Extended Laguerre Polynomials Associated with Hermite, Bernoulli

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, [email protected] 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


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


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


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


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


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


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


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


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


×

∞

i1 ···ir n−l


−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


i1 ···ir n−l

Bi1 · · · Bir

Bi1 · · · Bia


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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