Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 957350, 15 pages
doi:10.1155/2012/957350
Research Article
Extended Laguerre Polynomials
Associated with Hermite, Bernoulli, and
Euler Numbers and Polynomials
Taekyun Kim1 and Dae San Kim2
1
2
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
Correspondence should be addressed to Dae San Kim, dskim@sogang.ac.kr
Received 14 June 2012; Accepted 9 August 2012
Academic Editor: Pekka Koskela
Copyright q 2012 T. Kim and D. S. Kim. 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.
Let Pn {px ∈ Rx | deg px ≤ n} be an inner product space with the inner product px,
∞
qx 0 xα e−x pxqxdx, where px, qx ∈ Pn and α ∈ R with α > −1. In this paper we
study the properties of the extended Laguerre polynomials which are an orthogonal basis for Pn .
From those properties, we derive some interesting relations and identities of the extended Laguerre
polynomials associated with Hermite, Bernoulli, and Euler numbers and polynomials.
1. Introduction/Preliminaries
For α ∈ R with α > −1, the extended Laguerre polynomials are defined by the generating
function as follows:
exp−xt/1 − t
1 − tα1
∞
Lαn xtn ,
1.1
n0
see 1–6.
From 1.1, we can derive the following:
Lαn x n
−1r nα
n−r r0
see 1–9.
r!
xr ,
1.2
2
Abstract and Applied Analysis
As is well known, Rodrigues’ formula for Lαn x is given by
Lαn x
1 −α x dn −x nα
e x
,
x e
n!
dxn
1.3
see 1–6, 8, 9.
From 1.3, we note that
∞
0
xα e−x Lαm xLαn xdx 1
Γα n 1δm,n ,
n!
α > −1,
1.4
where δm,n is the Kronecker symbol.
From 1.1, 1.2, and 1.3, we can derive the following identities:
n 1Lαn1 x x − α − 2n − 1Lαn x n αLαn−1 x 0,
d α
d α
Ln x −
L x Lαn−1 x 0,
dx
dx n−1
x
n ∈ N,
for n ≥ 1,
d α
L x nLαn x − n αLαn−1 x 0,
dx n
n ≥ 1,
1.5
1.6
1.7
and Lαn x is a solution of xy α 1 − xy xy 0.
The derivatives of general Laguerre polynomials are given by
d α
L x −Lα1
n−1 x,
dx n
d −x α e Ln x −e−x Lα1
n x,
dx
d α α
x Ln x n αxα−1 Lα−1
n x,
dx
d α −x α x e Ln x n 1xα−1 e−x Lα−1
n1 x.
dx
1.8
The nth Bernoulli polynomials, Bn x, are defined by the generating function to be
et
∞
tn
t
ext eBxt Bn x ,
n!
−1
n0
1.9
see 10–17, with the usual convention about replacing Bn x by Bn x. In the special case,
x 0, Bn 0 Bn are called the nth Bernoulli numbers.
It is well known that the nth Euler polynomials are also defined by the generating
function to be
∞
tn
2
xt
Ext
,
e
e
E
x
n
n!
et 1
n0
see 18–22, with the usual convention about replacing En x by En x.
1.10
Abstract and Applied Analysis
3
The Hermite polynomials are given by
Hn x H 2xn n n
l0
l
2l xl Hn−l ,
1.11
see 23, 24, with the usual convention about replacing H n by Hn . In the special case, x 0,
Hn 0 Hn are called the nth Hermite numbers.
From 1.11, we note that
d
Hn x 2nH 2xn−1 2nHn−1 x,
dx
1.12
see 23, 24, and Hn x is a solution of Hermite differential equation which is given by
y − 2xy ny 0,
1.13
see 1–6, 23–32.
Throughout this paper we assume that α ∈ R with α > −1. Let Pn {px ∈ Rx|
deg
px
∞ α −x ≤ n}. Then Pn is an inner product space with the inner product px, qx x e pxqxdx, where px, qx ∈ Pn . By 1.4 the set of the extended Laguerre
0
polynomials {Lα0 x, Lα1 x, . . . , Lαn x} is an orthogonal basis for Pn . In this paper we study
the properties of the extended Laguerre polynomials which are an orthogonal basis for Pn .
From those properties, we derive some new and interesting relations and identities of the
extended Laguerre polynomials associated with Hermite, Bernoulli and Euler numbers and
polynomials.
2. On the Extended Laguerre Polynomials Associated with Hermite,
Bernoulli, and Euler Polynomials
For px ∈ Pn , px is given by
px n
Ck Lαk x,
for uniquely determined real numbers Ck .
2.1
k0
From 1.3, 1.4, and 2.1, we note that
px, Lαk x
Ck Lαk x, Lαk x
Ck
∞
0
xα e−x Lαk xLαk xdx Ck
Γα k 1
.
k!
2.2
4
Abstract and Applied Analysis
Thus, by 2.2, we get
k!
px, Lαk x
Γα k 1
1 ∞ dk kα −x
k!
pxdx
x e
Γα k 1 k! 0 dxk
∞ k
1
d kα −x
pxdx.
x e
Γα k 1 0 dxk
Ck 2.3
Therefore, by 2.1 and 2.3, we obtain the following proposition.
Proposition 2.1. For px ∈ Pn , let
px n
Ck Lαk x,
α > −1.
2.4
k0
Then one has the following:
1
Ck Γα k 1
∞
0
dk kα −x
pxdx.
x e
dxk
2.5
To derive inverse formula of 1.2, let take one px xn ∈ Pn . Then, by Proposition 2.1,
one gets
dk −x kα
xn dx
e x
k
dx
0
∞
k nn − 1 · · · n − k 1
e−x xαn dx
−1
Γα k 1
0
1
Ck Γα k 1
−1k
∞
nn − 1 · · · n − k 1
Γα n 1
Γα k 1
2.6
−1k n!α n · · · αΓα
α k · · · αΓαn − k!
α n · · · α k 1
αn
k
−1 n!
.
−1 n!
n−k
n − k!
k
Therefore, by 2.6, we obtain the following corollary.
Corollary 2.2 Inverse formula of Lαn x. For n ∈ Z , one has
xn n!
n αn
k0
n−k
−1k Lαk x.
2.7
Abstract and Applied Analysis
5
Let one takes Bernoulli polynomials of degree n with px Bn x ∈ Pn . Then Bn x
can be written as
Bn x n
Ck Lαk x,
α ∈ R with α > −1.
2.8
k0
From Proposition 2.1, one has
1
Ck Γα k 1
∞
0
dk −x kα
Bn xdx
e x
dxk
−1k nn − 1 · · · n − k 1
Γα k 1
∞
e−x xkα Bn−k xdx
0
∞
n−k −1k nn − 1 · · · n − k 1 n−k
e−x xkαl dx
Bn−k−l
l
Γα k 1
0
l0
2.9
n−k −1k nn − 1 · · · n − k 1 n−k
Bn−k−l Γα k l 1.
l
Γα k 1
l0
By the fundamental property of gamma function, one gets
αkl Γα k l 1
α l k · · · α k 1Γα k 1n − k!
n−k
l
.
l
Γα k 1n − k!
Γα k 1n − k!n − k − l!l!
n − k − l!
2.10
Therefore, by 2.8, 2.9, and 2.10, we obtain the following theorem.
Theorem 2.3. For n ∈ Z , α ∈ R with α > −1, one has
Bn x n!
n−k
n k0 l0
−1k
Bn−k−l
αkl
Lα x.
l
n − k − l! k
2.11
As is known, relationships between Hermite and Laguerre polynomials are given by
x2 ,
H2m x −1m 22m m!L−1/2
m
2.12
x2 ,
H2m1 x −1m 22m1 m!L−1/2
m
2.13
see 1–6. In the special case α −1/2, by 2.12 and 2.13, we obtain the following corollary.
6
Abstract and Applied Analysis
Corollary 2.4. For n ∈ Z , one has
⎞
⎛
1
n−k
n H
x
k
l
−
2k
⎠ Bn−k−l .
⎝ 2
Bn x2 n!
2k
n − k − l!
l
k0 l0 2 k!
2.14
By the same method as Theorem 2.3, one gets
En x n!
En−k−l
αkl
Lα x,
−1
l
− k − l! k
n
l0
n n−k
k0
k
2.15
where En x are the nth Euler polynomials. In the special case, x 0, En 0 En are called
the nth Euler numbers.
Let one considers the nth Hermite polynomials with px Hn x ∈ Pn . Then Hn x
can be written as
Hn x n
Ck Lαk x,
α ∈ R with α > −1.
2.16
k0
From Proposition 2.1, one notes that
1
Ck Γα k 1
−2n
Γα k 1
∞
0
∞
0
dk −x kα
Hn xdx
e x
dxk
dk−1 −x kα
Hn−1 xdx
e x
dxk−1
···
−2n−2n − 1 · · · −2n − k 1
Γα k 1
∞
e−x xkα Hn−k xdx
2.17
0
∞
n−k −1k 2k n!
n−k
l
e−x xkαl dx
Hn−k−l 2
l
Γα k 1n − k! l0
0
n−k −1k 2k n!
n−k
Hn−k−l 2l Γα k l 1.
l
Γα k 1n − k! l0
It is not difficult to show that
n−k αkl Γα k l 1 n − k!α k l · · · α k 1Γα k 1
l
.
Γα k 1n − k!
n − k − l!l!Γα k 1n − k!
n − k − l!
l
Therefore, by 2.16, 2.17, and 2.18, we obtain the following theorem.
2.18
Abstract and Applied Analysis
7
Theorem 2.5. For n ∈ Z , α ∈ R with α > −1, one has
Hn x n!
n−k
n −1k 2kl
k0 l0
Hn−k−l
αkl
Lα x.
l
n − k − l! k
2.19
In the special case, α −1/2, we obtain the following corollary.
Corollary 2.6. For n ∈ Z , one has
⎞
⎛
1
n−k l−k
n 2 H2k x ⎝− k l⎠ Hn−k−l
2
.
Hn x n!
2
k!
n − k − l!
l
k0 l0
2.20
For β ∈ R with β > −1, let one takes
β
2.21
px Ln x ∈ Pn .
β
Then Ln x is also written as
β
Ln x n
Ck Lαk x.
2.22
k0
From Proposition 2.1, one can determine the coefficients of 2.22 as follows:
∞ k
d −x kα
1
β
Ln xdx
e x
Ck Γα k 1 0 dxk
∞ k−1
d
1
β1
−x kα
Ln−1 xdx
e x
Γα k 1 0 dxk−1
···
1
Γα k 1
1
Γα k 1
∞
0
n−k −1
r
r0
r
2.23
βk
e−x xkα Ln−k xdx
nβ
n−k−r
r!
nβ
n−k−r
n−k −1
1
Γα k 1 r0
r!
∞
e−x xkαr dx
0
Γk α r 1.
By the fundamental property of gamma function, one gets
Γk α r 1 k α r · · · α k 1Γα k 1
r!Γα k 1
r!Γα k 1
kαr
.
r
Therefore, by 2.22, 2.23, and 2.24, we obtain the following theorem.
2.24
8
Abstract and Applied Analysis
Theorem 2.7. For β ∈ R with β > −1, and n ∈ Z , one has
β
Ln x n−k
n −1r
k0 r0
αkr α
Lk x.
r
2.25
αkr
Lαk x 0.
r
2.26
nβ
n−k−r
In the special case, α β, one has
n−1 n−k
k0
−1
r
r0
nα
n−k−r
Thus, by 2.26, we obtain the following corollary.
Corollary 2.8. For 0 ≤ k ≤ n − 1, α ∈ R with α > −1, one has
n−k
−1
r
r0
nα
n−k−r
αkr
r
0.
2.27
Let one assumes that
px n
Bl xBn−l x ∈ Pn .
2.28
l0
Then px can be rewritten as a linear combination of Lα0 x, Lα1 x, . . . , Lαn x as follows:
px n
Bl xBn−l x l0
n
Ck Lαk x.
2.29
k0
By Proposition 2.1, one can determine the coefficients of 2.29 as follows:
n
1
Ck Γα k 1 l0
∞
0
dk −x kα
Bl xBn−l xdx.
e x
dxk
2.30
It is known that
n
l0
see 25.
Bl xBn−l x n−2 2 n2
Bn−l Bl x n 1Bn x,
l
n 2 l0
2.31
Abstract and Applied Analysis
9
From 2.30 and 2.31, one notes that
∞
dn −x nα
e
x
Bn xdx
dxn
0
∞
n1
n 1!−1n
Γn α 1
e−x xnα dx −1n n!
Γα n 1
Γα n 1
0
n1
Cn Γα n 1
Cn−1
n 1!−1n ,
∞ n−1
d
n1
−x n−1α
Bn xdx
e x
Γα n 0 dxn−1
∞
n1
n−1
e−x xn−1α B1 xdx
−1 n!
Γα n
0
1
n1
−1n−1 n! Γα n 1 − Γα n
Γα n
2
1
n 1!−1n−1 n α −
.
2
2.32
For 0 ≤ k ≤ n − 2, one has
1
Ck Γα k 1
1
Γα k 1
∞ k
n−2 d −x kα
2 n2
Bl xdx
e x
Bn−l
l
n 2 l0
dxk
0
∞ k
d −x kα
Bn xdx
e x
n 1
dxk
0
∞
n−2 2 n2
k
e−x xkα Bl−k xdx
Bn−l −1 ll − 1 · · · l − k 1
l
n 2 lk
0
∞
e−x xkα Bn−k xdx
n 1−1k nn − 1 · · · n − k 1
0
⎧
∞
n−2 l−k ⎨ 2 l!
1
n2
l−k
k
e−x xkαj dx
Bn−l −1
Bl−k−j
l
j
Γα k 1 ⎩ n 2 lk
l − k! j0
0
n 1−1k
n!
n − k!
n−k n−k
j0
j
∞
Bn−k−j
0
e−x xkαj dx
⎫
⎬
⎭
⎧
n−2 l−k ⎨ 2 1
l!
l−k
n2
Bl−k−j Γ α k j 1
Bn−l −1k
j
l
Γα k 1 ⎩ n 2 lk j0
l − k!
10
Abstract and Applied Analysis
⎫
n−k ⎬
n!
n−k
n 1−1k
Bn−k−j Γ α k j 1
⎭
j
n − k! j0
l−k n−2 α k j · · · α k 1Bl−k−j
2 n2
k
Bn−l −1 l!
l
n 2 lk j0
j! l − k − j !
n−k α k j · · ·
α k 1
n 1−1 n!
Bn−k−j
j!
n
−
k
−j !
j0
k
l−k n−2 Bl−k−j
2 αkj
n2
Bn−l −1k l!
j
l
n 2 lk j0
l−k−j !
n 1−1k n!
n−k αkj
j0
j
Bn−k−j
.
n−k−j !
2.33
Therefore, by 2.29 and 2.32, we obtain the following theorem.
Theorem 2.9. For n ∈ Z , α ∈ R with α > −1, one has
n
k0
Bk xBn−k x
⎧
n−2 ⎨
n−2 Bl−k−j
2 αkj
k n2
l!Bn−l
−1
⎩
l
j
n
2
l
−
k−j !
k0
lk 0≤j≤n−k
⎫
⎬
B
n−k−j
α
k
j
n 1!
Lαk x
−1k
j
n − k − j !⎭
0≤j≤n−k
2.34
1 α
α
L x .
−1 n 1! Ln x − n α −
2 n−1
n
Let one takes the polynomial px in Pn as follows:
px Bi1 xBi2 x · · · Bir x ∈ Pn .
i1 ···ir n
2.35
From the orthogonality of {Lα0 x, . . . , Lαn x}, one notes that
px i1 ···ir n
Bi1 xBi2 x · · · Bir x n
Ck Lαk x,
2.36
k0
where
1
Ck Γα k 1
∞
0
dk −x kα
pxdx.
e x
dxk
2.37
Abstract and Applied Analysis
11
It is known in 25 that
i1 ···ir n
Bi1 x · · · Bir x
⎧
n−2 1
nr−1 ⎨
r
⎩max{0,kr−n}≤a≤r a i
k
2 k0
1 ···ia na−k−r
i1 ···ir n−k
Bi1 · · · Bir
⎫
⎬
Ek x ⎭
Bi1 Bi2 · · · Bia
2.38
nr−1
En x.
n
From 2.35, 2.37, and 2.38, one notes that
Cn Cn−1 nr−1 ∞
dn −x nα
e x
En xdx
Γα n 1 0 dxn
nr−1 ∞
n
xnα e−x dx
−1n n!
Γα n 1
0
nr−1 nr−1
n
n
−1 n!Γα n 1 −1n n!,
n
Γα n 1
nr−1 ∞ n−1
d
n
−x nα−1
e x
En xdx
Γα n 0 dxn−1
nr−1 ∞
n
n
Γα n
nr−1 2.39
e−x xnα−1 E1 xdx
−1n−1 n!
0
1
−1n−1 n! Γα n 1 − Γα n
Γα n
2
1
nr−1
.
−1n−1 n! n α −
n
2
n
For 0 ≤ k ≤ n − 2, by 2.37 and 2.38, one gets
⎧
⎛
n−2 ⎨1 1
nr−1 ⎝
r
Ck l
a
Γα k 1 ⎩ 2 l0
i
max{0,lr−n}≤a≤r
1 ···ia na−l−r
i1 ···ir n−l
Bi1 · · · Bir
∞
0
⎫
⎬
∞ k
d −x nα
nr−1
e x
En xdx
⎭
n
dxk
0
Bi1 Bi2 · · · Bia
dk −x nα
El xdx
e x
dxk
12
Abstract and Applied Analysis
⎧
⎛
n−2 ⎨1 1
nr−1 ⎝
r
l
a
Γα k 1 ⎩ 2 lk
i
max{0,lr−n}≤a≤r
1 ···ia na−l−r
×
∞
i1 ···ir n−l
Bi1 Bi2 · · · Bia
−1k ll − 1 · · · l − k 1
Bi1 · · · Bir
e−x xkα El−k xdx
0
⎫
∞
⎬
nr−1
e−x xkα En−k xdx
−1k nn − 1 · · · n − k 1
⎭
n
0
⎧
⎛
n−2 ⎨1 1
nr−1 ⎝
l
Γα k 1 ⎩ 2 lk
max{0,lr−n}≤a≤r
×
∞
i1 ···ir n−l
i1 ···ia na−l−r
Bi1 · · · Bia
l−k −1k l! l−k
El−k−j
j
l − k! j0
Bi1 · · · Bir
e−x xkαj dx
0
⎫
∞
k
n−k ⎬
n r − 1 −1 n! n−k
e−x xkαj dx
En−k−j
⎭
n
j
n − k! j0
0
⎛
n−2 1
nr−1 ⎝
l
2 lk
max{0,lr−n}≤a≤r
× −1k l!
l−k αkj
j0
j
i1 ···ia na−l−r
Bi1 · · · Bia i1 ···ir n−l
⎞
Bi1 · · · Bir ⎠
El−k−j
l−k−j !
n−k En−k−j
αkj
nr−1
k
.
−1 n!
j
n
n
−k−j !
j0
2.40
Therefore, by 2.36, 2.39, and 2.40, we obtain the following theorem.
Theorem 2.10. For n ∈ Z , r ∈ N, and α ∈ R with α > −1, one has
i1 ···ir n
Bi1 xBi2 x · · · Bir x
⎧
⎛
k ⎨
n−2
n−2 −1
nr−1 ⎝
r
l!
⎩
l
a
2
k0
lk
i
max{0,lr−n}≤a≤r
1 ···ia na−l−r
i1 ···ir n−l
Bi1 · · · Bir
Bi1 · · · Bia
Abstract and Applied Analysis
13
l−k αkj
nr−1
×
−1k n!
j
n
l−k−j !
j0
⎫
n−k En−k−j ⎬ α
αkj
nr−1
×
Lk x −1n−1 n!
⎭
j
n
n
−
k
−
j
!
j0
El−k−j
1 α
nr−1
Ln−1 x × nα−
−1n n!Lαn x.
n
2
2.41
For m, s ∈ Z with m s n, let one assumes that px Lαs xLαm x ∈ Pn .
By Proposition 2.1, one sees that px can be written as
px Lαs xLαm x n
α ∈ R with α > −1.
Ck Lαk x,
2.42
k0
From the orthogonality of {Lα0 x, Lα1 x, . . . , Lαn x}, one has
1
Ck Γα k 1
∞
0
dk −x kα
pxdx.
e x
dxk
2.43
By 1.2, 1.3, and 1.8, one gets
Lαs xLαm x
s
−1r1 s α r1
x
s − r1
r1 !
r1 0
r1
m
−1r2 m α r2
x
m − r2
r2 !
r2 0
r
x
r
sα
n−sα
.
−1
r
s
−
r
α
r
−
r
r!
1
1
1
0
n
r
r0
2.44
r
Thus, from 2.44, one has
Lαs xLαm x
n
r
r0
r
x
r
sα
n−sα
.
−1
l
α
l
α
r
−
l
r!
l0
r
By 2.44 and 2.45, one gets
n
r
1
1
r
sα
n−sα
r
Ck −1
l
αl
αr−l
Γα k 1 r0
r!
l0
∞ k
d −x kα
xr dx
e x
×
dxk
0
n
r
1
r
sα
n−sα 1
r
−1
l
αl
α r − l r!
Γα k 1 rk
l0
2.45
14
Abstract and Applied Analysis
× −1k rr − 1 · · · r − k 1
∞
e−x xrα dx
0
n
r
1
r
sα
n−sα 1
r
−1
l
αl
α r − l r!
Γα k 1 rk
l0
−1k r!
Γα r 1
r − k!
n
r
r
sα
n − s α α rα r − 1 · · · α k 1
rk
−1
l
αl
αr−l
r − k!
rk
l0
×
n
−1
rk
r
r
sα
n−sα
αr
rk
l0
αl
l
αr−l
r−k
.
2.46
Therefore, by 2.42 and 2.46, we obtain the following theorem.
Theorem 2.11. For s, m ∈ Z with s m n, α ∈ R with α > −1, one has
Lαs xLαm x
n
n k0 rk
−1
rk
r
r
sα
n−sα
αr
l0
l
αl
αr−l
r−k
Lαk x.
2.47
Acknowledgments
The authors would like to express their sincere gratitude to referee for his/her valuable
comments and information. 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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