Hermite Polynomials and their Applications Associated with Bernoulli and Euler Numbers

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Volume 2012, Article ID 974632,13pages 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 Lee

⁴

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

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

3Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea

4Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea

Correspondence should be addressed to Taekyun Kim,[email protected] Received 7 May 2012; Accepted 15 May 2012

Academic Editor: Garyfalos Papaschinopoulos

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 derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials from the orthogonality of Hermite polynomials. LetPn {px ∈ Qx | degpx ≤ n}be the n1-dimensional vector space overQ. Then we show that{H0x, H1x, . . . , Hnx}is a good basis for the spacePnfor our purpose of arithmetical and combinatorial applications.

1. Introduction

As is well known, the Euler polynomials,Enx, are defined by the generating function as follows:

2

e^t1e^xte^Ext^∞

n0

Enxtⁿ

n! 1.1

see1–8, with the usual convention about replacingEⁿxbyEnx.

In the special case,x 0,En0 En is called the nth Euler number. From1.1and definition of Euler numbers, we note that

Enx Exⁿⁿ

l0

n l

Elx^n−lⁿ

l0

n l

En−lx^l 1.2

with the usual convention about replacingEⁿbyEn.

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The Bernoulli numbers are defined as

B01, B1ⁿ−Bn δ1,n 1.3

see9–14, whereδk,nis a Kronecker symbol.

As is well known, Bernoulli polynomials are also defined by

Bnx Bxⁿⁿ

l0

n l

Blx^n−lⁿ

l0

n l

Bn−lx^l 1.4

with the usual convention about replacingBⁿbyBnsee1,15–18.

The Hermite polynomials are defined by the generating function as follows:

e^2xt−t²e^Hxt^∞

n0

Hnxtⁿ

n! 1.5

see5,19, with the usual convention about replacingHⁿxbyHnx.

From1.5, we can derive the following identities:

Hnx ∂

∂t _n

e^2xt−t²

t0 e^x² ∂

∂t _n

e^−x−t² t0

−1ⁿe^x² ∂

∂x _n

e^−x−t²

_t0 −1ⁿe^x² dⁿ

dxⁿe^−x²

.

1.6

Let us consider two operators as follows:

f −→O1f −

e^x² d dxe^−x²

f2xf− df dx, f−→O2f

e^x²^/2

x− d

dx

e^−x²^/2

f 2xf− df dx.

1.7

By1.7, we getO1O2. In particular, if we takef 1, then we have

−e^x² d dxe^−x²

e^x²^/2

x− d

dx

e^−x²^/2. 1.8

We note that

−1ⁿe^x² dⁿ

dxⁿe^−x²

−e^x² d dxe^−x²

_n

. 1.9

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From1.8, we note that

−1ⁿe^x²

dⁿe^−x² dxⁿ

−e^x²de^−x² dx

_n

e^x²^/2

x− d

dx

e^−x²^/2 _n

e^x²^/2

x− d dx

_n e^−x²^/2.

1.10

Thus, by1.10, we get

Hnx e^x²^/2

x− d dx

_n

e^−x²^/2 1.11

see5,19–23. In the special case,x0,Hn0 Hnare called the Hermite numbers.

From1.5, we can derive the following identities:

Hnx H2xⁿⁿ

l0

n l

Hn−l2^lx^l 1.12

cf.5,19, with the usual convention about replacingHⁿbyHn. It is easy to show that ∞

n0

Hntⁿ

n! e^−t²^∞

l0

−1ⁿ

n! t²ⁿ. 1.13

By comparing coeﬃcients on the both sides of1.13, we get

H2n −1ⁿ2n2n−1· · ·n1 −1ⁿ2n!

n! , H2n−10, 1.14 wheren∈N. From1.12, we have

dHnx

dx 2nH_n−1x n∈N. 1.15

LetPn{p∈Qx|degpx≤n}be then1-dimensional vector space overQ. Probably, {1, x, x², . . . , xⁿ}is the most natural basis for this space. But{H0x, H1x, H2x, . . . , Hnx}

is also a good basis for the space Pn, for our purpose of arithmetical and combinatorial applications.

Forpx∈Pn,

px ⁿ

k0

CkHkx, 1.16

for some uniquely determinedbl∈Q.

The purpose of this paper is to develop methods for computing Ck from the information ofpx. By using these methods, we define some interesting identities.

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2. Properties of Hermite Polynomials

From1.5and1.13, we note that

1 _∞

m0

Hmt^m m!

_∞

l0

t^2l l!

_∞

m0

H2m t^2m 2m!

_∞

l0

2l2l−1· · ·l1 2l! t^2l

^∞

n0

_n

l0

2l2l−1· · ·l1

2l!2n−2l! H2n−2l2n!

t²ⁿ 2n!

^∞

n0

_n

l0

l!

2l l

2n 2l

H2n−2l

t²ⁿ 2n!.

2.1

Thus, by2.1, we obtain the following recurrence formula.

Proposition 2.1. Forn∈ZN∪ {0}, one has

n l0

l!

2l l

2n 2l

H2n−2l

1, if n0

0, if n /0 . 2.2

By,1.5, we get

∞ n0

Hn−xtⁿ

n! e^2t−x−t²e^{2x−t−−t}² ^∞

n0

Hnx−1ⁿtⁿ

n!. 2.3

From2.3, we can derive the following reflection symmetric identity ofHnx:

Hn−x −1ⁿHnx. 2.4

By1.5, we easily see that

∂

∂t e^2xt−t²

2x−2te^2xt−t². 2.5

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Thus, by1.5and2.5, we get

∂

∂t _∞

n0

Hnxtⁿ n!

2x−2t _∞

n0

Hnxtⁿ n!

. 2.6

LHS of2.5 ^∞

n1

Hnx tⁿ⁻¹

n−1! ^∞

n0

Hn1xtⁿ

n!, 2.7

RHS of2.5 ^∞

n0

2xHnxtⁿ n!

−^∞

n0

2Hnxtⁿ¹ n!

^∞

n0

2xHnxtⁿ n!

−^∞

n1

2Hn−1x tⁿ n−1!

^∞

n0

2xHnxtⁿ n!−^∞

n1

2nHn−1xtⁿ n!.

2.8

Thus, by2.6and2.7, we get

H_n1x 2xHnx−2nH_n−1x, n∈N. 2.9 From1.15and2.9, we note that

H_n1x−2xHnx H_nx 0. 2.10 Diﬀerentiating on both sides, we have

2n1Hnx−2Hnx−2xH_nx H_nx 0. 2.11

Thus, we have

H_nx−2xH_nx 2nHnx 0. 2.12 From2.12, we note thatHnxis a solution of the following second-order linear diﬀerential equation:

u−2xu2nu0. 2.13

From1.5, we note that ∞ m0

Hnxtⁿ

n! e^2tx−t² _∞

l0

2x^l l! t^l

_∞

k0

−1^k k! t^2k

^∞

n0

_n/2

k0

−1^kn!2x^n−2k k!n−2k!

tⁿ n!.

2.14

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Thus, by2.14, we get

Hnx ^n/2

k0

−1^kn!

k!n−2k!2x^n−2k

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

n/2

l0

−1^n/2−ln!2^2l

n/2−l!2l!x^2l, ifn≡0mod 2,

n−1/2

l0

−1^n−1/2−ln!2^2l1

n−1/2−l!2l1!x^2l1, ifn≡1mod 2.

2.15

3. Main Results

By1.6, we easily get _∞

−∞e^−x²HnxHmxdx −1ⁿ _∞

−∞

dⁿ dxⁿe^−x²

Hmxdx. 3.1

From3.1, we note that _∞

−∞e^−x²HnxHmxdx2ⁿn!√

πδm,n. 3.2

It is easy to show that _∞

−∞e^−x²x^ldx

⎧⎪

⎪⎨

⎪⎪

⎩

0 ifl≡1mod 2, l!√

π

2^ll/2! ifl≡0mod 2, 3.3 wherel∈Z N∪ {0}. By3.3, we get

_∞

−∞

dⁿe^−x² dxⁿ

x^mdx

⎧⎪

⎪⎨

⎪⎪

⎩

0 ifn > morn≤mwithn−m≡1mod 2, m!−1ⁿ√

π

2^m−nm−n/2! ifn≤mwithn−m≡0mod 2.

3.4 From 3.2, we note thatH0x, H1x, . . . , Hnx are orthogonal basis for the spacePn {px∈Qx|degpx≤n}with respect to the inner product

px, qx

_∞

−∞e^−x²pxqxdx. 3.5

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Forpx∈Pn, the polynomialpxis given by px ^∞

k0

CkHkx, 3.6

where

Ck 1 2^kk!√

π

px, Hkx

−1^k 2^kk!√

π _∞

−∞

d^ke^−x² dx^k

pxdx.

3.7

Let us takepx xⁿ∈Pn. Forn≡0mod 2, we computeCkin3.6as follows

Ck −1^k 2^kk!√

π _∞

−∞

d^ke^−x² dx^k

xⁿdx

⎧⎪

⎪⎨

⎪⎪

⎩

−1^k 2^kk!√

π × −1^kn!√ π

2^n−kn−k/2! ifk≡0mod 2, 0 ifk≡1mod 2.

3.8

Letn≡1mod 2. Then we have

Ck −1^k 2^kk!√

π _∞

−∞

d^ke^−x² dx^k

xⁿdx

⎧⎪

⎨

⎪⎩

n!

2ⁿk!n−k/2! ifk≡1mod 2, 0 ifk≡0mod 2.

3.9

Therefore, by3.6,3.8, and3.9, we obtain the following proposition.

Proposition 3.1. One has

x²ⁿ 2n!

2²ⁿ n k0

1

2k!n−k!H2kx,

x²ⁿ¹ 2n1!

2²ⁿ¹ n k0

1

2k1!n−k!H_2k1x.

3.10

Let us takepx Bnx. From3.4,Pxcan be rewritten by Bnx ⁿ

k0

CkHkx, 3.11

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where

Ck −1^k 2^kk!√

π _∞

−∞

d^ke^−x² dx^k

Bnxdx. 3.12

By integrating by parts, we get _∞

−∞

d^ke^−x² dx^k

Bnx −n−n−1· · ·−n−k1 _∞

−∞e^−x²B_n−kxdx −1^k n!

n−k!

n−k

l0

n−k l

Bn−k−l

_∞

−∞e^−x²x^ldx −1^kn!

n−k!

0≤l≤n−k l≡0mod 2

n−k!Bn−k−l

l!n−k−l! × l!√ π 2^ll/2!

−1^kn!√

π

0≤l≤n−k l≡0mod 2

Bn−k−l

n−k−l!2^ll/2!.

3.13

Thus, from3.11and3.13, we have Ck n!

2^kk!

0≤l≤n−k l≡0mod 2

Bn−k−l

n−k−l!2^ll/2!. 3.14

Therefore, by3.11and3.14, we obtain the following theorem.

Theorem 3.2. Forn∈Z, one has

Bnx n!

n k0

0≤l≤n−k l≡0mod 2

Bn−k−l

2^klk!n−k−l!l/2!Hkx. 3.15

Remark 3.3. Let us takepx Enx. Then, by the same method, we obtain the following identity:

Enx n!

n k0

0≤l≤n−k l≡0mod 2

En−k−l

2^klk!n−k−l!l/2!Hkx. 3.16

Now, we considerpx Hnx. From3.6, we note thatpxcan be rewritten as

Hnx ⁿ

k0

CkHkx, 3.17

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where

Ck −1^k 2^kk!√

π _∞

−∞

d^ke^−x² dx^k

Hnxdx. 3.18

By integrating by parts, we get _∞

−∞

d^ke^−x² dx^k

Hnxdx −2n· · ·−2n−k1 _∞

−∞e^−x²Hn−kxdx −1^k2^kn!

n−k!

n−k

l0

n−k l

2^lHn−k−l

_∞

−∞e^−x²x^ldx

−1^k2^kn!

n−k!

n−k

l≡0mod 2l0

2^ln−k!

l!n−k−l!Hn−k−l l!√ π 2^ll/2!

−1^k2^kn!√ π

n−k

l≡0mod 2l0

Hn−k−l

n−k−l!l/2!.

3.19

From3.17and3.19, we note that

Ck

−1^k 2^kk!√

π

×

⎛

⎜⎜

⎝−1^k2^kn!√

π

0≤l≤n−k l≡0mod 2

Hn−k−l

n−k−l!l/2!

⎞

⎟⎟

⎠

n!

k!

0≤l≤n−k l≡0mod 2

Hn−k−l

n−k−l!l/2!.

3.20

Therefore, by3.17and3.20, we obtain the following theorem.

Hnx n!

n k0

0≤l≤n−k l≡0mod 2

Hn−k−l

k!n−k−l!l/2!Hkx. 3.21

From Theorem3.4, we note that

Hnx n!

n−1

k0

0≤l≤n−k l≡0mod 2

Hn−k−l

k!n−k−l!l/2!Hkx n!Hnx

n! . 3.22

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Thus, we have, for 0≤k≤n−k,

0≤l≤n−k l≡0mod 2

Hn−k−l

n−k−l!l/2! 0. 3.23

Letl, k∈Zwithk≤l. Then we easily see that _∞

−∞

d^ke^−x² dx^k

Blxdx −1^kl!√

π

0≤j≤l−k j≡0mod 2

Bl−k−j

l−k−j

!2^j j/2

!, 3.24

_∞

−∞

d^ke^−x² dx^k

Elxdx −1^kl!√

π

0≤j≤l−k j≡0mod 2

E_l−k−j l−k−j

!2^j j/2

!. 3.25

Let us consider the following polynomial of degreeninPn: px ⁿ

k0

BkxBn−kx. 3.26

From3.6, we note thatpxcan be rewritten as px ⁿ

k0

CkHkx, 3.27

where

Ck −1^k 2^kk!√

π _∞

−∞

d^ke^−x² dx^k

pxdx. 3.28

In15, it is known that

px ⁿ

k0

BkxB_n−kx

2 n2

n−2

l0

n2 l

B_n−lBlx n1Bnx.

3.29

From3.23and3.29, we have the following:

Ck −1^k 2^kk!√

π 2

n2

n−2

l0

n2 l

∞

−∞

d^ke^−x² dx^k

Blxdx n1 _∞

−∞

d^ke^−x² dx^k

Bnxdx

, 3.30

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By3.24and3.30, we get

Cn −1ⁿ 2ⁿn!√

π

×n1 _∞

−∞

dⁿe^−x² dxⁿ

Bnxdx

−1ⁿ 2ⁿn!√

π

×

n1−1ⁿn!√ πB0

0!2⁰0!

n1 2ⁿ , C_n−1

−1ⁿ⁻¹ 2ⁿ⁻¹n−1!√

π

×

n1 _∞

−∞

dⁿ⁻¹e^−x² dxⁿ⁻¹

Bnxdx

−1ⁿ⁻¹ 2ⁿ⁻¹n−1!√

π

×

⎛

⎜⎜

⎝n1−1ⁿ⁻¹n!√ π

1 j≡0mod 2j0

B_1−j 1−j

!2^j j/2

!

⎞

⎟⎟

⎠

−1ⁿ⁻¹ 2ⁿ⁻¹n−1!√

π

× n1−1ⁿ⁻¹n!√ πB1

−nn1 2ⁿ .

3.31

For 0≤k≤n−2, we have

Ck

−1^k 2^kk!√

π 2

n2 n−2

lk

n2 l

B_n−l

_∞

−∞

d^ke^−x² dx^k

Blxdx n1 _∞

−∞

d^ke^−x² dx^k

Bnxdx

−1^k 2^kk!√

π

⎧⎪

⎪⎨

⎪⎪

⎩ 2 n2

n−2

lk

n2 l

Bn−l−1^kl!√

π×

0≤j≤l−k j≡0mod 2

Bl−k−j

l−k−j

!2^j j/2

!

n1−1^kn!√

π

0≤j≤n−k j≡0mod 2

Bn−k−j

n−k−j

!2^j j/2

!

⎫⎪

⎪⎬

⎪⎪

⎭ 2

n2

n−2

lk

0≤j≤l−k j≡0mod 2

n2 l

B_n−lB_l−k−jl!

2^kjk!

l−k−j

! j/2

!

n1!

0≤j≤n−k j≡0mod 2

B_n−k−j k!

n−k−j

! j/2

!2^kj.

3.32 Therefore, by3.27and3.32, we obtain the following theorem.

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n k0

BkxB_n−kx

ⁿ⁻²

k0

⎧⎪

⎪⎨

⎪⎪

⎩ 2 n2

n−2 lk

0≤j≤n−k j≡0mod 2

n2 l

l!B_n−lB_l−k−j 2^kjk!

l−k−j

! j/2

!

n1!

0≤j≤n−k j≡0mod 2

Bn−k−j

2^kjk!

n−k−j

! j/2

!

⎫⎪

⎪⎬

⎪⎪

⎭Hkx

−nn1

2ⁿ H_n−1x n1 2ⁿ Hnx.

3.33

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.

References

1 H. Ozden, I. N. Cangul, and Y. Simsek, “On the behavior of two variable twistedp-adic Eulerq-l- functions,” Nonlinear Analysis, vol. 71, no. 12, pp. e942–e951, 2009.

2 S.-H. Rim, A. Bayad, E.-J. Moon, J.-H. Jin, and S.-J. Lee, “A new construction on theq-Bernoulli polynomials,” Advances in Diﬀerence Equations, vol. 2011, article 34, 2011.

3 C. S. Ryoo, “Some relations between twistedq-Euler numbers and Bernstein polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 217–223, 2011.

4 Y. Simsek, “Construction a new generating function of Bernstein type polynomials,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1072–1076, 2011.

5 Y. Simsek and M. Acikgoz, “A new generating function ofq-Bernstein-type polynomials and their interpolation function,” Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010.

6 Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–

278, 2008.

7 C. Vignat, “Old and new results about relativistic Hermite polynomials,” Journal of Mathematical Physics, vol. 52, no. 9, Article ID 093503, 16 pages, 2011.

8 T. Kim, “A note onq-Bernstein polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 1, pp. 73–82, 2011.

9 S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionicp-adicq-integral representation onZp

associated with weightedq-Bernstein andq-Genocchi polynomials,” Abstract and Applied Analysis, Article ID 649248, 10 pages, 2011.

10 A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.

11 A. Bayad and T. Kim, “Identities involving values of Bernstein,q-Bernoulli, andq-Euler polynomials,”

Russian Journal of Mathematical Physics, vol. 18, no. 2, pp. 133–143, 2011.

(13)

12 L. Carlitz, “Note on the integral of the product of several Bernoulli polynomials,” Journal of the London Mathematical Society, vol. 34, pp. 361–363, 1959.

13 L. Carlitz, “Multiplication formulas for products of Bernoulli and Euler polynomials,” Pacific Journal of Mathematics, vol. 9, pp. 661–666, 1959.

14 L. Carlitz, “Arithmetic properties of generalized Bernoulli numbers,” Journal f ¨ur die Reine und Angewandte Mathematik, vol. 202, pp. 174–182, 1959.

15 D. S. Kim, D. V. Dolgy, T. Kim, and S. H. Rim, “Some formulae for the product of two Bernoulli and Euler polynomials,” Abstract and Applied Analysis. In press.

16 T. Kim, “Some identities on theq-Euler polynomials of higher order andq-Stirling numbers by the fermionicp-adic integral onZp,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009.

17 H. Y. Lee, N. S. Jung, and C. S. Ryoo, “A note on theq-Euler numbers and polynomials with weak weightα,” Journal of Applied Mathematics, vol. 2011, Article ID 497409, 14 pages, 2011.

18 H. Ozden, “p-adic distribution of the unification of the Bernoulli, Euler and Genocchi polynomials,”

Applied Mathematics and Computation, vol. 218, no. 3, pp. 970–973, 2011.

19 T. Kim, J. Choi, Y. H. Kim, and C. S. Ryoo, “Onq-Bernstein andq-Hermite polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 2, pp. 215–221, 2011.

20 K. Coulembier, H. De Bie, and F. Sommen, “Orthogonality of Hermite polynomials in superspace and Mehler type formulae,” Proceedings of the London Mathematical Society. Third Series, vol. 103, no. 5, pp.

786–825, 2011.

21 H. Chaggara and W. Koepf, “On linearization and connection coeﬃcients for generalized Hermite polynomials,” Journal of Computational and Applied Mathematics, vol. 236, no. 1, pp. 65–73, 2011.

22 H. E. J. Curzon, “On a connexion between the functions of Herimite and the functions of Legendre,”

Proceedings of the London Mathematical Society, vol. 12, no. 1, pp. 236–259, 1913.

23 S. Fisk, “Hermite polynomials,” Journal of Combinatorial Theory. Series A, vol. 91, no. 1-2, pp. 334–336, 2000.

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