
1
α
k
-
X
i and also using (52) we can immediately find Corollary 8.Corollary 9.
For integersn
≥
0
,
b
≥
c, b, and c are positive integers,(53)
F
3
2
1
1
n
+
b
-
c
+
2
n
+
2
n
+
b
+
2
;
1
=
-
1
n
n
+
1
n
+
b
+
1
c
∑
k
=
1
n
-
1
k
n
k
k
+
b
c
H
n
1
-
H
n
-
k
1
-
-
1
n
n
+
1
n
+
b
+
1
c
-
c
n
+
c
2
n
+
b
b
-
c
+
c
n
+
c
n
+
b
b
-
c
H
n
-
c
1
-
H
n
+
b
1
.Using the following identity (54) given in Volume 5 of Gould’s book [13] and considering Theorem 2 for x
=
-
1
,
m
=
1
;
f
(
k
,
b
,
c
)
=
1
/
k
+
b
c
(54)
∑
k
=
1
n
-
1
k
n
k
k
+
b
c
=
c
n
+
c
n
+
b
b
-
c
.Terminating version of these kinds of hypergeometric series goes back to Bailey [15].Differentiation of Laguerre Polynomials with respect to Its Order. Let L
n
(
x
) be the Laguerre Polynomials defined by(55)
L
n
x
=
∑
k
=
1
n
-
1
k
n
k
x
k
k
!
.Corollary 10.
For integersn
≥
0,(56)
d
d
r
L
r
x
r
=
n
=
-
x
n
+
1
n
+
1
n
+
1
!
F
2
2
1
1
n
+
2
n
+
2
;
x
+
∑
k
=
1
n
-
x
k
k
!
n
k
H
n
1
-
H
n
-
k
1
.Using the previous identity (55) for Laguerre Polynomials and considering Theorem 2 for x
=
-
x
,
m
=
1
;
f
(
k
)
=
1
/
k
! we can easily derive Corollary 10.Corollary 11.
For integersm
≥
1
,
n
≥
0, and p
=
m
+
n, (57)
F
4
3
1
1
p
p
n
+
2
p
+
1
p
+
1
;
1
=
-
1
n
+
1
n
+
1
m
+
n
2
1
m
p
n
H
p
1
-
H
m
-
1
1
H
n
1
-
H
p
1
+
ζ
2
-
H
p
2
+
-
1
n
n
+
1
m
+
n
2
∑
k
=
1
n
-
1
k
k
+
m
2
n
k
H
n
1
-
H
n
-
k
1
.Making use of the following identity (58) derived in [17] and considering Theorem 2 for x
=
-
1
,
m
=
1
;
f
(
k
,
m
)
=
1
/
(
k
+
m
)
2,(58)
∑
k
=
1
n
-
1
k
k
+
m
2
n
k
=
1
m
m
+
n
n
H
m
+
n
1
-
H
m
-
1
1
.Corollary 12.
For integersm
≥
1
,
n
≥
0, and p
=
m
+
n, (59)
F
5
4
1
1
p
p
p
n
+
2
p
+
1
p
+
1
p
+
1
;
1
=
-
1
n
n
+
1
m
+
n
3
∑
k
=
1
n
-
1
k
k
+
m
3
n
k
H
n
1
-
H
n
-
k
1
-
-
1
n
n
+
1
p
3
m
p
n
ζ
2
,
p
+
1
H
p
1
-
H
m
-
1
1
+
ζ
3
,
p
+
1
-
-
1
n
n
+
1
p
3
2
m
p
n
H
n
1
-
H
p
1
H
p
1
-
H
m
-
1
1
2
+
H
p
2
-
H
m
-
1
2
.Making use of the following identity (60) derived in [17] and considering Theorem 2 for x
=
-
1
,
m
=
1
;
f
(
k
,
m
)
=
1
/
(
k
+
m
)
3,(60)
∑
k
=
1
n
-
1
k
k
+
m
3
n
k
=
1
2
m
m
+
n
n
H
m
+
n
1
-
H
m
-
1
1
2
+
H
m
+
n
2
-
H
m
-
1
2
.Corollary 13.
Forx
>
-
1 and integers n
,
p
≥
0, (61)
p
+
2
F
p
+
1
1
1
n
+
2
n
+
2
n
+
2
⋯
n
+
2
︷
p
-
1
n
+
1
⋯
n
+
1
︸
p
;
-
x
=
-
1
n
+
1
n
+
1
p
-
1
∑
k
=
0
n
x
k
n
k
k
p
H
n
1
-
H
n
-
k
1
+
-
1
n
n
+
1
p
-
1
∑
j
=
0
p
x
j
1
+
x
n
-
j
n
j
j
!
S
p
,
j
ln
⁡
1
+
x
+
H
n
1
-
H
n
-
j
1
,where S
(
p
,
j
) are Stirling numbers of the second kind. Considering the following identity (62) given in [8] for x
=
y and applying Theorem 2 for, x
=
1
,
m
=
1
;
f
(
k
,
p
)
=
k
p,(62)
∑
k
=
0
n
x
k
n
k
k
p
=
∑
j
=
0
p
x
j
1
+
x
n
-
j
n
j
j
!
S
p
,
j
.Corollary 14.
For|
x
|
<
1 and integers n
≥
0,(63)
x
n
+
1
n
+
1
F
2
1
1
1
n
+
2
;
x
=
-
1
n
+
1
1
-
x
n
ln
⁡
1
+
x
+
-
1
n
∑
k
=
0
n
-
x
k
n
k
H
n
1
-
H
n
-
k
1
.Proof.
Substitutingm
=
1 in Theorem 3,(64)
∑
k
=
1
∞
-
1
k
x
k
+
n
k
n
+
1
=
-
1
n
!
F
1
1
n
,
x
-
∑
k
=
0
n
x
k
n
k
H
n
1
-
H
n
-
k
1
.Using Newton’s binomial theorem we have(65)
F
1
r
,
x
=
∑
k
=
0
r
x
k
r
k
.This implies(66)
F
1
1
n
,
x
=
d
d
r
F
1
r
,
x
r
=
n
=
d
d
r
1
+
x
r
r
=
n
=
1
+
x
n
ln
⁡
1
+
x
.Substituting the value of F
1
(
1
)
(
n
,
x
) from (66) in (64) and after some modifications we can obtain Corollary 14. The modified version of Corollary 14 can be found in [18]. Zeilberger proved quite interesting properties of these types of hypergeometric functions in [19].Corollary 15.
For integersn
≥
0,(67)
1
n
+
1
2
F
3
2
1
1
1
n
+
2
n
+
2
;
1
=
2
n
n
2
H
2
n
1
-
H
n
1
2
+
H
n
2
-
2
H
2
n
2
+
ζ
2
-
∑
k
=
0
n
n
k
2
2
H
n
1
-
H
n
-
k
1
2
-
H
n
2
-
H
n
-
k
2
.Proof.
Substitutingm
=
2 and x
=
1 in Theorem 3 and using the property of Bell polynomials,(68)
∑
k
=
1
∞
1
k
n
+
1
2
=
-
1
2
n
!
2
F
2
2
n
,
1
-
∑
k
=
0
n
n
k
2
4
H
n
1
-
H
n
-
k
1
2
-
2
H
n
2
-
H
n
-
k
2
.Another classical result, special case of the Vandermonde Convolution, is given by(69)
F
2
r
,
1
=
∑
k
=
0
r
r
k
2
=
2
r
r
.Hence we have(70)
F
2
2
n
,
1
=
d
2
d
r
2
2
r
r
r
=
n
=
2
n
n
4
H
2
n
1
-
H
n
1
2
+
2
H
n
2
-
4
H
2
n
2
+
2
ζ
2
.Considering the value of F
2
(
2
)
(
n
,
1
) from (70) in (68) we finally recover Corollary 15.Corollary 16.
For integersn
≥
0,(71)
1
n
+
1
2
F
3
2
1
1
1
n
+
2
n
+
2
;
-
1
=
-
1
n
2
n
-
3
π
Γ
n
+
2
/
2
Γ
1
-
n
/
2
ζ
2
,
1
-
n
2
+
ζ
2
,
n
+
2
2
+
∑
k
=
0
n
-
1
n
+
k
n
k
2
2
H
n
1
-
H
n
-
k
1
2
-
H
n
2
-
H
n
-
k
2
+
-
1
n
+
1
2
n
-
1
π
Γ
n
+
2
/
2
Γ
1
-
n
/
2
ln
⁡
2
2
+
ln
⁡
2
H
-
n
+
1
/
2
1
-
H
n
/
2
1
+
H
-
n
+
1
/
2
1
-
H
n
/
2
1
2
.Proof.
Substitutingm
=
2 and x
=
-
1 in Theorem 3 and using the property of Bell polynomials,(72)
∑
k
=
1
∞
-
1
k
+
n
k
n
+
1
2
=
-
1
2
n
!
2
F
2
2
n
,
-
1
-
∑
k
=
0
n
-
1
k
n
k
2
4
H
n
1
-
H
n
-
k
1
2
-
2
H
n
2
-
H
n
-
k
2
.Considering another classical result illustrated in Gould’s book [13] Volume 5 which first appeared in American Math Monthly, for the case z
=
r, we have(73)
F
2
r
,
-
1
=
∑
k
=
0
r
-
1
k
r
k
2
=
2
r
π
Γ
r
+
2
/
2
Γ
1
-
r
/
2
. Hence we have(74)
F
2
2
n
,
-
1
=
d
2
d
r
2
2
r
π
Γ
r
+
2
/
2
Γ
1
-
r
/
2
r
=
n
=
-
1
n
2
n
-
2
π
Γ
n
+
2
/
2
Γ
1
-
n
/
2
ζ
2
,
1
-
n
2
+
ζ
2
,
n
+
2
2
+
2
n
π
Γ
n
+
2
/
2
Γ
1
-
n
/
2
ln
⁡
2
2
+
ln
⁡
2
H
-
n
+
1
/
2
1
-
H
n
/
2
1
+
H
-
n
+
1
/
2
1
-
H
n
/
2
1
2
.Substituting the value of F
2
(
2
)
(
n
,
-
1
) from (74) in (72) we can immediately obtain Corollary 16. Various special cases of Corollary 16 can be found in several works of Wimp [20, 21].Corollary 17.
For integersn
≥
0,(75)
1
n
+
1
3
F
4
3
1
1
1
1
n
+
2
n
+
2
n
+
2
;
-
1
=
-
1
n
6
F
3
3
n
,
-
1
-
∑
k
=
0
n
-
1
n
+
k
2
n
k
3
9
H
n
1
-
H
n
-
k
1
3
-
9
H
n
1
-
H
n
-
k
1
H
n
2
-
H
n
-
k
2
+
2
H
n
3
-
H
n
-
k
3
.Proof.
Substitutingm
=
3 and x
=
-
1 in Theorem 3 and using the property of Bell polynomials,(76)
∑
k
=
1
∞
-
1
k
+
n
k
n
+
1
3
=
-
1
3
!
n
!
3
F
3
3
n
,
-
1
-
∑
k
=
0
n
-
1
k
n
k
3
Y
3
m
B
1
,
m
B
2
,
m
B
3
.Considering another classical result illustrated in volume 5 of Gould’s book [13] which first appeared in American Math Monthly, considering for the case z
=
r that(77)
F
3
r
,
-
1
=
∑
k
=
0
r
-
1
k
r
k
3
=
Γ
r
+
1
Γ
-
r
π
2
Γ
3
r
+
2
/
2
Γ
-
3
r
/
2
1
+
-
1
r
=
Γ
r
+
1
Γ
-
r
2
Γ
3
r
+
2
/
2
Γ
-
3
r
/
2
+
π
Γ
2
r
+
1
Γ
-
r
2
r
+
1
Γ
4
r
+
2
/
2
Γ
-
3
r
/
2
Γ
r
-
1
/
2
=
y
1
+
y
2
, hence we have (78)
ln
⁡
y
1
=
ln
⁡
Γ
r
+
1
+
ln
⁡
Γ
-
r
-
ln
⁡
2
-
3
ln
⁡
Γ
r
+
2
2
-
ln
⁡
Γ
-
3
r
2
,
ln
⁡
y
2
=
1
2
ln
⁡
π
+
2
ln
⁡
Γ
r
+
1
+
ln
⁡
Γ
-
r
-
r
+
1
ln
⁡
2
-
4
ln
⁡
Γ
r
+
2
2
-
ln
⁡
Γ
-
3
r
2
-
ln
⁡
Γ
r
-
1
2
.Now differentiating the above identities with respect to r and using the definition of polygamma function,(79)
d
d
r
y
1
=
y
1
ψ
r
+
1
-
ψ
-
r
-
3
2
ψ
r
+
2
2
+
3
2
ψ
-
3
r
2
=
y
1
θ
0
r
,
d
d
r
y
2
=
y
2
2
ψ
r
+
1
-
ψ
-
r
-
ln
⁡
2
-
2
ψ
r
+
2
2
+
3
2
ψ
-
3
r
2
-
1
2
ψ
r
-
1
2
=
y
2
ρ
0
r
. We also define(80)
θ
1
r
=
d
d
r
θ
0
r
=
ψ
1
r
+
1
+
ψ
1
-
r
-
3
4
ψ
1
r
+
2
2
-
9
4
ψ
1
-
3
r
2
,
θ
2
r
=
d
2
d
r
2
θ
0
r
=
ψ
2
r
+
1
-
ψ
2
-
r
-
3
8
ψ
2
r
+
2
2
+
27
8
ψ
2
-
3
r
2
,
ρ
1
r
=
d
d
r
ρ
0
r
=
2
ψ
1
r
+
1
+
ψ
1
-
r
-
ψ
1
r
+
2
2
-
1
4
ψ
1
r
-
1
2
-
9
4
ψ
1
-
3
r
2
,
ρ
2
r
=
d
2
d
r
2
ρ
0
r
=
2
ψ
2
r
+
1
-
ψ
1
-
r
-
1
2
ψ
2
r
+
2
2
-
1
8
ψ
2
r
-
1
2
+
27
8
ψ
2
-
3
r
2
.Finally we have(81)
F
3
3
n
,
-
1
=
y
1
θ
0
3
n
+
3
θ
0
n
θ
1
n
+
θ
2
n
+
y
2
ρ
0
3
n
+
3
ρ
0
n
ρ
1
n
+
ρ
2
n
,
Y
3
m
B
1
,
m
B
2
,
m
B
3
=
27
H
n
1
-
H
n
-
k
1
3
-
27
H
n
1
-
H
n
-
k
1
H
n
2
-
H
n
-
k
2
+
6
H
n
3
-
H
n
-
k
3
.Compiling (76), (77), and (81) we can obtain Corollary 17.Some
q
Series Identities. q-Pochhammer symbol, also called q-shifted factorial, is a q-analog of the common Pochhammer symbol. It is defined as(82)
a
;
q
n
=
1
-
a
1
-
a
q
⋯
1
-
a
q
n
-
1
,
q
;
q
n
=
q
n
=
1
-
q
1
-
q
2
⋯
1
-
q
n
.And the q binomial coefficients also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients are(83)
n
m
q
=
q
n
q
n
-
k
q
k
.Let us define(84)
Q
q
,
n
,
m
=
d
m
d
n
m
ln
⁡
q
n
m
,
H
a
;
q
n
m
=
∑
k
=
0
n
-
1
1
1
-
a
q
k
m
;
S
q
,
m
=
∑
k
=
1
∞
q
k
1
-
q
k
k
m
-
1
,
R
q
,
a
=
∑
k
=
1
∞
a
k
1
-
q
k
.The operator q
D, used extensively in several references, was recently used by authors in [3, 8, 22].(85)
q
D
m
f
q
=
∑
k
=
0
m
m
k
q
k
f
k
q
,
q
D
m
q
k
=
k
m
q
k
.Now by the properties of Bell Polynomials we have(86)
d
m
d
n
m
q
n
m
=
d
m
d
n
m
e
ln
⁡
q
n
m
=
q
n
m
Y
m
m
Q
q
,
n
,
1
,
m
Q
q
,
n
,
2
,
…
,
m
Q
q
,
n
,
m
.We can further deduce (87)
ln
⁡
q
n
m
=
m
ln
⁡
q
n
=
m
ln
⁡
1
-
q
+
ln
⁡
1
-
q
2
+
⋯
+
ln
⁡
1
-
q
n
=
-
m
∑
k
=
1
∞
q
k
k
1
+
q
k
+
⋯
+
q
k
n
-
1
=
-
m
∑
k
=
1
∞
q
k
1
-
q
n
k
k
1
-
q
k
. This implies (88)
d
m
d
n
m
l
n
q
n
m
=
m
ln
⁡
q
m
∑
k
=
1
∞
q
k
n
+
1
k
m
k
1
-
q
k
=
m
ln
⁡
q
m
S
q
,
m
-
m
ln
⁡
q
m
∑
k
=
1
∞
k
m
-
1
q
k
1
+
q
k
+
⋯
+
q
k
n
-
1
. From the geometric series we have(89)
∑
k
=
1
∞
q
k
=
1
1
-
q
-
1
. Applying the operator q
D on both sides m times,(90)
∑
k
=
1
∞
k
m
q
k
=
1
1
-
q
-
1
+
∑
k
=
1
m
m
k
q
k
k
!
1
-
q
k
+
1
.By means of this identity we have(91)
∑
k
=
1
∞
k
m
-
1
q
k
+
⋯
+
q
n
k
=
∑
v
=
1
n
∑
k
=
1
∞
k
m
-
1
q
v
k
=
H
q
n
1
-
n
+
∑
v
=
1
n
∑
k
=
1
m
-
1
m
-
1
k
q
v
k
k
!
1
-
q
v
k
+
1
. Finally (92)
d
m
d
n
m
ln
⁡
q
n
m
=
Q
q
,
n
,
m
=
m
ln
⁡
q
m
S
q
,
m
-
m
ln
⁡
q
m
∑
k
=
1
∞
k
m
-
1
q
k
1
+
q
k
+
⋯
+
q
k
n
-
1
=
m
ln
⁡
q
m
S
q
,
m
-
m
ln
⁡
q
m
H
q
n
1
-
n
-
m
ln
⁡
q
m
∑
v
=
1
n
∑
k
=
1
m
-
1
m
-
1
k
q
v
k
k
!
1
-
q
v
k
+
1
. Now(93)
Q
q
,
n
,
k
,
m
=
d
m
d
n
m
ln
⁡
n
k
q
m
=
d
m
d
n
m
ln
⁡
q
n
m
-
ln
⁡
q
k
m
-
ln
⁡
q
n
-
k
m
=
m
ln
⁡
q
m
H
q
n
-
k
1
-
H
q
n
1
+
k
-
m
ln
⁡
q
m
∑
v
=
n
-
k
+
1
n
∑
u
=
1
m
-
1
m
-
1
u
q
v
u
u
!
1
-
q
v
u
+
1
=
m
ln
⁡
q
m
H
q
n
-
k
1
-
H
q
n
1
+
k
-
m
ln
⁡
q
m
∑
v
=
1
k
∑
u
=
1
m
-
1
m
-
1
u
q
n
-
k
+
v
u
u
!
1
-
q
n
-
k
+
v
u
+
1
. By virtue of Bell Polynomials we have(94)
d
m
d
n
m
n
k
q
m
=
d
m
d
n
m
e
ln
⁡
n
k
q
m
=
n
k
q
m
Y
m
m
Q
q
,
n
,
k
,
1
,
m
Q
q
,
n
,
k
,
2
,
…
,
m
Q
q
,
n
,
k
,
m
.Case 1.
Substitutingm
=
1 in (94),(95)
d
d
n
n
k
q
=
n
k
q
ln
⁡
q
k
+
H
q
n
-
k
1
-
H
q
n
1
. And for r is an integer,(96)
d
d
n
n
k
q
n
=
r
=
r
k
q
ln
⁡
q
k
+
H
q
r
-
k
1
-
H
q
r
1
. Finally using the definition of (93) we can derive(97)
l
i
m
k
→
r
+
v
Q
q
,
r
,
k
,
1
=
l
i
m
k
→
r
+
v
r
k
q
ln
⁡
q
k
+
H
q
r
-
k
1
-
H
q
r
1
=
ln
⁡
q
-
1
v
q
r
q
v
-
1
q
v
v
-
1
/
2
q
r
+
v
.Case 2.
Substitutingm
=
2 in (94), we already derived the expression for Q
(
q
,
r
,
k
,
1
). (98)
Q
q
,
r
,
k
,
2
=
2
ln
⁡
q
2
k
+
H
q
r
-
k
1
-
H
q
r
1
-
∑
v
=
1
k
q
n
-
k
+
v
1
-
q
n
-
k
+
v
2
=
2
ln
⁡
q
2
k
+
H
q
r
-
k
2
-
H
q
r
2
. Finally using (94) for m
=
2 we have (99)
d
2
d
n
2
n
k
q
2
=
n
k
q
2
Y
2
2
Q
q
,
n
,
k
,
1
,
2
Q
q
,
n
,
k
,
2
=
n
k
q
2
4
Q
q
,
n
,
k
,
1
2
+
2
Q
q
,
n
,
k
,
2
=
4
ln
⁡
q
2
n
k
q
2
k
+
H
q
n
-
k
1
-
H
q
n
1
2
+
k
+
H
q
n
-
k
2
-
H
q
n
2
.From previous calculation discussed in (97) we have(100)
l
i
m
k
→
r
+
v
r
k
q
k
+
H
q
r
-
k
1
-
H
q
r
1
2
=
q
r
2
q
v
-
1
2
q
v
v
-
1
q
r
+
v
2
.By the same procedure we have(101)
l
i
m
k
→
r
+
v
r
k
q
2
k
+
H
q
r
-
k
2
-
H
q
r
2
=
-
q
r
2
q
v
-
1
2
q
v
v
-
1
q
r
+
v
2
.So finally(102)
l
i
m
k
→
r
+
v
4
ln
⁡
q
2
n
k
q
2
k
+
H
q
n
-
k
1
-
H
q
n
1
2
+
k
+
H
q
n
-
k
2
-
H
q
n
2
=
0
.Theorems Closely Related to Roger-Ramanujan IdentitiesRoger-Ramanujan Identities. Rogers Ramanujan identities are given in(103)
∑
k
=
0
∞
q
k
2
q
;
q
k
=
1
q
;
q
5
∞
q
4
;
q
5
∞
,
∑
k
=
0
∞
q
k
k
+
1
q
;
q
k
=
1
q
2
;
q
5
∞
q
3
;
q
5
∞
. Roger discovered these identities in 1894 [23, 24], but they were entirely ignored until Ramanujan rediscovered them about 20 years later. A detailed history of these identities can be found in great detail in the survey article written by Andrews [25].Theorem 18.
One has(104)
∑
v
=
1
∞
q
v
-
1
q
r
+
v
a
q
r
v
=
a
;
q
r
-
a
r
q
r
q
r
r
-
1
/
2
R
q
,
a
+
r
-
H
a
;
q
r
1
-
∑
k
=
0
r
-
a
k
-
r
q
k
-
r
k
+
r
-
1
/
2
q
k
q
r
-
k
k
+
H
q
r
-
k
1
-
H
q
r
1
.Proof.
From theq binomial theorem,(105)
a
;
q
n
=
∑
k
=
0
∞
n
k
q
-
a
k
q
k
k
-
1
/
2
.Differentiating both sides of (105) with respect to n at the point n
=
r (integer), (106)
d
d
n
a
;
q
n
n
=
r
=
ln
⁡
q
a
;
q
r
∑
v
=
1
∞
a
q
r
v
1
-
q
v
=
ln
⁡
q
∑
k
=
0
∞
-
a
k
q
k
k
-
1
/
2
r
k
q
k
+
H
q
r
-
k
1
-
H
q
r
1
=
ln
⁡
q
∑
k
=
0
r
-
a
k
q
k
k
-
1
/
2
r
k
q
k
+
H
q
r
-
k
1
-
H
q
r
1
+
ln
⁡
q
-
a
r
q
r
q
r
r
-
1
/
2
∑
v
=
1
∞
q
v
-
1
q
r
+
v
a
q
r
v
.After some calculations we have Theorem 18: (107)
∑
v
=
1
∞
q
v
-
1
q
r
+
v
a
q
r
v
=
a
;
q
r
-
a
r
q
r
q
r
r
-
1
/
2
R
q
,
a
+
r
-
H
a
;
q
r
1
-
∑
k
=
0
r
-
a
k
-
r
q
k
-
r
k
+
r
-
1
/
2
q
k
q
r
-
k
k
+
H
q
r
-
k
1
-
H
q
r
1
.Theorem 19.
One has(108)
∑
k
=
0
n
q
k
2
r
k
q
2
k
+
H
q
r
-
k
1
-
H
q
r
1
2
+
k
+
H
q
r
-
k
2
-
H
q
r
2
=
2
r
r
q
r
+
H
q
r
1
-
H
q
2
r
1
2
+
1
2
S
q
,
2
+
H
q
2
r
1
-
H
q
2
r
2
+
1
2
H
q
r
2
-
H
q
r
1
.Proof.
The proof is similar to the previous one. For this case we have to consider the identity(109)
∑
k
=
0
∞
q
k
2
n
k
q
2
=
2
n
n
q
.Differentiating (109) with respect to n two times when n
=
r (integer) and considering (99) and (102) together we have,(110)
∑
k
=
0
r
q
k
2
r
k
q
2
k
+
H
q
r
-
k
1
-
H
q
r
1
2
+
k
+
H
q
r
-
k
2
-
H
q
r
2
=
d
2
d
n
2
2
n
n
q
n
=
r
.Using the same idea stated above we can calculate(111)
d
d
n
ln
⁡
2
n
n
q
n
=
r
=
2
ln
⁡
q
r
+
H
q
r
1
-
H
q
2
r
1
,
d
2
d
n
2
ln
⁡
2
n
n
q
n
=
r
=
4
ln
⁡
q
2
H
q
2
r
1
-
H
q
2
r
2
+
1
2
H
q
r
2
-
H
q
r
1
+
1
2
S
q
,
2
.By virtue of Bell polynomials we can derive further(112)
d
2
d
n
2
2
n
n
q
n
=
r
=
2
n
n
q
r
+
H
q
r
1
-
H
q
2
r
1
2
+
1
2
S
q
,
2
+
H
q
2
r
1
-
H
q
2
r
2
+
1
2
H
q
r
2
-
H
q
r
1
.Finally plugging back the above identity in (110) we can obtain Theorem 19.


## 3. Conclusion

We would like to enrich the subject by doing further research in this field. In our upcoming paper, we will use the theorems of this paper to prove some series identities related to the Hurwitz Zeta function and also generalize a new theorem. Also we will derive some closed form identities related to double differentiation of Legendre Polynomials with respect to their order and we will derive some new generating function identities involving generalized harmonic numbers and Stirling polynomials.By using the same techniques derived in this paper one can find various hypergeometric transformation formulas and theirq-analog. We will consider a curious generalization expression such as(113)
∑
v
=
1
∞
q
v
-
1
m
q
r
+
v
m
a
q
r
vin our upcoming paper.



---

*Source: 4546509-2016-09-27.xml*