2. Main Results
Theorem 1.
For integers
m
>
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 function
F
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
m
th 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
j
th 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 integers
m
>
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 integers
m
>
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 take
f
(
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 integers
n
⩾
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 Theorem 2 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 integers
n
≥
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.
Setting
m
=
2
,
y
=
2
,
x
=
1
, and
f
(
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 Corollaries
Corollary 6.
For integers
n
≥
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 integers
n
≥
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 integers
n
≥
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 integers
n
≥
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 integers
n
≥
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 integers
m
≥
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 integers
m
≥
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.
For
x
>
-
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.
Substituting
m
=
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 integers
n
≥
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.
Substituting
m
=
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 integers
n
≥
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.
Substituting
m
=
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 integers
n
≥
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.
Substituting
m
=
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.
Substituting
m
=
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.
Substituting
m
=
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 Identities
Roger-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 the
q
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.