Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 584643, 11 pages
doi:10.1155/2012/584643
Research Article
Some Identities on Bernoulli and Hermite
Polynomials Associated with Jacobi Polynomials
Taekyun Kim,1 Dae San Kim,2 and Dmitry V. Dolgy3
1
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Department of Mathematics, Sogang University, Seoul, Republic of Korea
3
Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea
2
Correspondence should be addressed to Dae San Kim, dskim@sogong.ac.kr
Received 17 July 2012; Accepted 9 August 2012
Academic Editor: Josef Diblı́k
Copyright q 2012 Taekyun 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 investigate some identities on the Bernoulli and the Hermite polynomials arising from the
orthogonality of Jacobi polynomials in the inner product space Pn .
1. Introduction
α,β
For α, β ∈ R with α > −1 and β > −1, the Jacobi polynomials Pn
α,β
Pn
x are defined as
α 1n
1−x
2F1 −n, 1 α β n; α 1;
n!
2
n n 1 α β n
α 1n x−1 k
k
k
,
n! k0
2
α 1k
x 1.1
see 1–4, where αn αα 1 · · · α n − 1 Γα n/Γα.
From 1.1, we note that
α,β
Pn x
n
nk Γ α β n k 1 x − 1 k
Γα 1 n .
Γα k 1
2
n!Γ α β n 1 k0
1.2
2
Discrete Dynamics in Nature and Society
α,β
By 1.2, we see that Pn x is polynomial of degree n with real coefficients. It is not difficult
α,β
α,β
to show that the leading coefficient of Pn x is 2−n αβ2n
. From 1.2, we have Pn 1 n
αn
n .
By 1.1, we get
d
dx
k
α,β
Pn x
−k
2
Γ n α β k 1 αk,βk
Pn−k
x
Γ nαβ1
αk,βk
1
k n α β k n α β k − 1 · · · n α β 1 Pn−k
x,
2
1.3
where k is a positive integer see 1–4.
α,β
The Rodrigues’ formula for Pn x is given by
α,β
1 − xα 1 xβ Pn
α,β
It is easy to show that u Pn
x −1n
2n n!
d
dx
k 1 − xnα 1 xnβ .
x is a solution of the following differential equation:
1 − x2 u β − α − α β 2 x u n n α β 1 u 0.
α,β
As is well known, the generating function of Pn
∞
α,β
Pn xtn Fx, t 1.4
n0
1.5
x is given by
2αβ
R1 − t Rα 1 t Rβ
,
1.6
√
where R 1 − 2xt t2 , see 1–4.
From 1.3, 1.4, and 1.6, we can derive the following identity:
1
α,β
−1
Pm
α,β
xPn
x1 − xα 1 xβ dx
2αβ1 Γn α 1Γ n β 1
δn,m ,
2n α β 1 Γ n α β 1 Γn 1
1.7
where δn,m is the Kronecker symbol.
Let Pn {px ∈ Rx | deg px ≤ n}. Then Pn is an inner product space with respect
1
to the inner product q1 x, q2 x −1 1 − xα 1 xβ q1 xq2 xdx, where q1 x, q2 x ∈
α,β
α,β
α,β
Pn . From 1.7, we note that {P0 x, P1 x, . . . , Pn x} is an orthogonal basis for Pn .
The so-called Euler polynomials En x may be defined by means of
∞
tn
2
xt
Ext
,
e
e
E
x
n
n!
et 1
n0
1.8
Discrete Dynamics in Nature and Society
3
see 5–22, with the usual convention about replacing En x by En x. In the special case,
x 0, En 0 En are called the Euler numbers.
The Bernoulli polynomials are also defined by the generating function to be
∞
tn
t
xt
Bxt
,
e
e
B
x
n
n!
et − 1
n0
1.9
see 11–21, with the usual convention about replacing Bn x by Bn x.
From 1.8 and 1.9, we note that
Bn x n n
k0
k
En x Bn−k xk ,
n n
k0
k
En−k xk .
1.10
For n ∈ Z , we have
dBn x
nBn−1 x,
dx
dEn x
nEn−1 x
dx
1.11
see 23–29 By the definition of Bernoulli and Euler polynomials, we get
B0 1,
Bn 1 − Bn δ1,n ,
E0 1,
En 1 En 2δ0,n .
1.12
In this paper we give some interesting identities on the Bernoulli and the Hermite polynomials arising from the orthogonality of Jacobi polynomials in the inner product space Pn .
2. Bernoulli, Euler and Jacobi Polynomials
From 1.4, we have
α,β
Pn x
n x − 1 k x 1 n−k
nα
nβ
k0
n−k
k
2
2
.
2.1
By 2.1, we have
∞
n0
α,β
Pn xtn
1
2πi
1 x 1/2znα 1 x − 1/2znβ
dz,
zn1
2.2
where we assume x /
± 1 and circle around 0 is taken so small that −2x ± 1−1 lie neither on
α,β
β,α
it nor in its interior. It is not so difficult to show that Pn −x −1n Pn x.
For qx ∈ Pn , let
qx n
α,β
Ck Pk x,
k0
Ck ∈ R.
2.3
4
Discrete Dynamics in Nature and Society
From 1.7, we note that
α,β
qx, Pk
α,β
α,β
x Ck Pk x, Pk x
Ck
1
−1
α,β 2
1 − xα 1 xβ Pk x dx
2.4
2αβ1 Γk α 1Γ k β 1
Ck .
2k α β 1 Γ α β k 1 k!
Thus, by 2.4, we get
2k α β 1 Γ α β k 1 k! 1
α,β
Ck 1 − xα 1 xβ Pk xqxdx.
2αβ1 Γk α 1Γ k β 1
−1
2.5
Therefore, by 1.7, 2.3, and 2.5, we obtain the following proposition.
Proposition 2.1. For qx ∈ Pn n ∈ N, one has
qx n
α,β
Ck Pk x,
2.6
k0
where
k
−1k 2k α β 1 Γ k α β 1 1
d
kα
kβ
Ck qxdx.
1 − x 1 x
dxk
2αβ1k Γα k 1Γ β k 1
−1
Let us take qx xn ∈ Pn . First, we consider the following integral:
1 −1
d k
1 − xkα 1 xkβ qxdx
dx
1 k d
1 − xkα 1 xkβ xn dx
dx
−1
1 k−1 d
−n
1 − xkα 1 xkβ xn−1 dx
dx
−1
···
n!
−1
n − k!
k
1
−1
1 − xkα 1 xkβ xn−k dx
2.7
Discrete Dynamics in Nature and Society
−1k n!22kαβ1
n − k!
1
5
kα n−k
ykβ 1 − y
dy
2y − 1
0
n−k −1k n! 2kαβ1 n−k l
2
2 −1n−k−l B k l β 1, k α 1
l
n − k!
l0
n−k −1k n! 2kαβ1 n−k l
n−k−l Γ k l β 1 Γk α 1
.
2
2 −1
l
n − k!
Γ 2k α β l 2
l0
2.8
From 2.5 and 16, we have
−1k 2k α β 1 Γ k α β 1
Ck 2αβ1k Γα k 1Γ β k 1
1 k d
×
1 − xkα 1 xkβ xn dx
dx
−1
−1k 2k α β 1 Γ k α β 1 −1k n!22kαβ1
·
n − k!
2αβ1k Γα k 1Γ β k 1
n−k n−k l
n−k−l Γ k l β 1 Γk α 1
×
2 −1
l
Γ 2k α β l 2
l0
2.9
n−k −1n−k−l n−k 2l Γ k l β 1
2k α β 1 Γ k α β 1 n!2k .
l
Γ β k 1 n − k!
Γ 2k α β l 2
l0
By Proposition 2.1, we get
2k α β 1 Γ k α β 1 k
2
x n!
Γ k β 1 n − k!
k0 l0
l 2Γ klβ1
−1n−k−l n−k
α,β
l
Pk x.
×
Γ 2k α β l 2
n
n−k
n 2.10
From 1.9, we have
ext ∞ Bn1 x 1 − Bn1 x tn
1 t
xt t
.
e
−
1
e
t et − 1
n1
n!
n0
2.11
By 2.11, we get
xn Bn1 x 1 − Bn1 x
,
n1
n ∈ Z .
Therefore, by 2.10 and 2.12, we obtain the following theorem.
2.12
6
Discrete Dynamics in Nature and Society
Theorem 2.2. For n ∈ Z , one has
1
{Bn1 x 1 − Bn1 x}
n 1!
n
n−k
−1n−k−l 2kl 2k α β 1 n−k
l
k0
l0 Γ k β 1 Γ 2k α β l 2 n − k!
α,β
×Γ k α β 1 Γ k l β 1 Pk x.
2.13
Let us take qx Bn x ∈ Pn . Then we evaluate the following integral:
1 −1
d
dx
k 1 − xkα 1 xkβ Bn xdx
n n
lk
l
Bn−l
−1k l! 2kαβ1
2
l − k!
1
kα l−k
ykβ 1 − y
dy
2y − 1
0
n n
l−k −1k l! 2kαβ1 l−k m
2
Bn−l
2 −1l−k−m
l
m
−
k!
l
m0
lk
Γ k m β 1 Γk α 1
×
Γ 2k α β m 2
m l−k n
n l Bn−l −1l−m l!22kαβ1 l−k
m 2 Γ k m β 1 Γk α 1
.
l − k!Γ 2k α β m 2
lk m0
2.14
Finding 2.5 and 21, we have
−1k 2k α β 1 Γ α β k 1
Ck 2αβk1 Γα k 1Γ β k 1
1 k
d
kα
kβ
Bn xdx
×
1 − x 1 x
dxk
−1
l−k 2km n
n l Bn−l −1l−m−k l! 2k
α β 1 l−k
m
Γ
β
k
1
−
k!Γ
2k
α
β
m
2
l
lk m0
×Γ kmβ1 Γ kαβ1 .
2.15
Discrete Dynamics in Nature and Society
7
Theorem 2.3. For n ∈ Z , one has
l−k 2km n
n
n l Bn−l −1l−m−k l! 2k α β 1 l−k
m
Bn x Γ
β
k
1
−
k!Γ
2k
α
β
m
2
l
k0 lk m0
α,β
×Γ k m β 1 Γ k α β 1 Pk x.
α,β
Let qx Pn
integral:
2.16
x ∈ Pn . From Proposition 2.1 , we firstly evaluate the following
1 −1
d
dx
k α,β
1 − xkα 1 xkβ Pn xdx
1 Γ nαβk1 1
αk,βk
−1 k
1 − xkα 1 xkβ Pn−k
xdx.
2
Γ nαβ1
−1
2.17
k
By 2.1 and 2.17, we get
1 −1
d
dx
k α,β
1 − xkα 1 xkβ Pn xdx
n−k
−1k Γ n α β k 1 nα
nβ
n−k−l
l
2k
Γ nαβ1
l0
×
1
−1
1 − xkα 1 xkβ
x−1
2
l x1
2
n−k−l
dx
n−k
−1k Γ n α β k 1 nα
nβ
−1l 22kαβ1
n
−
k
−
l
l
2k
Γ nαβ1
l0
×
1
0
1−y
kαl
ynβ−l dy
n−k
nαβk1 nα
nβ
−1 2
−1l
n
−
k
−
l
l
Γ nαβ1
l0
× B k α l 1, n β − l 1
n−k
n α n β
k αβk1 Γ n α β k 1
−1 2
−1l
n
−
k
−
l
l
Γ nαβ1
l0
k αβk1 Γ
8
Discrete Dynamics in Nature and Society
Γα k l 1Γ n β − l 1
×
Γ αβkn2
n−k 1
nα
nβ
−1 2
−1l
l
Γ n α β 1 l0 n − k − l
Γα k l 1Γ n β − l 1
×
.
αβkn1
k αβk1
2.18
It is easy to show that
Γ nβ−l1
n β − l · · · βΓ β
n β − l ··· β k 1
Γ βk1
β k · · · βΓ β
nβ−l
n − k − l!.
n−k−l
2.19
From 2.5, 2.18, and 2.19, we can derive the following equation:
−1k 2k α β 1 Γ k α β 1
2αβk1 Γα k 1Γ β k 1
1 k α,β
d
×
1 − xkα 1 xkβ Pn xdx
dx
−1
n−k
2k α β 1 Γ α β k 1 nα
nβ
αkl
n−k−l
l
l
Γ βk1 Γ nαβ1
l0
Ck l
× l!−1 Γ nβ−l1
αβkn1
n−k
2k α β 1 Γ α β k 1
l0
×
nα
n−k−l
nβ
l
n − k − l!l!
−1l .
αβkn1
Therefore, by Proposition 2.1, we obtain the following theorem.
nβ−l
n−k−l
2.20
Discrete Dynamics in Nature and Society
9
Theorem 2.4. For n ∈ Z , one has
α,β
n
n−k
Γ n α β 1 Pn x αβk
nα
2k α β 1
k
n−k−l
Γ αβ1
l0
k0
l
n β α k l n β − l −1 n − k − l!k!l! α,β
Pk x.
×
l
l
n−k−l αβnk1
2.21
Let Hn x be the Hermite polynomial with
Hn x qx n
k0
α,β
Ck Pk
x,
2.22
where
−1k 2k α β 1 Γ k α β 1
Ck 2αβk1 Γα k 1Γ β k 1
1 k d
×
1 − xkα 1 xkβ Hn xdx.
dx
−1
2.23
Integrating by parts, one has
1 −1
d
dx
k 1 − xkα 1 xkβ Hn xdx
1
2k −1k n!
n − k!
1
n−k 2k −1k n! n−k
Hn−k−l 2l
1 − xkα 1 xkβ xl dx
l
n − k! l0
−1
n−k l 2k −1k n! n−k
l
Hn−k−l 22kαβl1
−1l−m 2m
l
m
n − k! l0
m0
×
1
1−y
−1
1 − xkα 1 xkβ Hn−k xdx
kα
ykβm dy
0
l n−k 2k −1k n! n−k
l
Hn−k−l −1l−m 22kαβml1
l
m
n − k! l0 m0
Γk α 1Γ β k m 1
.
×
Γ 2k α β m 2
2.24
10
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By 2.23 and 29, we get
Ck l
n−k k!
Hn−k−l −1l−m 2k α β 1 αβk
k
α β 1 2kαβm1
m 2k!n − k!
m2k
n−k l l
l0 m0
m
× 22kml n!
2.25
βkm
m!.
m
Therefore, by 2.22 and 2.25, we obtain the following theorem.
Theorem 2.5. For n ∈ Z , one has
⎧
n−k l l−m l
n ⎨
n−k 2k α β 1
α β 1 Hn x m Hn−k−l −1
l
⎩ l0 m0
2kαβm1
n!
m 2k!n − k!
k0
m2k
αβk
α,β
2kml β k m
×
m! Pk x,
k!2
m
k
2.26
where Hn is the nth Hermite number.
Remark 2.6. By the same method as Theorem 2.3 , we get
n
n−k
2kl 2k α β 1 n−k
−1n−k−l
1
l
{En x 1 En x} 2n!
k0
l0 Γ k β 1 Γ 2k α β l 2 n − k!
α,β
×Γ k α β 1 Γ k l β 1 Pk x.
2.27
Acknowledgments
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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