Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 974632, 13 pages
doi:10.1155/2012/974632
Research Article
Hermite Polynomials and their Applications
Associated with Bernoulli and Euler Numbers
Dae San Kim,1 Taekyun Kim,2
Seog-Hoon Rim,3 and Sang Hun Lee4
1
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
3
Department of Mathematics Education, Kyungpook National University,
Taegu 702-701, Republic of Korea
4
Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea
2
Correspondence should be addressed to Taekyun Kim, taekyun64@hotmail.com
Received 7 May 2012; Accepted 15 May 2012
Academic Editor: Garyfalos Papaschinopoulos
Copyright q 2012 Dae San Kim et al. 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.
We derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials
from the orthogonality of Hermite polynomials. Let Pn {px ∈ Qx | deg px ≤ n} be the
n 1-dimensional vector space over Q. Then we show that {H0 x, H1 x, . . . , Hn x} is a good
basis for the space Pn for our purpose of arithmetical and combinatorial applications.
1. Introduction
As is well known, the Euler polynomials, En x, are defined by the generating function as
follows:
et
∞
tn
2
ext eExt En x
n!
1
n0
1.1
see 1–8, with the usual convention about replacing En x by En x.
In the special case, x 0, En 0 En is called the nth Euler number. From 1.1 and
definition of Euler numbers, we note that
n
En x E x n n
l0
l
El x
n−l
with the usual convention about replacing En by En .
n n
En−l xl
l
l0
1.2
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Discrete Dynamics in Nature and Society
The Bernoulli numbers are defined as
B 1n − Bn δ1,n
B0 1,
1.3
see 9–14, where δk,n is a Kronecker symbol.
As is well known, Bernoulli polynomials are also defined by
n
Bn x B x n n
l
l0
Bl x
n−l
n n
Bn−l xl
l
l0
1.4
with the usual convention about replacing Bn by Bn see 1, 15–18.
The Hermite polynomials are defined by the generating function as follows:
2
e2xt−t eHxt ∞
tn
Hn x
n!
n0
1.5
see 5, 19, with the usual convention about replacing H n x by Hn x.
From 1.5, we can derive the following identities:
Hn x ∂
∂t
n
e
n x2
2xt−t2 −1 e
∂
∂x
e
x2
t0
n
2
e−x−t ∂
∂t
n
e
−x−t2 t0
n x2
−1 e
t0
dn −x2
e
.
dxn
1.6
Let us consider two operators as follows:
df
2 d
2
e−x f 2xf −
,
f −→ O1 f − ex
dx
dx
df
d
2
2
f −→ O2 f ex /2 x −
.
e−x /2 f 2xf −
dx
dx
1.7
By 1.7, we get O1 O2 . In particular, if we take f 1, then we have
−e
x2
d −x2
e
dx
e
x2 /2
d
2
x−
e−x /2 .
dx
1.8
We note that
n x2
−1 e
dn −x2
e
dxn
d −x2
e
−e
dx
x2
n
.
1.9
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From 1.8, we note that
n x2
−1 e
dn e−x
dxn
2
de−x
−e
dx
2
n
x2
n
d
2
2
e−x /2
ex /2 x −
dx
d n −x2 /2
2
ex /2 x −
e
.
dx
1.10
Thus, by 1.10, we get
Hn x e
x2 /2
d
x−
dx
n
e−x
2
1.11
/2
see 5, 19–23. In the special case, x 0, Hn 0 Hn are called the Hermite numbers.
From 1.5, we can derive the following identities:
Hn x H 2xn n n
l0
l
Hn−l 2l xl
1.12
cf. 5, 19, with the usual convention about replacing H n by Hn . It is easy to show that
∞
∞
tn
−1n 2n
2
t .
Hn e−t n!
n!
n0
l0
1.13
By comparing coefficients on the both sides of 1.13, we get
H2n −1n 2n2n − 1 · · · n 1 −1n 2n!
,
n!
H2n−1 0,
1.14
where n ∈ N. From 1.12, we have
dHn x
2nHn−1 x n ∈ N.
dx
1.15
Let Pn {p ∈ Qx | deg px ≤ n} be the n 1-dimensional vector space over Q. Probably,
{1, x, x2 , . . . , xn } is the most natural basis for this space. But {H0 x, H1 x, H2 x, . . . , Hn x}
is also a good basis for the space Pn , for our purpose of arithmetical and combinatorial
applications.
For px ∈ Pn ,
px n
Ck Hk x,
1.16
k0
for some uniquely determined bl ∈ Q.
The purpose of this paper is to develop methods for computing Ck from the
information of px. By using these methods, we define some interesting identities.
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Discrete Dynamics in Nature and Society
2. Properties of Hermite Polynomials
From 1.5 and 1.13, we note that
∞
Hm tm
1
m0
∞
m!
∞ 2l
t
l0
t2m
H2m
2m!
m0
l!
∞
2l2l − 1 · · · l 1
l0
2l!
t
2l
∞
n
t2n
2l2l − 1 · · · l 1
H2n−2l 2n!
2l!2n − 2l!
2n!
n0
l0
2.1
∞
n t2n
2l
2n
.
l!
H2n−2l
l
2l
2n!
n0
l0
Thus, by 2.1, we obtain the following recurrence formula.
Proposition 2.1. For n ∈ Z N ∪ {0}, one has
n 1, if n 0
2l
2n
l!
H2n−2l .
l
2l
0, if n /
0
l0
2.2
∞
∞
2
tn
tn
2
Hn −x e2t−x−t e2x−t−−t Hn x−1n .
n!
n!
n0
n0
2.3
By, 1.5, we get
From 2.3, we can derive the following reflection symmetric identity of Hn x:
Hn −x −1n Hn x.
2.4
∂ 2xt−t2 2
e
2x − 2te2xt−t .
∂t
2.5
By 1.5, we easily see that
Discrete Dynamics in Nature and Society
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Thus, by 1.5 and 2.5, we get
∂
∂t
∞
tn
Hn x
n!
n0
LHS of 2.5 RHS of 2.5 ∞
tn
Hn x
2x − 2t
.
n!
n0
∞
∞
tn−1
tn
Hn x
Hn1 x ,
n!
n − 1! n0
n1
∞ n0
tn
2xHn x
n!
−
∞
2Hn x
n0
∞
2xHn x
n0
2.7
tn1
n!
∞ ∞
tn
tn
2xHn x
− 2Hn−1 x
n!
n − 1!
n0
n1
2.6
2.8
∞
tn tn
− 2nHn−1 x .
n! n1
n!
Thus, by 2.6 and 2.7, we get
Hn1 x 2xHn x − 2nHn−1 x,
n ∈ N.
2.9
From 1.15 and 2.9, we note that
Hn1 x − 2xHn x Hn x 0.
2.10
Differentiating on both sides, we have
2n 1Hn x − 2Hn x − 2xHn x Hn x 0.
2.11
Hn x − 2xHn x 2nHn x 0.
2.12
Thus, we have
From 2.12, we note that Hn x is a solution of the following second-order linear differential
equation:
u − 2xu 2nu 0.
2.13
From 1.5, we note that
∞
tn
2
Hn x e2tx−t n!
m0
∞
n0
∞
∞
2xl l
−1k 2k
t
t
l!
k!
l0
k0
n/2
−1k n!2xn−2k tn
k0
k!n − 2k!
n!
.
2.14
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Discrete Dynamics in Nature and Society
Thus, by 2.14, we get
Hn x n/2
k0
−1k n!
2xn−2k
k!n − 2k!
⎧
n/2
−1n/2−l n!22l 2l
⎪
⎪
⎪
x ,
⎪
⎪
⎨ l0 n/2 − l!2l!
⎪
⎪
n−1/2
⎪
⎪
⎪
⎩
l0
if n ≡ 0 mod 2,
2.15
−1n−1/2−l n!22l1
x2l1 , if n ≡ 1 mod 2.
n − 1/2 − l!2l 1!
3. Main Results
By 1.6, we easily get
∞
−∞
e−x Hn xHm xdx −1n
2
∞ −∞
dn −x2
e
Hm xdx.
dxn
3.1
From 3.1, we note that
∞
−∞
√
2
e−x Hn xHm xdx 2n n! πδm,n .
3.2
It is easy to show that
∞
−∞
e
−x2 l
x dx ⎧
0
⎪
⎪
⎨
if l ≡ 1 mod 2,
√
l! π
⎪
⎪
if l ≡ 0 mod 2,
⎩ l
2 l/2!
3.3
where l ∈ Z N ∪ {0}. By 3.3, we get
∞ −∞
n −x2
d e
dxn
x dx m
⎧
0
⎪
⎪
⎨
⎪
⎪
⎩
if n > m or n ≤ m with n − m ≡ 1 mod 2,
n√
m!−1 π
if n ≤ m with n − m ≡ 0 mod 2.
2m−n m − n/2!
3.4
From 3.2, we note that H0 x, H1 x, . . . , Hn x are orthogonal basis for the space Pn {px ∈ Qx | deg px ≤ n} with respect to the inner product
px, qx ∞
−∞
e−x pxqxdx.
2
3.5
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For px ∈ Pn , the polynomial px is given by
px ∞
Ck Hk x,
3.6
k0
where
1
√ px, Hk x
π
∞ k −x2 d e
−1k
k √
pxdx.
2 k! π −∞
dxk
Ck 2k k!
3.7
Let us take px xn ∈ Pn . For n ≡ 0 mod 2, we compute Ck in 3.6 as follows
−1k
Ck k √
2 k! π
∞ −∞
dk e−x
dxk
2
xn dx
⎧
√
⎪
−1k n! π
−1k
⎪
⎨
×
if k ≡ 0 mod 2,
√
2k k! π 2n−k n − k/2!
⎪
⎪
⎩
0
if k ≡ 1 mod 2.
3.8
Let n ≡ 1 mod 2. Then we have
∞ k −x2 d e
−1k
xn dx
Ck k √
2 k! π −∞
dxk
⎧
n!
⎪
⎨
if k ≡ 1 mod 2,
n
2 k!n − k/2!
⎪
⎩0
if k ≡ 0 mod 2.
3.9
Therefore, by 3.6, 3.8, and 3.9, we obtain the following proposition.
Proposition 3.1. One has
x2n x2n1
n
1
2n! H2k x,
22n k0 2k!n − k!
n
1
2n 1! H2k1 x.
22n1 k0 2k 1!n − k!
3.10
Let us take px Bn x. From 3.4, P x can be rewritten by
Bn x n
k0
Ck Hk x,
3.11
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Discrete Dynamics in Nature and Society
where
−1k
Ck k √
2 k! π
∞ −∞
dk e−x
dxk
2
3.12
Bn xdx.
By integrating by parts, we get
∞ −∞
dk e−x
dxk
2
Bn x −n−n − 1 · · · −n − k 1
−1k
∞
−∞
e−x Bn−k xdx
2
∞
n−k n! 2
n−k
e−x xl dx
Bn−k−l
l
n − k! l0
−∞
−1k n!
n − k!
√
l! π
n − k!Bn−k−l
× l
l!n − k − l!
2 l/2!
0≤l≤n−k
l≡0 mod 2
√
−1k n! π
0≤l≤n−k
l≡0 mod 2
3.13
Bn−k−l
.
n − k − l!2l l/2!
Thus, from 3.11 and 3.13, we have
Ck n!
2k k!
0≤l≤n−k
l≡0 mod 2
Bn−k−l
.
n − k − l!2l l/2!
3.14
Therefore, by 3.11 and 3.14, we obtain the following theorem.
Theorem 3.2. For n ∈ Z , one has
Bn x n!
n
k0 0≤l≤n−k
l≡0 mod 2
Bn−k−l
Hk x.
2kl k!n − k − l!l/2!
3.15
Remark 3.3. Let us take px En x. Then, by the same method, we obtain the following
identity:
En x n!
n
k0 0≤l≤n−k
l≡0 mod 2
2kl k!n
En−k−l
Hk x.
− k − l!l/2!
3.16
Now, we consider px Hn x. From 3.6, we note that px can be rewritten as
Hn x n
Ck Hk x,
k0
3.17
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where
−1k
Ck k √
2 k! π
∞ −∞
dk e−x
dxk
2
3.18
Hn xdx.
By integrating by parts, we get
∞ −∞
dk e−x
dxk
2
Hn xdx −2n · · · −2n − k 1
∞
−∞
e−x Hn−k xdx
2
∞
n−k −1k 2k n! 2
n−k l
e−x xl dx
2 Hn−k−l
l
n − k! l0
−∞
−1k 2k n!
n − k!
n−k
l0
l≡0 mod 2
√
2l n − k!
l! π
Hn−k−l l
l!n − k − l!
2 l/2!
n−k
√
−1k 2k n! π
l0
l≡0 mod 2
3.19
Hn−k−l
.
n − k − l!l/2!
From 3.17 and 3.19, we note that
Ck n!
k!
k
−1
√
π
2k k!
0≤l≤n−k
l≡0 mod 2
⎛
⎞
⎜
k k √
×⎜
⎝−1 2 n! π
0≤l≤n−k
l≡0 mod 2
⎟
Hn−k−l
⎟
n − k − l!l/2! ⎠
3.20
Hn−k−l
.
n − k − l!l/2!
Therefore, by 3.17 and 3.20, we obtain the following theorem.
Theorem 3.4. For n ∈ Z , one has
Hn x n!
n
Hn−k−l
Hk x.
k!n − k − l!l/2!
3.21
Hn−k−l
n!Hn x
Hk x .
k!n − k − l!l/2!
n!
3.22
k0 0≤l≤n−k
l≡0 mod 2
From Theorem 3.4, we note that
Hn x n!
n−1
k0 0≤l≤n−k
l≡0 mod 2
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Discrete Dynamics in Nature and Society
Thus, we have, for 0 ≤ k ≤ n − k,
0≤l≤n−k
l≡0 mod 2
Hn−k−l
0.
n − k − l!l/2!
3.23
Let l, k ∈ Z with k ≤ l. Then we easily see that
∞ −∞
∞ −∞
dk e−x
dxk
2
dk e−x
dxk
√
Bl xdx −1k l! π
Bl−k−j
,
l − k − j !2j j/2 !
3.24
El−k−j
.
l − k − j !2j j/2 !
3.25
0 ≤ j ≤ l−k
j ≡ 0 mod 2
2
√
El xdx −1k l! π
0≤j≤l−k
j≡0 mod 2
Let us consider the following polynomial of degree n in Pn :
px n
Bk xBn−k x.
3.26
k0
From 3.6, we note that px can be rewritten as
px n
Ck Hk x,
3.27
k0
where
−1k
Ck k √
2 k! π
∞ −∞
dk e−x
dxk
2
pxdx.
3.28
In 15, it is known that
px n
Bk xBn−k x
k0
n−2 2 n2
Bn−l Bl x n 1Bn x.
l
n 2 l0
3.29
From 3.23 and 3.29, we have the following:
−1k
Ck k √
2 k! π
∞ k −x2 ∞ k −x2 n−2 d e
d e
2 n2
Bl xdx n 1
Bn xdx ,
l
n 2 l0
dxk
dxk
−∞
−∞
3.30
Discrete Dynamics in Nature and Society
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By 3.24 and 3.30, we get
2
dn e−x
Bn xdx
× n 1
Cn dxn
−∞
√
n1
−1n n! πB0
−1n
×
1
n ,
n
√
0
n
2
0!2 0!
2 n! π
2
∞
dn−1 e−x
−1n−1
Cn−1 Bn xdx
× n 1
√
dxn−1
2n−1 n − 1! π
−∞
⎛
⎞
n−1
1
⎜
⎟
B1−j
√
−1
×⎜
j
⎟
n 1−1n−1 n! π
√
⎝
n−1
2 n − 1! π
1 − j !2 j/2 ! ⎠
j0
−1n
√
2n n! π
∞ 3.31
j≡0 mod 2
−1n−1
√
2n−1 n − 1! π
−nn 1
√
× n 1−1n−1 n! πB1 .
2n
For 0 ≤ k ≤ n − 2, we have
Ck
∞ k −x2 ∞ k −x2 n−2 d e
d e
2 n2
Bl xdx n 1
Bn xdx
Bn−l
l
n 2 lk
dxk
dxk
−∞
−∞
⎧
⎪
⎪
n−2 Bl−k−j
√
−1k ⎨ 2 n2
k √
Bn−l −1k l! π ×
j
l
n
2
2 k! π ⎪
⎪
lk
0 ≤ j ≤ l−k l − k − j !2 j/2 !
⎩
−1k
k √
2 k! π
j≡0 mod 2
√
n 1−1k n! π
0 ≤ j ≤ n−k
j≡0 mod 2
n−2
2 n 2 lk
0 ≤ j ≤ l−k
j≡0 mod 2
n 1!
⎫
⎪
⎪
⎬
Bn−k−j
n − k − j !2j j/2 ! ⎪
⎪
⎭
Bn−l Bl−k−j l!
n2
kj
l
2 k! l − k − j ! j/2 !
0 ≤ j ≤ n−k
j≡0 mod 2
Bn−k−j
.
k! n − k − j ! j/2 !2kj
3.32
Therefore, by 3.27 and 3.32, we obtain the following theorem.
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Discrete Dynamics in Nature and Society
Theorem 3.5. For n ∈ Z , one has
n
Bk xBn−k x
k0
⎧
⎪
n−2 ⎪
⎨
n−2
n 2
l!Bn−l Bl−k−j
2 kj
⎪
l
n 2 lk 0 ≤ j ≤ n−k
2 k! l − k − j ! j/2 !
k0 ⎪
⎩
j≡0 mod 2
n 1!
0 ≤ j ≤ n−k
j≡0 mod 2
−
2kj k!
Bn−k−j
⎫
⎪
⎪
⎬
3.33
Hk x
n − k − j ! j/2 ! ⎪
⎪
⎭
n1
nn 1
Hn−1 x n Hn x.
2n
2
Acknowledgment
This research was supported by Basic Science Research Program through the National
Research Foundation of Korea NRF funded by the Ministry of Education, Science and
Technology 2012R1A1A2003786.
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