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

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

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

3_{Department 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

Zpfxdμ−qx Nlim→ ∞

1q

1qpN

pN₋₁

x0

fx−qx, see1. 1.1

From1.1, we note that

qn_I

−q

fn

−1nI−q

f1q

n−1

l0

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

2

et₁e

xt_eExt∞ n0

Enxt n

n!, 1.3

with the usual convention about replacingEn_x_by_E

nx 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 et₋_qe

xt∞ n0

Hn

q, xt

n

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

qxn−lHqxn, 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:

Zpe

xyt_dμ

−q

y 1q

−1

et_q−1ext ∞

n0

Hn

−q−1, x t

n

n!. 2.1

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

Zp

xyndμ−q

yHn

−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

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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−xndμ−q

y −1n

Zp

yxndμ−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

qH−q−1 1 l−q

−q

n

l0

n l

Hl

−q−1

−1qδ0,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−21qδ0,nq−2Hn

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

I1

ZpHn

−q−1, x dμ−qx

n

l0

n l

Hl

−q−1

Zpx

n−l_dμ

−qx

n

l0

n l

Hl

−q−1 Hn−l

−q−1 , wheren∈Z.

2.9

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

I1

ZpHn

−q−1, x dμ−qx −1n

ZpHn

−q,1−xdμ−qx

−1n

n

l0

n l

Hn−l

−q

Zp1−x

l_dμ

−qx

n

l0

n l

−1n−lHn−l

−q

Zpx−1

l

dμ−qx

n

l0

n

l

−1n−lHn−l

−qHl

−q−1,−1 .

2.10

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

I1

n

l0

n l

−1n−lHn−l

−qHl

−q−1,−1

n

l0

n l

−1nHn−l

−qHl

−q,2

n

l0

n

l

−1nHn−l

−q1q−q21q−1 δ0,lq2Hl

−q

−1n1q1qδ0,n−qHn

−q−Hn

−qqq2 −1n

−1nq2

n

l0

n l

Hn−l

−qHl

−q.

2.11

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n

l0 n l Hl

−q−1 Hn−l

−q−1 −1n1q1qδ0,n−2qHn

−q

−1nq2

n

l0

n l

Hn−l

−qHl

−q.

2.12

In particular, forn∈N, one has

n

l0 n l Hl

−q−1 Hn−l

−q−1 2−1n1q1qHn

−q

−1nq2

n

l0

n l

Hn−l

−qHl

−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

ZpBmxHn

−q−1, x dμ−qx

m

k0

n

l0 m k n l

Bm−kHn−l

−q−1

Zpx

kl_dμ

−qx

m

k0

n

l0 m k n l

Bm−kHn−l

−q−1 Hkl

−q−1 .

2.14

It is known thatBnx −1nBn1−x.

On the other hand, byLemma 2.1, we get

I2 −1mn

ZpBm1−xHn

−q,1−xdμ−qx

−1mn

m

k0

n

l0 m k n l

Bm−kHn−l

−q

Zp1−x

kl_dμ

−qx

−1mn

m

k0

n

l0 m k n l

Bm−kHn−l

−q1q−q21q−1 δ0,klq2Hkl

−q

−1mn1qBm1Hn

−q,1−q2_q ₋₁mn_B

mHn

−q

q2−1mn

m

k0

n

l0 m k n l

Bm−kHn−l

−qHkl

−q.

2.15

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Theorem 2.4. Form, n∈Z, one has

m

k0

n

l0

m k

n l

Bm−kHn−l

−q−1 Hkl

−q−1

−1mn1qBm1Hn

−q,1−q2q −1mnBmHn

−q

q2−1mn

m

k0

n

l0

m k

n l

Bm−kHn−l

−qHkl

−q.

2.16

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

m

k0

n

l0

m

k

n

l

Bm−kHn−l

−q−1 Hkl

−q−1

2−1mn1q2q BmHn

−q

q2−1mn

m

k0

n

l0

m k

n l

Bm−kHn−l

−qHkl

−q.

2.17

It is known that xn 1/n 1n_l₀n1

l

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

Zpx

n_dμ

−qx 1

n1 n

l0

n1

l ZpBlxdμ−qx.

1

n1 n

l0

n1

l

_l

k0

l

k

Bl−k

Zpx

k_dμ

−qx

1

n1 n

l0

n1

l

_l

k0

l k

Bl−kHk

−q−1 .

2.18

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Hn

−q−1 1

n1 n

l0

n1

l

_l

k0

l k

Bl−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:

Zpx

n_dμ

−qx

ZpEnxdμ−qx

1 2

n−1

l0

n

l ZpElxdμ−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

En−lHl

−q−1 1

2 n−1

l0

n l

_l

k0

l k

El−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

Bl−kHk

−q−1

n

l0

n

l

En−lHl

−q−1 1

2 n−1

l0

n

l

_l

k0

l

k

El−kHk

−q−1 .

2.23

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Thus, we have

Zpx

n_dμ

−qx

−1n

ZpEn1−xdμ−qx

1 2

n−1

l0

n l

−1l

ZpEl1−xdμ−qx

−1n

n

l0

n l

En−l

Zp1−x

l_dμ

−qx

1

2 n−1

l0

n

l

−1l

l

k0

l

k

El−k

Zp1−x

k

dμ−qx

n

l0

n l

En−l−1n−lHl

−q−1,−1 1

2 n−1

l0

n l

_l

k0

l k

El−k−1l−kHk

−q−1,−1

n

l0

n l

En−l−1nHl

−q,21

2 n−1

l0

n l

_l

k0

l k

El−k−1lHk

−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

El−k−1lHk

−q,2.

2.25

Acknowledgment

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

References

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

2 L. Carlitz, “The product of two eulerian polynomials,” Mathematics Magazine, vol. 36, no. 1, pp. 37–41, 1963.

3 A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in

Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.

4 C.-P. Chen and L. Lin, “An inequality for the generalized-Euler-constant function,” Advanced Studies

in Contemporary Mathematics, vol. 17, no. 1, pp. 105–107, 2008.

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6 H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on q-Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.

7 S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin, “On the q-Genocchi numbers and polynomials associated with q-zeta function,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 3, pp. 261–267, 2009.

8 C. S. Ryoo, “A note on the weighted q-Euler numbers and polynomials,” Advanced Studies in

Contemporary Mathematics, vol. 21, pp. 47–54, 2011.

9 C. S. Ryoo, “On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity,” Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 2, pp. 255–263, 2010.

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