Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 605313, 6 pages
doi:10.1155/2009/605313
Research Article
2n−1
A Recurrence Formula for D Numbers D2n
Guodong Liu
Department of Mathematics, Huizhou University, Huizhou, Guangdong 516015, China
Correspondence should be addressed to Guodong Liu, gdliu@pub.huizhou.gd.cn
Received 28 June 2009; Accepted 16 October 2009
Recommended by Binggen Zhang
2n−1
we establish a recurrence formula for D numbers D2n
2n−1
is also presented.
D2n
. A generating function for D numbers
Copyright q 2009 Guodong Liu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction and Results
k
The Bernoulli polynomials Bn x of order k, for any integer k, may be defined by see 1–4
t
t
e −1
k
k
ext ∞
tn
k
Bn x ,
n!
n0
|t| < 2π.
k
1.1
1
The numbers Bn Bn 0 are the Bernoulli numbers of order k, Bn Bn are the
ordinary Bernoulli numbers see 2, 5. By 1.1, we can get see 4, page 145
d k
k
Bn x nBn−1 x,
dx
k1
Bn
k1
Bn
x x 1 1.2
n k
k − n k
Bn x x − k Bn−1 x,
k
k
1.3
nx k
n − k k
B x −
Bn x,
k n−1
k
1.4
where n ∈ N, with N being the set of positive integers.
2
Discrete Dynamics in Nature and Society
n
The numbers Bn are called the Nörlund numbers see 2, 4, 6. A generating function
n
for the Nörlund numbers Bn is see 4, page 150
∞
n
t
n t
.
Bn
n!
1 t log1 t n0
1.5
k
The D numbers D2n may be defined by see 4, 7, 8
t csc tk ∞
2n
k t
,
−1n D2n
2n!
n0
|t| < π.
1.6
By 1.1, 1.6, and note that csc t 2i/eit − e−it where i2 −1, we can get
k
k
D2n 4n B2n
k
.
2
1.7
1
2
Taking k 1, 2 in 1.7, and note that B2n 1/2 21−2n − 1B2n , B2n 1 1 − 2nB2n
see 4, page 22, page 145, we have
1
D2n 2 − 22n B2n ,
2
D2n 4n 1 − 2nB2n .
1.8
k
The D numbers D2n satisfy the recurrence relation see 7
k
D2n 2n − k 22n − k 1 k−2 2n2n − 1k − 2 k−2
D2n−2 .
D2n −
k−1
k − 2k − 1
1.9
By 1.9, we may immediately deduce the following see 4, page 147:
2n1
D2n
2n3
D2n
−1n 2n!
4n
−1n 2n!
2 · 42n
2n
The numbers D2n
relation see 7
n
0
2
2n
n
,
2n2
D2n
−1n 4n
n!2 ,
2n 1
2n 2
1
1
1
1 2 2 ··· .
3
5
n1
2n 12
1.10
1.11
are called the D-Nörlund numbers that satisfy the recurrence
−1j
j
j0 4 2j 1
so we find D0 1, D2
−18940160/33, . . . .
2n−2j 2j D2n−2j
−1n 2n
,
4n
2n − 2j !
j
n
4
−2/3, D4
6
88/15, D6
8
−3056/21, D8
1.12
10
319616/45, D10
Discrete Dynamics in Nature and Society
3
2n
A generating function for the D-Nörlund numbers D2n is see 7
√
∞
2n
t
2n t
,
D2n
√
2n!
1 t2 log t 1 t2
n0
2n
2n−1
These numbers D2n and D2n
4, page 246
π/2
0
1.13
|t| < 1.
have many important applications. For example see
2n
∞ −1n D
sin t
2n
dt ,
t
1!
2n
n0
π/2
0
2n−1
∞ −1n1 D
π
sin t
2n
dt ,
t
2 n0 22n 2n − 1n!2
2n−1
1.14
∞ −1n1 D
2
2n
.
π n0 2n − 12n!
The main purpose of this paper is to prove a recurrence formula for D numbers
2n−1
2n−1
D2n
and to obtain a generating function for D numbers D2n . That is, we will prove
the following main conclusion.
Theorem 1.1. Let n ∈ N. Then
n
2n
2j
j1
j−1 j−1 −1
1
2 2n−1−2j −1n−1 22n!
j − 1 ! D2n−2j
4n
4
3
so one finds D2 −1/3, D4
−3887409/11, . . . .
5
17/5, D6
2n − 2
n−1
7
,
1.15
9
−1835/21, D8
195013/45, D10
Theorem 1.2. Let t be a complex number with |t| < 1. Then
∞
2n−1
D2n
n0
t2n
1
√
2n!
1 t2
logt t
√
2
1 t2 .
1.16
2. Proof of the Theorems
Proof of Theorem 1.1. Note the identity see 4, page 203
k
B2n
k
x
2
k−2j n
2n D2n−2j
j0
2j
22n−2j
2 x2 x2 − 12 x2 − 22 · · · x2 − j − 1 ,
2.1
we have
k−2j k
k
n
2n D2n−2j 2
B2n x k/2 − B2n k/2 2 2
2
2
2
x
x
·
·
·
x
.
−
1
−
2
−
j
−
1
x2
22n−2j
2j
j1
2.2
4
Discrete Dynamics in Nature and Society
Therefore,
k−2j k
k
n
2n D2n−2j
B2n x k/2 − B2n k/2 2
lim
−1j−1 j − 1 ! .
2n−2j
x→0
x2
2
2j
j1
2.3
By 2.3 and 1.2, we have
k−2j k
n
2n D2n−2j
2n2n − 1B2n−2 x k/2 2
lim
−1j−1 j − 1 ! .
2n−2j
x→0
2
2
2j
j1
2.4
That is,
k−2j n
2n D2n−2j
2
k
−1j−1 j − 1 ! .
2n−2j
2
2
2j
j1
2.5
n
2n
2 k−2j 1
−1j−1 4j−1 j − 1 ! D2n−2j .
n2n − 1 j1 2j
2.6
k
1B2n−2
n2n −
By 2.5 and 1.7, we have
k
D2n−2
Setting k 2n−1 in 2.6, and note 1.10, we immediately obtain Theorem 1.1. This completes
the proof of Theorem 1.1.
Remark 2.1. Setting k 2n in 2.6, and note 1.10, we may immediately deduce the following
2n
recurrence formula for D-Nörlund numbers D2n :
n
2n
j1
2j
−1j 4j
2 2n−2j j − 1 ! D2n−2j −1n n4n n − 1!2
n ∈ N.
2.7
Proof of Theorem 1.2. Note the identity see 9
∞
t2n
1
t
2
,
1
−
log
t
1
t
−1n 4n n!2
√
2n! 1 t2
1 t2
n0
2.8
where |t| < 1. We have
∞
−1n 4n n!2
n0
2
t2n2
1 log t 1 t2
,
2n 2! 2
2.9
Discrete Dynamics in Nature and Society
5
That is,
∞
−1n−1 4n−1 n − 1!2
n1
2
t2n
1 log t 1 t2
.
2n! 2
2.10
On the other hand,
∞
−1n−1 22n!
n1
4n
2n − 2
n−1
∞
t2n
1
−1n
2n! 2 n0 4n
2n
n
t2
t2n2 √
.
2 1 t2
2.11
Thus, by 2.10, 2.11, and Theorem 1.1, we have
∞
−1
n−1 n−1
4
n1
∞
∞
2n
t2n −1n−1 22n!
2n−1 t
D2n
n − 1!
4n
2n! n1
2n! n1
2
2n − 2
t2n
.
n − 1 2n!
2.12
That is,
∞
2 2n
1 t2
2n−1 t
log t 1 t2
√
D2n
.
2
2n! 2 1 t2
n1
2.13
By 2.13, and note that
lim
t→0
log t t
√
1
t2
1,
−1
D0
1,
2.14
we immediately obtain Theorem 1.2. This completes the proof of Theorem 1.2.
Acknowledgment
This work was Supported by the Guangdong Provincial Natural Science Foundation no.
8151601501000002.
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Discrete Dynamics in Nature and Society
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