INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
A COMMON GENERALIZATION OF SOME IDENTITIES1
Zhizheng Zhang
Department of Mathematics, Luoyang Teachers’ College, Luoyang 471022, P. R. China
College of Mathematics and Information Science, Henan University, Kaifeng 475001, P. R. China
zhzhzhang-yang@163.com
Jun Wang
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P. R. China
junwang@dlut.edu.cn
Received: 8/12/05, Revised: 11/18/05, Accepted: 12/10/05, Published: 12/21/05
Abstract
By means of partial fraction decomposition, this paper provides an algebraic identity from
which a lot of interesting identities follow.
1. Notation and Introduction
In [2], by the WZ method, the following identity was confirmed successfully.
#2
n " #2 "
!
n
n+k
{1 + 2kHn+k + 2kHn−k − 4kHk } = 0,
(1)
k
k
k=1
$
where H0 = 0 and Hn = nk=1 k1 . Ahlgren and Ono [1] have shown that Beukers’ conjecture is
implied by this beautiful binomial identity. See [3] or [1, Theorem 7] for Beukers’ conjecture.
In [4] and [6], by means of partial fraction decomposition, W.-C. Chu gave beautiful proofs
of (1) and a number of interesting combinatorial identities. At the same time, Identity (1)
was also extended.
The purpose of this paper is to obtain an algebraic identity by means of partial fraction
decomposition. As applications, we show a number of interesting identities.
Throughout the paper, f (x) is known for an arbitrary polynomial of degree < λn and
the function A(T1 , T2 , . . . , Tn ) is defined by
%
1,
if n = 0,
&
&
'
&
'
'
A(T1 , T2 , . . . , Tn ) = $
k
k
k
n
1
2
T
T
n!
1
2
. . . Tnn
, if n > 0,
k1 !k2 !...kn !
1
2
Supported by the National Natural Science Foundation of China (Grant No. 10471016), the Natural
Science Foundation of Henan Province (Grant No. 0511010300) and the Natural Science Foundation of the
Education Department of Henan Province (Grant No. 200510482001).
1
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
2
where the sum is taken over all nonnegative integers ki ’s such that k1 + 2k2 + . . . + nkn = n.
For example, A(T1 , T2 , . . . , Tn ) for n ≤ 4 are:
A(T1 ) =
A(T1 , T2 ) =
A(T1 , T2 , T3 ) =
A(T1 , T2 , T3 , T4 ) =
T1
T12 + T2
T13 + 3T1 T2 + 2T3
T14 + 8T1 T3 + 6T12 T2 + 3T22 + 6T4 .
2. Main Result
# j) and f (ai ) #= 0,
Theorem 2.1 Let a1 , a2 , . . . , an be a real sequence with ai #= aj (i =
(i = 1, . . . , n). Suppose that λ and r are two integers with λ ≥ 1 and r ≥ 0. Then
n
!
k=1
1
n
(
j=1
j "= k
(aj − ak )λ
" #
"
λ−1
!
1!
s "
(−1)
f ("−s) (−ak )
"! s=0
s
"=0
&λ−"+r−1'
r
×A (λT1 (x), λT2 (x), . . . , λTs (x))
(x + ak )λ−"+r
" #
r
!
(−1)r
j r
=
(−1)
f (r−j) (x)
r!(x + a1 )λ (x + a2 )λ . . . (x + an )λ j=0
j
×A (λS1 (x), λS2 (x), . . . , λSj (x)) ,
where
Tm (x) =
n
!
i=1
i "= k
and
Sm (x) =
n
!
i=1
1
, (1 ≤ m ≤ λ),
(x + ai )m
1
,
(x + ai )m
(1 ≤ m ≤ r).
Proof. Applying the partial fraction decomposition, let
n
λ−1
! ! G(k, ")
f (x)
=
.
(x + a1 )λ (x + a2 )λ . . . (x + an )λ
(x + ak )λ−"
k=1 "=0
Multiply both sides in the preceding equation by (x + ak )λ−" and take x → ak . Then
%
)
"−1
!
f
(x)
G(k,
i)
−
G(k, ") = lim (x + ak )λ−"
x→−ak
(x + a1 )λ (x + a2 )λ . . . (x + an )λ
(x + ak )λ−i
i=0
(2)
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33















"−1
i
!
f (x)
G(k, i)(x + ak )
−
= lim
n
(
x→−ak 

(x + ak )"
"
λ


i=0
(x
+
a
)
(x
+
a
)


k
j






j
=
1


j "= k
f (x)
n
(
=
lim
(x+aj )λ
j=1
j "= k
−
"−1
$
G(k, i)(x + ak )i
i=0
.
(x + ak )"
x→−ak
Repeatedly applying L’Hôspital’s rule we obtain the coefficient as follows:
G(k, ") =
1 d"
"! dx"
lim
x→−ak
f (x)
n
(
(x + aj )λ
j=1
j "= k
=
1
lim
"! x→−ak
" "
!
r=0

(r)






" ("−r)
1

f
(x) 
n
 (

r
λ

(x + aj ) 


#
.
j=1
j "= k
By using differentiation of composite functions (see [12]), we have

(r)






1


n
 (

λ

(x
+
a
)
j


=
j=1
j "= k
where
(−1)r
n
(
(x + aj
A (λT1 (x), λT2 (x), . . . , λTr (x)) ,
)λ
j=1
j "= k
Tm (x) =
n
!
i=1
i "= k
1
, (1 ≤ m ≤ ").
(x + ai )m
Hence,
1
G(k, ") =
"!
n
(
j=1
j "= k
(aj − ak )λ
" #
" ("−r)
(−1)
(−ak )A (λT1 (x), λT2 (x), . . . , λTr (x)) ,
f
r
r=0
"
!
r
3
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
where
Tm (x) =
n
!
i=1
i "= k
4
1
, (1 ≤ m ≤ ").
(ai − ak )m
So we get
n
!
f (x)
=
(x + a1 )λ (x + a2 )λ . . . (x + an )λ
k=1
1
n
(
j=1
j "= k
(aj − ak )λ
" #
λ−1
"
!
1!
s "
(−1)
s
"!
s=0
"=0
×f ("−s) (−ak )A (λT1 (x), λT2 (x), . . . , λTs (x))
1
.
(x + ak )λ−"
By differentiating the above equation r times with respect to x, we have
n
!
1
k=1
n
(
j=1
j "= k
(aj − ak )λ
" #
"
λ−1
!
1!
s "
(−1)
f ("−s) (−ak )A (λT1 (x), λT2 (x), . . . , λTs (x))
s
"! s=0
"=0
(−1)r (λ − " + r − 1)r
(x + ak )λ−"+r
"
#
dr
f (x)
=
dxr (x + a1 )λ (x + a2 )λ . . . (x + an )λ
"
#(j)
r " #
!
1
r (r−j)
=
(x)
f
(x + a1 )λ (x + a2 )λ . . . (x + an )λ
j
j=0
" #
r
!
1
j r
f (r−j) (x)A (λS1 (x), λS2 (x), . . . , λSj (x)) ,
=
(−1)
(x + a1 )λ (x + a2 )λ . . . (x + an )λ j=0
j
×
where
Sm (x) =
n
!
i=1
This completes the proof.
3. Some Corollaries
Corollary 3.2
n
!
k=1
f (−ak )
n
(
(x + ak )r+1
(aj − ak )
j=1
j "= k
1
,
(x + ai )m
(1 ≤ m ≤ r).
!
5
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
" #
r
!
(−1)r
j r
=
(−1)
f (r−j) (x)A (S1 (x), S2 (x), . . . , Sj (x)) ,
r!(x + a1 )(x + a2 ) . . . (x + an ) j=0
j
(3)








n
n

$
!
!
f (−ak )
f (−ak )
1 
r+1
+
−2
n
(

x + ak
f (−ak )
x + ai 
2


k=1 (x + ak )r+1
(aj − ak ) 

i=1


i "= k
j=1
j "= k
" #
r
!
(−1)r
j r
(−1)
=
f (r−j) (x)A (2S1 (x), 2S2 (x), . . . , 2Sj (x))
r!(x + a1 )2 (x + a2 )2 . . . (x + an )2 j=0
j
(4)
and
n
!
k=1







 (r + 2)(r + 1)
f (−ak )
n
(

(x + ak )r+1
(aj − ak )3 




j=1

(x + ak )2
j "= k

f ! (−ak )
f (−ak )
+ 2(r + 1)
−3
n
$
i=1
i "= k
1
x+ai
x + ak





2
n
n
n
 !
!
f $$ (−ak )
f $ (−ak ) !
1
1 
1


+
+ 9
−6
 +3
f (−ak )
f (−ak )
x + ai
x + ai 
(x + ai )2 



i=1
i=1
i=1

i "= k
i "= k
i "= k
" #
r
!
2(−1)r
j r
(−1)
=
f (r−j) (x)A (3S1 (x), 3S2 (x), . . . , 3Sj (x)) .
r!(x + a1 )3 (x + a2 )3 . . . (x + an )3 j=0
j
(5)
Proof. In Theorem 2.1, take λ = 1, 2, 3.
!
Note. The first formula (3) was given in [14] and an extension of (3) was obtained in [15].
Corollary 3.3
n
!
k=1
1
n
(
j=1
j "= k
(aj − ak )λ
" #
"
λ−1
!
1!
s "
(−1)
f ("−s) (−ak )
"! s=0
s
"=0
×A (λT1 (x), λT2 (x), . . . , λTs (x))
=
f (x)
,
(x + a1 )λ (x + a2 )λ . . . (x + an )λ
1
(x + ak )λ−"
(6)
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
n
!
1
n
(
k=1
and
j=1
j "= k
(aj − ak )λ
" #
"
λ−1
!
1!
s "
(−1)
f ("−s) (−ak )
"!
s
s=0
"=0
λ−"
×A (λT1 (x), λT2 (x), . . . , λTs (x))
(x + ak )λ−"+1
%
)
n
!
1
1
= −
f $ (x) − λf (x)
(x + a1 )λ (x + a2 )λ . . . (x + an )λ
x + ai
i=1
n
!
k=1
1
n
(
j=1
j "= k
(aj − ak )λ
(λ − " + 1)(λ − ")
×A (λT1 (x), λT2 (x), . . . , λTs (x))
2(x + ak )λ−"+2
%
n
!
1
1
$$
$
=
f (x) − 2λf (x)
λ
λ
λ
2(x + a1 ) (x + a2 ) . . . (x + an )
x + ai
i=1

7 n
82
n

! 1
!
1
2
+λ f (x)
+ λf (x)
,
x + ai
(x + ai )2 
Tm (x) =
n
!
i=1
i "= k
k=1
(8)
i=1
1
, (1 ≤ m ≤ λ).
(x + ai )m
Proof. In Theorem 2.1, take r = 0, 1, 2.
Corollary 3.4
n
!
(7)
" #
λ−1
"
!
1!
s "
(−1)
f ("−s) (−ak )
s
"!
s=0
"=0
i=1
where
6
!
f (x)
f (−ak )
=
,
n
(
(x + a1 )(x + a2 ) . . . (x + an )
(x + ak )
(aj − ak )
(9)
j=1
j "= k








n
n

$
!
!
f (−ak )
f (−ak )
1 
r+1
+
−2
n
(

x
+
a
f
(−a
x + ai 
k
k)
2
2


k=1 (x + ak )
(aj − ak ) 

i=1


j=1
j "= k
1
= −
(x + a1 )2 (x + a2 )2 . . . (x + an )2
i "= k
%
f $ (x) − 2f (x)
n
!
i=1
1
x + ai
)
(10)
7
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
and
n
!
k=1
f (−ak )
n
(
(x + ak )3
(aj − ak )3
j=1
j "= k







 (r + 2)(r + 1)







(x + ak )2

f ! (−ak )
f (−ak )
+ 2(r + 1)
2
−3
n
$
i=1
i "= k
1
x+ai
x + ak





n
n
n
 !
!
f $$ (−ak )
f $ (−ak ) !
1
1 
1


+
−6
+ 9
 +3
f (−ak )
f (−ak )
x + ai
x + ai 
(x + ai )2 



i=1
i=1
i=1

i "= k
i "= k
i "= k
%
n
!
1
1
$$
$
=
(x)
−
6f
(x)
f
2(x + a1 )3 (x + a2 )3 . . . (x + an )3
x + ai
i=1

7 n
82
n

! 1
!
1
+9f (x)
+ 3f (x)
.
(11)
x + ai
(x + ai )2 
i=1
i=1
Proof. Take r = 0, 1, 2 in Corollary 3.2 or λ = 1, 2, 3 in Corollary 3.3.
!
4. Some Applications
It is seen that the identities in Corollaries 3.2, 3.3 and 3.4 are independent of the choice of
ak and f (x). So we can obtain some interesting identities by taking special values for ak and
f (x).
4.1 The case: ak = −k
In [5, 11], some interesting identities involving the harmonic numbers were studied. In this
section, by taking ak = −k, we show a number of identities involving the harmonic numbers.
Theorem 4.5
n
!
k=1
(k+1)λ
(−1)
&λ−"+r−1'
" #λ !
λ−1
" " #
(−1)" ! " ("−s)
n
r
(k)A (λT1 , λT2 , . . . , λTs ) λ−"+r−1
f
k
s
"!
k
s=0
"=0
r " #
1 ! r (r−j)
=
(0)A (λS1 , λS2 , . . . , λSj ) ,
f
r! j=0 j
(12)
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
where
n
!
Tm =
i=1
i "= k
and
8
1
, (1 ≤ m ≤ λ),
im
n
!
1
Sm =
,
im
i=1
(1 ≤ m ≤ r).
Proof. Take ak = −k and x = 0 in Theorem 2.1.
!
When λ = 1, then we can obtain the main results of paper [9] by Mercier:
& '
n
r " #
!
(−1)k+1 nk f (k)
1 ! r (r−j)
=
(0)A (S1 , S2 , . . . , Sj ) .
f
kr
r! j=0 j
k=1
(13)
In [9], from this identity, Mercier obtained a lot of combinatorial identities. For example,
" #
n
!
k+1 n
(−1)
f (k) = f (0),
(14)
k
k=1
n
!
(−1)k+1
k
k=1
n
!
k=1
& '
(−1)k+1 nk
k2
&n'
k
f (k) = f $ (0) + f (0)
n
!
1
i=1
i
,
(15)

7

82
n
n
n

!1
!1
!1 
1
 .
f $$ (0) + 2f $ (0)
+ f (0) 
f (k) =
+
2
i
i
i2 
i=1
i=1
(16)
i=1
For λ = 2, 3, we have




n &n'2
!
f (k)  r + 1
k
kr
k=1

k



−
f $ (k)
−2
f (k)




n
!
1
i=1
i "= k
i



r " #
1 ! r (r−j)
(0)A (2S1 , 2S2 , . . . , 2Sj )
=
f
r! j=0 j
and



& '

n
k+1 n 3
!
(−1)
f (k)  (r + 2)(r + 1)
k=1
k
kr




k2

(17)

n
!
r+1
1
 f $ (k)
 f $$ (k)
−2
+3

+
k  f (k)
i
f (k)
i=1
i "= k
9
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33





n
!
1
2
n
n
 !
f $ (k) ! 1
1


+ 9
+6
 +3
f (k)
i
i

i=1
i "= k
i=1
i "= k
i=1
i "= k
r " #
2 ! r (r−j)
=
(0)A (3S1 , 3S2 , . . . , 3Sj ) .
f
r! j=0 j
i2 



(18)
In particular,
n " #2
!
n




1
−
f $ (k)
−2
f (k)




n
!
1
= f (0),

k
i




i=1


i "= k








n &n'2
n
n
 2 f $ (k)

!
!
!
1
1
$
k
= f (0) + 2f (0)
f (k)
−
−2
,


k
k
f
(k)
i
i


i=1
k=1


i=1


i "= k








n &n'2
n
 3 f $ (k)

!
!
1
k
f
(k)
−
−
2

k2
k
f (k)
i


k=1


i=1


k=1
k
f (k)
(19)
(20)
i "= k

7

82
n
n
n

!1
!1
!1 
1
$$
$

 ,
=
f (0) + 4f (0)
+ f (0)
4
+2
2
i
i
i2 
i=1
i=1
(21)
i=1






" #3
n
n
n
2
!
!
2
1
n
f $ (k) ! 1
 f $ (k)
 f $$ (k)
k+1
(−1)
f (k)
− 
+3
+6
+
2

k
k
k
f
(k)
i
f
(k)
f (k)
i



k=1

i=1
i=1

i "= k
i "= k

2




n
n

 !
!
1
1

+9 
 +3
i
i2 



i=1
i=1

i "= k
= 2f (0),
i "= k
(22)
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
10





& '3

n
n
n
6
!
!
(−1)k+1 nk
1
4
f $ (k) ! 1
 f $ (k)
 f $$ (k)
− 
f (k)
+3
+6
+
2

k
k
k
f
(k)
i
f
(k)
f (k)
i



k=1

i=1
i=1

i "= k
i "= k


2



n
n
 !

!
1
1

+9 
 +3
i
i2 



i=1
i=1

(23)
= 2f $ (0) + 6f (0)
n
!
i=1
n
!
(−1)k+1
k=1
k2
&n'3
k
i "= k
i "= k
1
.
i







n
n
!
12 6 
1
f $ (k) ! 1
 f $ (k)
 f $$ (k)
f (k)
− 
+3
+6
+

k 2 k  f (k)
i
f (k)
f (k)
i


i=1
i=1

i "= k
i "= k


2



n
n
 !


!
1
1


+9 
+
3

i
i2 



i=1
i=1

i "= k
i "= k
 7

82
n
n
n
!
!
!
1
1
1
= f $$ (0) + 6f $ (0)
+3
+ f (0) 9
.
i
i
i2
i=1
i=1
i=1
(24)
In identities (15)-(16) and (19)-(24), according to different choices of f (x), we can get a
lot of identities. For example, if f (x) = 1, then we have
& '
n
n
!
!
(−1)k+1 nk
1
=
,
(25)
k
i
i=1
k=1
n
!
(−1)k+1
k=1
k2
&n'
k
7

82
n
n

!
!
1
1 
1 
=
+
,
2

2
i
i
i=1
i=1



n " #2 
!
n 1
k=1
k

k



−2




n
!
1
i=1
i "= k
i



= 1,
(26)
(27)
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33


&n'2 

n
2
!




n
! 1
k
n
!
1
=2
−2
,
k 
k
i
i


i=1


i
=
1


i "= k






7 n 82

n &n'2 
n
n
3

!
!
!1
!
1
1
k
+
,
=2
−2
2
k 
k
i
i
i2


i=1
i=1
k=1


i
=
1


k=1
11
(28)
(29)
i "= k



2





" #3 
n
n
n
n




!
!
!
!
1
1
1
6
n
2


k+1
+
9
(−1)
−
+
3
= 2,


2
2

k
k
i
i
i
k




k=1


i=1
i=1
i=1


i "= k
i "= k
i "= k



2




&n'3 

n
n
n
n
n



k+1
! (−1)
!
!
!
!
2
1
1
1
1
4

k
=2
−
+ 3
.
 +
2
2


k
k
k
i
i
i 
i


i=1
k=1


i
=
1
i
=
1
i
=
1


i "= k
(30)
i "= k
(31)
i "= k



2



7 n 82
& ' 


n
n
n
n
n



k+1 n 3 
!
!
!
!
!1
!
(−1)
1
1
1
1
4
6


k
+ 3
=3
−
+
.
 +
2
2
2

k
k
k
i
i
i 
i
i2



i=1
i=1
k=1


i=1
i=1
i=1


i "= k
i "= k
i "= k
(32)
k
4.2 The case: ak = − 1−q
1−q
k
In this section, taking ak = − 1−q
we obtain some identities involving the Gaussian binomial
1−q
coefficient.
Theorem 4.6
n
!
k=1
(k+1)λ
(−1)
q λ(nk− 2 k(k+1))
1
9
;
n
k
1−q k
1−q
:λ
<λ+r−1
"
#
λ−1
" " #
!
(−1)" ! " ("−s) 1 − q k
f
"! s=0 s
1−q
"=0
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
#"
#"
λ−"+r−1
1 − qk
×A (λT1 , λT2 , . . . , λTs )
1−q
r
"
#
r
1 ! r (r−j)
=
(0)A (λS1 , λS2 , . . . , λSj )
f
r! j=0 j
12
"
where
(33)
"
#m
n
!
1−q
, (1 ≤ m ≤ λ),
Tm =
1 − qi
i=1
i "= k
and
Sm =
#m
n "
!
1−q
i=1
1 − qi
,
(1 ≤ m ≤ r).
k
Proof. Take ak = − 1−q
and x = 0 in Theorem 2.1.
1−q
!
When λ = 1, this reduces to an identity due to Mercier [10]:
9 : ;
<
n
1−q k
k+1
(−1)
f
n
r " #
1−q
!
k
1 ! r (r−j)
;
<r =
(0)A (S1 , S2 , . . . , Sj ) .
f
j
r!
nk− 12 k(k+1) 1−q k
q
j=0
k=1
1−q
In particular, for r = 0, 1, we have
9 : ;
<
n
1−q k
k+1
(−1)
f
n
1−q
!
k
1
q nk− 2 k(k+1)
k=1
(35)
: ;
<
1−q k
f 1−q
n (−1)
n
!
!
1−q
$
; k < = f (0) + f (0)
.
1
1 − qi
q nk− 2 k(k+1) 1−q
i=1
k=1
k+1
9
= f (0),
(34)
n
k
(36)
1−q
Take f (x) = 1 and let q → 1/q. Then (36) becomes the result of Van Hamme [13]:
k+1
:
n 9
n
!
n (−1)k−1 q ( 2 ) ! q i
=
.
k
1 − qk
1 − qi
i=1
k=1
When λ = 2, we obtain
9 :2 ;
<
n
1−q k
f
n
1−q
!
k
; k <r
2nk−k(k+1) 1−q
k=1 q
1−q
=
1
r!
r "
!
j=0
#


;
<






$ 1−q k
n

f
!
1−q
1−q
1−q
− ; k< − 2
(r + 1)

1 − qk
1 − qi 


f 1−q


i
=
1
1−q


r (r−j)
(0)A (2S1 , 2S2 , . . . , 2Sj ) .
f
j
(37)
i "= k
(38)
13
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
In particular, for r = 0, 1 in (38), the following identities hold.


; k<






#
9
:
"
$ 1−q
n
n
2

f
k
! n
!
1−q
1−q
1−q
1−q
k(k+1)−2nk
q
f
− ; k< − 2
= f (0),
k
 1 − qk
1
−
q
1 − qi 


f 1−q
k=1


i
=
1
1−q


(39)
i "= k
and

:2 ;
< 
; k<


1−q k



$ 1−q
f
n
n


f
1−q
!
! 1−q
1−q
1−q
; k< 2
;
<
−
−
2
k

1 − qk
1 − qi 
2nk−k(k+1) 1−q


f 1−q
k=1 q


i=1
1−q
1−q


9
n
k
= f $ (0) + 2f (0)
i "= k
n
!
1−q
.
1 − qi
i=1
Specifically, for f (x) = 1, we have





(40)





:2
n 9
n
!
!
1
1
1
n
k(k+1)−2nk
=
q
−2
,
k
i
k

1−q
1−q 
1−q


k=1


i=1







i "= k
(41)





:2 k(k+1)−2nk
n 9
n
n
!
!
q
1
1
1 ! 1
n
−
.
=
k
k
i
i
k

1
−
q
1
−
q
1
−
q
1
−
q
1
−
q


i=1
k=1


i=1


i "= k
(42)
4.3 An identity of Fu and Lascoux
Theorem 4.7
'λ
&
9 :λ
n
!
anλ bqa ; q n
n
λ(k+1
−λnk
(k−1)λ
2 )
(−1)
q
(1 − q k )(a − bq k )λ(n−2)
k
(az − bc)(n−1)λ (q; q)n k=1
" #
"
#
λ−1
"
!
c − zq k
1!
s "
("−s)
×
(−1)
f
−
A (λT1 (x), λT2 (x), . . . , λTs (x)) ;
s
"! s=0
a − bq k
"=0
=
;
r! x +
c−zq
a−bq
<λ ;
(−1)r
x+
c−zq 2
a−bq 2
<λ
;
... x +
c−zq n
a−bq n
<λ
" #
r (r−j)
×
(−1)
(x)A (λS1 (x), λS2 (x), . . . , λSj (x)) ,
f
j
j=0
r
!
j
&λ−"+r−1'
r
x+
c−zq k
a−bq k
<λ−"+r
(43)
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
where
n
!
Tm (x) =
i=1
i "= k
and
Sm (x) =
n
!
i=1
14
(a − bq i )m
, (1 ≤ m ≤ λ),
(ax + c − q i (bx + z))m
(a − bq i )m
,
(ax + c − q i (bx + z))m
Proof. In Theorem 2.1, take ak =
(1 ≤ m ≤ r).
c−zq k
.
a−bq k
!
When λ = 1, r = 0, x → −x and f (x) = 1, this reduces to
&
'
9 :
n
k n−1
!
cn zqc ; q n
1
n
−nk
k−1
k (a − bq )
(k+1
2 )
(−1)
(1
−
q
)
·
q
k
n−1
k
a−bq
k
(az − bc) (q; q)n k=1
c − zq
1 − c−zqk x
1
<;
< ;
<.
= ;
(44)
a−bq 2
a−bq n
1 − a−bq
x
1
−
x
.
.
.
1
−
x
c−zq
c−zq 2
c−zq n
Comparing the coefficients of xτ on both sides of the above equation, we have an identity of
Fu and Lascoux [8]:
&
'
9 :
n
k n−1+τ
!
cn zqc ; q n
n
−nk
k−1
k (a − bq )
(k+1
2 )
(−1)
(1
−
q
)
q
k
(az − bc)n−1 (q; q)n k=1
(c − zq k )τ +1
"
#
a − bq n
a − bq a − bq 2
,
= hτ
,...,
,
(45)
c − zq c − zq 2
c − zq n
where hτ (a1 , a2 , . . . , an ) is the τ -th complete symmetric function defined by
!
hτ (a1 , a2 , . . . , an ) =
ai1 ai2 . . . aiτ .
1≤i1 ≤i2 ≤...≤iτ ≤n
Taking a = 0, c = 1 and b = −1 in (45) we have
n
!
k−1
(−1)
k=1
9
:
n
k
k
q (2)+τ k
(q; q)n
1 − qk
=
hτ
k
τ
+1
(1 − zq )
(zq; q)n
"
qn
q
q2
,...,
,
1 − zq 1 − zq 2
1 − zq n
#
. (46)
When z = 1, this identity reduces to Dilcher’s identity [7]:
n
!
k=1
k−1
(−1)
9
n
k
:
k
q (2)+τ k
1
= hτ
(1 − q k )τ
"
qn
q2
q
,
.
.
.
,
,
1 − q 1 − q2
1 − qn
#
.
(47)
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A33
15
Acknowledgements
We would like to thank the anonymous referee for his valuable suggestions.
References
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J. Reine Angew. Math. 518(2000): 187-212.
[2] S. Ahlgren, S. B. Ekhad, K. Ono and D. Zeilberger, A binomial coefficient identity associated to a
conjecture of Beukers, The Electronic J. Combinatorics 5(1998), #R10.
[3] F. Beukers, Another congruence for Apéry numbers, J. Number Theory 25(1987): 201-210.
[4] W.-C. Chu, A binomial coefficient identity associated with Beukers’ Conjecture on Apéry numbers,
The Electronic J. Combinatorics 11(2004), #N15.
[5] W. Chu and L. De. Donno, Hypergeometric series and harmonic number identities, Adv. Appl. math.,
34(2005): 123-137.
[6] W.-C. Chu, Partial-fraction decompositions and harmonic number identities, arXiv:math.CO/0502537
vl.
[7] K. Dilcher, Some q-series identities related to divisor functions, Discrete Math., 145(1995): 83-93.
[8] A. M. Fu and A. Lascoux, q-identities from Lagrange and Newton interpolation, Adv. Appl. math.,
31(2003): 527-531.
[9] A. Mercier, Quelques identites de L’analyse combinatoire, Discrete math., 49(1984): 139-149.
[10] A. Mercier, Identities containing Gaussian binomial coefficients, Discrete Math., 76(1989): 67-73.
[11] P. Paule and C. Schneider, Computer proofs of a new family of harmonic number identities, Adv. Appl.
math., 31(2003): 359-378.
[12] J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NY,
1978.
[13] L. Van Hamme, Advanced problem 6407, American Math. Monthly, 40(1982): 703-704.
[14] Zhizheng Zhang, Several identities relatrd to Mercier’s, Mathematica Montisnigri, 11(1999): 159-171.
[15] Zhizheng Zhang, Generalizations of several identities, Northeast. Math. J., 10.1(1994): 93-98.