Some Identities on Laguerre Polynomials in Connection with Bernoulli and Euler Numbers

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Volume 2012, Article ID 619197,10pages doi:10.1155/2012/619197

Research Article

Some Identities on Laguerre Polynomials in Connection with Bernoulli and Euler Numbers

Dae San Kim,

¹

Taekyun Kim,

²

and Dmitry V. Dolgy

³

1Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

2Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

3Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea

Correspondence should be addressed to Dae San Kim,[email protected] Received 15 May 2012; Accepted 28 June 2012

Academic Editor: Lee Chae Jang

Copyrightq2012 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 study some interesting identities and properties of Laguerre polynomials in connection with Bernoulli and Euler numbers. These identities are derived from the orthogonality of Laguerre polynomials with respect to inner productf, g_∞

0 e^−x²fxgxdx.

1. Introduction/Preliminaries

As is well known, Laguerre polynomials are defined by the generating function as

exp−xt/1−t

1−t ^∞

n0

L_nxtⁿ 1.1

see1,2. By1.1, we get

∞ n0

Lnxtⁿ exp−xt/1−t

1−t ^∞

r0

−1^rx^rt^r

r! 1−t^−r1 ^∞

n0

∞ r0

−1^rx^rrs!

r!r!s! t^rs^∞

n0

_n

r0

−1^rⁿr r! x^r

tⁿ.

1.2

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Thus, from1.2, we have

L_nx ⁿ

r0

−1^rⁿ_r

r! x^r. 1.3

By 1.3, we see that L_nx is a polynomial of degree nwith rational coeﬃcients and the leading coeﬃcient−1ⁿ/n!. It is well known that Rodrigues’ formula is given by

L_nx 1 n!e^x

dⁿ dxⁿe^−xxⁿ

1.4

see1–27. From1.1, we can derive the following of Laguerre polynomials:

L₀x 1, L₁x −x1,

n1Ln1x 2n1−xLnx−nL_n−1x, n≥1, 1.5 L_nx L_n−1x−L_n−1x 0, n≥1, 1.6 xL_nx nLnx−nL_n−1x 0, n≥1. 1.7 By1.7, we easily see thatu L_nxis a solution of the following diﬀerential equation of order 2:

xux 1−xux nux 0. 1.8

The Bernoulli numbers,Bn, are defined by the generating function as t

e^t−1 e^Bt^∞

n0

Bntⁿ

n! 1.9

see1–28,28, with the usual convention about replacingBⁿbyB_n. It is well known that Bernoulli polynomials of degreenare given by

Bnx Bxⁿⁿ

l0

n l

B_n−lx^l 1.10

see2,26. Thus, from1.10, we have

B_nx dBnx

dx nB_n−1x 1.11

see3–12. From1.9and1.10, we can derive the following recurrence relation:

B₀1, B1ⁿ−B_n δ_1,n 1.12 whereδ_n,kis Kronecker’s symbol.

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The Euler polynomialsEnxare also defined by the generating function as 2

e^t1e^xte^Ext^∞

n0

E_nxtⁿ

n! 1.13

see27,28, with the usual convention about replacingEⁿxbyEnx.

In this special case,x 0,E_n0 E_nare called thenth Euler numbers. From1.13, we note that the recurrence formula ofE_nis given by

E01, E1ⁿEn2δ0,n 1.14 see24. Finally, we introduce Hermite polynomials, which are defined by

e^2xt−t² e^Hxt^∞

n0

Hnxtⁿ

n! 1.15

see29. In the special case,x0,H_n0 H_nis called then-th Hermite number. By1.15, we get

Hnx H2xⁿⁿ

l0

n l

H_n−l2^lx^l 1.16

see29. It is not diﬃcult to show that _∞

0

e^−xLmxLnxdxδm,n, m, n∈ZN∪ {0}. 1.17

In the present paper, we investigate some interesting identities and properties of Laguerre polynomials in connection with Bernoulli, Euler, and Hermite polynomials. These identities and properties are derived from1.17.

2. Some Formulae on Laguerre Polynomials in Connection with Bernoulli, Euler, and Hermite Polynomials

Let

Pn px∈Qx|degpx≤n

. 2.1

ThenPnis an inner product space with the inner product p₁x, p2x

_∞

0

e^−xp₁xp2xdx,

p₁x, p2x∈Pn

. 2.2

By1.17,2.1, and2.2, we see thatL₀x, L1x, . . . , Lnxare orthogonal basis forPn.

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Forpx∈Pn, it is given by

px ⁿ

k0

C_kL_kx, 2.3

where

C_k

px, Lkx

_∞

0

e^−xL_kxpxdx 1 k!

_∞

0

d^k dx^ke^−xx^k

pxdx. 2.4

Let us takepx xⁿ∈Pn. From2.3and2.4, we note that

C_k 1 k!

_∞

0

d^k dx^ke^−xx^k

xⁿdx −n k!

_∞

0

d^k−1 dx^k−1e^−xx^k

xⁿ⁻¹dx

−n−n−1 k!

_∞

0

xⁿ⁻²dx

· · ·

−1^knn−1· · ·n−k1 k!

_∞

0

e^−xxⁿdx −1^k n

k

n!.

2.5

Therefore, by2.3,2.4, and2.5, we obtain the following theorem.

Theorem 2.1. Forn∈Z, one has

xⁿn!

n k0

−1^k n

k

Lkx. 2.6

Let us considerpx B_nx∈Pn. Then, by2.3and2.4, we get

Ck 1 k!

_∞

0

d^k dx^ke^−xx^k

Bnxdx −n k!

_∞

0

B_n−1xdx

−n−n−1 k!

_∞

0

B_n−2xdx · · ·

−1^knn−1· · ·n−k1 k!

n−k

l0

n−k l

B_n−k−l

_∞

0

e^−xx^kldx

−1^k n

k _n−k

l0

n−k l

B_n−k−lkl!n!−1^k^n−k

l0

kl k

B_n−k−l n−k−l!.

2.7

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Theorem 2.2. Forn∈Z, one has Bnx n!

n k0

n−k

l0

−1^k kl

k

B_n−k−l

n−k−l!Lkx. 2.8

Let us takepx Enx∈Pn. By the same method, we easily see that

Enx n!

n k0

n−k

l0

−1^k kl

k

E_n−k−l

n−k−l!Lkx. 2.9

Forpx Hnx∈Pn, we have

H_nx ⁿ

k0

C_kL_kx, 2.10

where

C_k 1 k!

_∞

0

d^k dx^ke^−xx^k

H_nxdx −2n k!

_∞

0

H_n−1xdx

−2n−2n−1 k!

_∞

0

H_n−2xdx

· · ·

−2n−2n−1· · ·−2n−k1 k!

_∞

0

e^−xx^kH_n−kxdx

−1^k2^knn−1· · ·n−k1 k!

n−k

l0

n−k l

H_n−k−l2^l _∞

0

e^−xx^kldx

−1^k n

k _n−k

l0

n−k l

2^klH_n−k−lkl!n!−1^k^n−k

l0

2^kl kl

k

H_n−k−l n−k−l!.

2.11

Therefore, by2.10and2.11, we obtain the following theorem.

Theorem 2.3. Forn∈Z, one has H_nx n!

n k0

n−k

l0

−1^k2^kl kl

k

H_n−k−l

n−k−l!L_kx. 2.12 Letpx _n

k0B_kxBn−kx∈Pn. Then we have px ⁿ

k0

B_kxB_n−kx ⁿ

k0

C_kL_kx, 2.13

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where

C_k 1 k!

_∞

0

d^k dx^ke^−xx^k

pxdx. 2.14

In15, it is known that n k0

BkxBn−kx 2 n2

n−2 l0

n2 l

B_n−lBlx n1Bnx. 2.15

By2.14and2.15, we get

Ck 1 k!

2 n2

n−2

l0

n2 l

B_n−l

_∞

0

d^k dx^ke^−xx^k

Blxdx

n1 _∞

0

d^k dx^ke^−xx^k

B_nxdx

.

2.16

From2.16, we can derive the following equations2.17-2.18:

C_n −1ⁿn1!, C_n−1nn1!−1ⁿ⁻¹−1

2−1ⁿ⁻¹n1!. 2.17 For 0≤k≤n−2, we have

C_k 2 n2

n−2 lk

l−k m0

n2 l

km k

l!−1^kB_n−l B_l−k−m l−k−m!

−1^kn1!^n−k

m0

km k

B_n−k−m

n−k−m!.

2.18

n k0

BkxBn−kx ⁿ⁻²

k0

2 n2

n−2 lk

l−k m0

−1^kl!

n2 l

km k

B_n−l B_l−k−m l−k−m!

−1^kn1!^n−k

m0

km k

B_n−k−m

n−k−m!

L_kx

nn1!−1ⁿ⁻¹−1

2−1ⁿ⁻¹n1!

L_n−1x −1ⁿn1!Lnx.

2.19

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Let us takepx _n

k0EkxEn−kx∈Pn. By2.3and2.4, we get

px ⁿ

k0

E_kxEn−kx ⁿ

k0

C_kL_kx, 2.20

where

C_k 1 k!

_∞

0

d^k dx^ke^−xx^k

pxdx. 2.21

It is knownsee15that n

k0

E_kxEn−kx ⁿ⁻¹

k0

n1ⁿ_k n−k1

_n

lk

E_l−kE_n−l−2E_n−k

E_kx 2n1Enx. 2.22

From2.20,2.21, and2.22, we can derive the following equations2.23-2.24:

Cn −1ⁿ

n! 2n1n!²2−1ⁿn1!. 2.23 For 0≤k≤n−1, we have

C_kⁿ⁻¹

lk

n1ⁿ_l n−l1!

_n

ml

E_m−lE_n−m−2E_n−l _l−k

p0

−1^kl!

kp k

E_l−k−p l−k−p

! 2n1!−1^k^n−k

p0

kp k

E_n−k−p n−k−p

!.

2.24

n k0

E_kxEn−kx ⁿ⁻¹

k0

⎧⎨

⎩ n−1

lk

n1ⁿ_l n−l1!

_n

ml

E_m−lE_n−m−2E_n−l _l−k

p0

−1^kl!

kp k

× E_l−k−p l−k−p

!2n1!−1^k^n−k

p0

kp k

E_n−k−p n−k−p

!

⎫⎬

⎭L_kx 2−1ⁿn1!Lnx.

2.25

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It is known that n k0

EkxEn−kx px − 4 n2

n k0

n2 k

E_n−k1Bkx 2.26

see15. From2.20,2.21, and2.23, we have C_k 1

k!

_∞

0

d^k dx^ke^−xx^k

pxdx − 4

n2 n lk

n2 l

E_n−l11

k!

_∞

0

d^ke^−xx^k dx^k

Blxdx

− 4 n2

n lk

l−k m0

n2 l

−1^kE_n−l1 B_l−k−m l−k−m!l!

mk k

.

2.27

n k0

E_kxE_n−kx

− 4 n2

n k0

n lk

l−k m0

n2 l

−1^kE_n−l1 B_l−k−m l−k−m!l!

mk k

L_kx.

2.28

Remark 2.7. Laguerre’s diﬀerential equation

ty 1−tyny0 2.29

is known to possess polynomial solutions whennis a nonnegative integer. These solutions are naturally called Laguerre polynomials and are denoted byL_nt. That is,y L_ntare solutions of2.29which are given by

yLnt ⁿ

r0

ⁿ_r−1^r

r! t^r, L01 1. 2.30

From2.30, we note that Laplace transform ofyL_ntis given by

L y

LLnt 1 s

n r0

n r

−1^r 1

s r

s−1ⁿ

sⁿ¹ . 2.31

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It is not diﬃcult to show that L

e^t n!

dⁿ dtⁿe^−ttⁿ

L

y

s−1ⁿ

sⁿ¹ . 2.32

Thus, we conclude that

Lnt e^t n!

dⁿ dtⁿe^−ttⁿ

, forn∈Z. 2.33

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