A Note on Some Identities of Frobenius-Euler Numbers and Polynomials

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Volume 2012, Article ID 861797,9pages doi:10.1155/2012/861797

Research Article

A Note on Some Identities of Frobenius-Euler Numbers and Polynomials

J. Choi,

¹

D. S. Kim,

²

T. Kim,

³

and Y. H. Kim

¹

1Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

2Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

3Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Correspondence should be addressed to D. S. Kim,[email protected] and T. Kim,[email protected]

Received 28 November 2011; Accepted 9 January 2012 Academic Editor: Feng Qi

Copyrightq2012 J. Choi 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.

The purpose of this paper is to give some identities on the Frobenius-Euler numbers and polynomials by using the fermionic p-adic q-integral equation onZp.

1. Introduction

Letpbe a fixed odd prime number. Throughout this paper,Zp,Qp, andCp will denote the ring ofp-adic rational integers, the field ofp-adic rational numbers, and the completion of algebraic closure ofQp, respectively. LetNbe the set of natural numbers andZ N∪ {0}.

Thep-adic absolute value onCp is normalized so that|p|_p 1/p. Assume thatq ∈Cpwith

|1−q|_p<1.

Letfbe a continuous function onZp. Then the fermionicp-adicq-integral onZpforf is defined by Kim as follows:

I_−q f

Zp

fxdμ_−qx lim

N→ ∞

1q 1q^p^N

p^N−1 x0

fx

−q_x

, see1. 1.1

From1.1, we note that

qⁿI_−q fn

−1ⁿI_−q f

1qⁿ⁻¹

l0

−1^n−1−lflq^l, 1.2

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wheren∈Nandfnx fxn see1. The ordinary Euler polynomialsEnxare defined by

2

e^t1e^xte^Ext^∞

n0

Enxtⁿ

n!, 1.3

with the usual convention about replacingEⁿxbyEnx see1–10. In the special case, x0,En0 Enis called thenth Euler number.

As the extension of1.3, the Frobenius-Euler polynomials are defined by 1−q

e^t−qe^xt^∞

n0

Hn

q, xtⁿ

n!, see2. 1.4

In the special case,x0,Hnq,0 Hnqis called thenth Frobenius-Euler number.

By1.3and1.4, we easily getHn−1, x Enx.

From1.4, we note that

Hn

q, x ⁿ

l0

n l

Hl

q x^n−l

H q

x_n

, see2, 1.5

with the usual convention about replacingHqⁿbyHnq.

In this paper, we consider the fermionicp-adicq-integral on Zp for the Frobenius- Euler numbers and polynomials. From thesep-adicq-integrals onZp, we derive some new and interesting identities on the Frobenius-Euler numbers and polynomials.

2. Identities on the Frobenius-Euler Numbers

From1.2and1.4, we can derive the following:

Zp

e^xytdμ−q y

1q⁻¹

e^tq⁻¹e^xt^∞

n0

Hn

−q⁻¹, x tⁿ

n!. 2.1

Thus, by2.1, we get Witt’s formula forHn−q⁻¹, xas follows:

Zp

xy_n dμ−q

y Hn

−q⁻¹, x , n∈Z. 2.2

By1.5and2.1, we get

H

−q⁻¹ 1 ⁿq⁻¹Hn

−q⁻¹

⎧⎨

⎩

1q⁻¹, ifn0,

0, ifn >0, 2.3 with the usual convention about replacingH−q⁻¹ⁿbyHn−q⁻¹.

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From1.5and2.3, we note that

H0

−q⁻¹ 1, Hn

−q⁻¹,1 q⁻¹Hn

−q⁻¹ 0, ifn >0. 2.4

By2.1and2.2, we get

Zp

y1−x_n dμ−q

y

−1ⁿ

Zp

yx_n dμ−q⁻¹

y

. 2.5

Therefore, by2.5, we obtain the following lemma.

Lemma 2.1. Forn∈Z, one has

Hn

−q⁻¹,1−x −1ⁿHn

−q, x

. 2.6

From2.3, we can derive the following:

q²Hn

−q⁻¹,2 −q²−qq² n

l0

n l

H

−q⁻¹ 1 ^l−q²−q

q n l1

n l

q

H

−q⁻¹ 1 ^l−q

−qⁿ

l0

n l

Hl

−q⁻¹

− 1q

δ0,nHn

−q⁻¹ ,

2.7

whereδk,nis the Kronecker symbol.

Therefore, by2.7, we obtain the following theorem.

Theorem 2.2. Forn∈Z, one has

Hn

−q⁻¹,2 1q⁻¹−q⁻² 1q

δ0,nq⁻²Hn

−q⁻¹ . 2.8

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First we consider the fermionicp-adic q-integral onZp for the nth Frobenius-Euler polynomials as follows:

I1

Zp

Hn

−q⁻¹, x dμ−qx

ⁿ

l0

n l

Hl

−q⁻¹

Zp

x^n−ldμ−qx

ⁿ

l0

n l

Hl

−q⁻¹ H_n−l

−q⁻¹ , wheren∈Z.

2.9

On the other hand, by2.5andLemma 2.1, we get I1

Zp

Hn

−q⁻¹, x dμ_−qx −1ⁿ

Zp

Hn

−q,1−x

dμ_−qx

−1ⁿⁿ

l0

n l

Hn−l

−q Zp

1−x^ldμ−qx

ⁿ

l0

n l

−1^n−lHn−l

−q Zp

x−1^ldμ−qx

ⁿ

l0

n l

−1^n−lHn−l

−q Hl

−q⁻¹,−1 .

2.10

FromLemma 2.1,Theorem 2.2, and2.10, we note that

I1ⁿ

l0

n l

−1^n−lHn−l

−q Hl

−q⁻¹,−1

ⁿ

l0

n l

−1ⁿH_n−l

−q Hl

−q,2

ⁿ

l0

n l

−1ⁿHn−l

−q

1q−q²

1q⁻¹ δ0,lq²Hl

−q

−1ⁿ

1q 1q

δ0,n−qHn

−q

−Hn

−q

qq² −1ⁿ −1ⁿq²

n l0

n l

Hn−l

−q Hl

−q .

2.11

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

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Theorem 2.3. Forn∈Z, one has n

l0

n l

Hl

−q⁻¹ H_n−l

−q⁻¹ −1ⁿ

1q 1q

δ0,n−2qHn

−q

−1ⁿq² n

l0

n l

Hn−l

−q Hl

−q .

2.12

In particular, forn∈N, one has n

l0

n l

Hl

−q⁻¹ H_n−l

−q⁻¹ 2−1ⁿ¹q 1q

Hn

−q

−1ⁿq² n

l0

n l

Hn−l

−q Hl

−q .

2.13

Let us consider the following fermionic p-adic q-integral on Zp for the product of Bernoulli and Frobenius-Euler polynomials as follows:

I2

Zp

BmxHn

−q⁻¹, x dμ−qx

^m

k0

n l0

m k

n l

Bm−kHn−l

−q⁻¹

Zp

x^kldμ−qx

^m

k0

n l0

m k

n l

B_m−kH_n−l

−q⁻¹ H_kl

−q⁻¹ .

2.14

It is known thatBnx −1ⁿBn1−x.

On the other hand, byLemma 2.1, we get I2 −1^mn

Zp

Bm1−xHn

−q,1−x

dμ_−qx

−1^mn^m

k0

n l0

m k

n l

B_m−kH_n−l

−q Zp

1−x^kldμ_−qx

−1^mn^m

k0

n l0

m k

n l

Bm−kHn−l

−q 1q

−q²

1q⁻¹ δ0,klq²Hkl

−q −1^mn

1q

Bm1Hn

−q,1

−

q²q −1^mnBmHn

−q

q²−1^mn^m

k0

n l0

m k

n l

Bm−kHn−l

−q Hkl

−q .

2.15 Therefore, by2.14and2.15, we obtain the following theorem.

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Theorem 2.4. Form, n∈Z, one has

m k0

n l0

m k

n l

B_m−kH_n−l

−q⁻¹ H_kl

−q⁻¹ −1^mn

1q

Bm1Hn

−q,1

−

q²q −1^mnBmHn

−q

q²−1^mn^m

k0

n l0

m k

n l

Bm−kHn−l

−q Hkl

−q .

2.16

In particular, form∈N− {1},n∈N, one has

m k0

n l0

m k

n l

B_m−kH_n−l

−q⁻¹ H_kl

−q⁻¹ 2−1^mn1

q²q BmHn

−q

q²−1^mn^m

k0

n l0

m k

n l

Bm−kHn−l

−q Hkl

−q .

2.17

It is known that xⁿ 1/n 1_n

l0

_n1

l

Blx. Let us consider the following fermionicp-adicq-integral onZp:

Zp

xⁿdμ−qx 1 n1

n l0

n1 l Zp

Blxdμ−qx.

1 n1

n l0

n1 l

_l

k0

l k

B_l−k

Zp

x^kdμ_−qx

1 n1

n l0

n1 l

_l

k0

l k

B_l−kHk

−q⁻¹ .

2.18

Therefore by2.18, we obtain the following theorem.

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Theorem 2.5. Forn∈Z, one has

Hn

−q⁻¹ 1 n1

n l0

n1 l

_l

k0

l k

B_l−kHk

−q⁻¹ . 2.19

From1.3, we can derive the following:

xⁿEnx 1 2

n−1 l0

n l

Elx. 2.20

Let us take the fermionicp-adicq-integral onZpin2.20as follows:

Zp

xⁿdμ_−qx

Zp

Enxdμ_−qx 1 2

n−1

l0

n l Zp

Elxdμ_−qx

ⁿ

l0

n l

En−lHl

−q⁻¹ 1 2

n−1

l0

n l

_l

k0

l k

El−kHk

−q⁻¹ .

2.21

Theorem 2.6. Forn∈N, one has

Hn

−q⁻¹ ⁿ

l0

n l

E_n−lHl

−q⁻¹ 1 2

n−1

l0

n l

_l

k0

l k

E_l−kHk

−q⁻¹ . 2.22

By Theorems2.5and2.6, we obtain the following corollary.

Corollary 2.7. Forn∈N, one has

1 n1

n l0

n1 l

_l

k0

l k

B_l−kHk

−q⁻¹

ⁿ

l0

n l

E_n−lHl

−q⁻¹ 1 2

n−1 l0

n l

_l

k0

l k

E_l−kHk

−q⁻¹ .

2.23

By1.3, we easily getEnx −1ⁿEn1−x.

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Thus, we have

Zp

xⁿdμ−qx −1ⁿ

Zp

En1−xdμ−qx 1 2

n−1 l0

n l

−1^l

Zp

El1−xdμ−qx

−1ⁿⁿ

l0

n l

E_n−l

Zp

1−x^ldμ_−qx 1

2 n−1

l0

n l

−1^l^l

k0

l k

El−k

Zp

1−x^kdμ−qx

ⁿ

l0

n l

En−l−1^n−lHl

−q⁻¹,−1 1 2

n−1 l0

n l

_l

k0

l k

El−k−1^l−kHk

−q⁻¹,−1

ⁿ

l0

n l

En−l−1ⁿHl

−q,2 1

2 n−1

l0

n l

_l

k0

l k

El−k−1^lHk

−q,2 .

2.24

Theorem 2.8. Forn∈N, one has Hn

−q⁻¹ ⁿ

l0

n l

En−l−1ⁿHl

−q,2

1 2

n−1

l0

n l

_l

k0

l k

E_l−k−1^lHk

−q,2 .

2.25

Acknowledgment

The second author was supported by National Research Foundation of Korea Grant funded by the Korean Government 2009-0072514.

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A Note on Some Identities of Frobenius-Euler Numbers and Polynomials

Research Article