Symmetry Fermionic

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Volume 2012, Article ID 424189,7pages doi:10.1155/2012/424189

Research Article

Symmetry Fermionic

p

-Adic

q

-Integral on

Zp

for

Eulerian Polynomials

Daeyeoul Kim

1

_{and Min-Soo Kim}

2

1_{National Institute for Mathematical Sciences, Yuseong-daero 1689-gil, Yuseong-gu,}

Daejeon 305-811, Republic of Korea

2_{Division of Cultural Education, Kyungnam University, Changwon 631-701, Republic of Korea}

Correspondence should be addressed to Min-Soo Kim,[email protected]

Received 18 June 2012; Accepted 14 August 2012

Academic Editor: Cheon Ryoo

Copyrightq2012 D. Kim and M.-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.

Kim et al.2012introduced an interesting p-adic analogue of the Eulerian polynomials. They studied some identities on the Eulerian polynomials in connection with the Genocchi, Euler, and tangent numbers. In this paper, by applying the symmetry of the fermionic p-adic q-integral onZp,

defined by Kim2008, we show a symmetric relation between the q-extension of the alternating sum of integer powers and the Eulerian polynomials.

1. Introduction

The Eulerian polynomialsAnt, n0,1, . . ., which can be defined by the generating function

1−t

et−1x−t

∞

n0

Antxn

n!, 1.1

have numerous important applications in number theory, combinatorics, and numerical

analysis, among other areas. From1.1, we note that

At t−1n₋_{tAnt 1}₋_tδ0,n, _1.2

where δn,k is the Kronecker symbol see 1. Thus far, few recurrences for the Eulerian

polynomials other than1.2have been reported in the literature. Other recurrences are of

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and they can help simplify the proofs of known properties. For more important properties,

see, for instance,1or2.

Letpbe a fixed odd prime number. LetZp,Qp, andCpbe the ring ofp-adic integers, the

field ofp-adic numbers, and the completion of the algebraic closure ofQp, respectively. Let

| · |pbe thep-adic valuation onQ, where|p|pp−1. The extended valuation onCpis denoted

by the same symbol| · |p. Letqbe an indeterminate, where|1−q|_p<1. Then, theq-number is

defined by

xq

1−qx

1−q, x−q

1−−qx

1q . 1.3

For a uniformlyor strictlydiﬀerentiable functionf : Zp → Cp see1,3–6, the

fermionicp-adicq-integral onZpis defined by

I−qf

Zp

fxdμ−qx lim

N→ ∞

1

pN

−q

pN₋₁

x0

fx−qx. 1.4

Then, it is easy to see that

1

qI−1/q

f1I−1/qf 21/qf0, 1.5

wheref1x fx1.

By using the same method as that described in1, and applying1.5tof, where

fx q1−ωxe−x1qωt 1.6

forω∈Z>0, we consider the generalized Eulerian polynomials onZpby using the fermionic

p-adicq-integral onZpas follows:

Zp

q1−ωxe−x1qωtdμ−1/qx 1q

q1−ωe−1qωtq

∞

n0

An−q, ωt

n

n!.

1.7

By expanding the Taylor series on the left-hand side of1.7and comparing the coeﬃcients

of the termstn/n!, we get

Zp

q1−ωxxndμ−1/qx −1

n

ωn₁_qnAn

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We note that, by substitutingω1 into1.8,

An−q,1An−q −1n1qn

Zp

xndμ−1/qx 1.9

is the Witt’s formula for the Eulerian polynomials in1, Theorem 1. Recently, Kim et al.1

investigated new properties of the Eulerian polynomialsAn−qatq1 associated with the

Genocchi, Euler, and tangent numbers.

LetTk,1/qndenote theq-extension of the alternating sum of integer powers, namely,

Tk,1/qn n

i0

−1i

ikq−i0kq0−1kq−1· · · −1nnkq−n, 1.10

where 00 _{1. If}_q _→ _1,_Tk,qn _→ _Tkn n

i0−1iik is the alternating sum of integer

powerssee4. In particular, we have

Tk,1/q0 1, fork0,

0, fork >0. 1.11

Letω1, ω2be any positive odd integers. Our main result of symmetry between the

q-extension of the alternating sum of integer powers and the Eulerian polynomials is given in

the following theorem, which is symmetric inω1andω2.

Theorem 1.1. Letω1, ω2be any positive odd integers andn≥0. Then, one has

n

i0

n i

Ai−q, ω1Tn−i,q−ω2ω1−1ω₂n−i

−1−qn−i

n

i0

n i

Ai−q, ω2Tn−i,q−ω1ω2−1ωn−i₁

−1−qn−i.

1.12

Observe that Theorem1.1can be obtained by the same method as that described in

4. Ifq1, Theorem1.1reduces to the form stated in the remark in4, page 1275.

Using1.11, if we takeω21 in Theorem1.1, we obtain the following corollary.

Corollary 1.2. Letω1be any positive odd integer andn≥0. Then, one has

An−q

n

i0

n i

Ai−q, ω1Tn−i,q−1ω1−1

−1−qn−i. 1.13

2. Proof of Theorem

1.1

For the proof of Theorem1.1, we will need the following two identitiessee2.4and2.5

related to the Eulerian polynomials and the q-extension of the alternating sum of integer

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Letω1, ω2be any positive odd integers. From1.7, we obtain

Zpq

1−ω1x_e−x1qω1t_dμ−1/qx

Zpq

1−ω1ω2xe−x1qω1ω2tdμ−1/qx

1q−ω1_e−1qω1tω2

1q−ω1e−1qω1t . 2.1

This has an interesting p-adic analytic interpretation, which we shall discuss below see

Remark2.1. It is easy to see that the right-hand side of2.1can be written as

1q−ω1e−1qω1t

ω2−1

i0

−1i

q−ω1i_e−1qω1ti

∞

k0

_ω

2−1

i0

−1i_ik_qω1−i_ωk

1−1k

1qk

tk

k!.

2.2

In1.10, letqqω1_{. The left-hand, right-hand side, by definition, becomes}

1q−ω1e−1qω1t

∞

k0

Tk,q−ω1ω2−1ωk₁−1k

1qkt

k

k!. 2.3

A comparison of2.1and2.3yields the identity

Zpq

1−ω1x_e−x1qω1tdμ−1/qx

Zpq

∞

k0

Tk,q−ω₁ω2−1ωk

1−1k

1qkt

k

k!. 2.4

By slightly modifying the derivation of2.4, we can obtain the following identity:

Zpq

1−ω2x_e−x1qω2tdμ−1/qx

Zpq

∞

k0

Tk,q−ω2ω1−1ωk₂−1k

1qkt

k

k!. 2.5

Remark 2.1. The derivations of identities are based on the fermionic p-adic q-integral

expression of the generating function for the Eulerian polynomials in1.7and the quotient

of integrals in2.4,2.5that can be expressed as the exponential generating function for the

q-extension of the alternating sum of integer powers.

Observe that similar identities related to the Eulerian polynomials and theq-extension

of the alternating sum of integer powers in2.4and2.5can be found, for instance, in3,

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Proof of Theorem1.1. Letω1, ω2be any positive odd integers. Using the iterated fermionic

p-adicq-integral onZpand1.7, we have

Zpq

1−ω1x11−ω2x2_e−1qω1x1ω2x2tdμ−1/qx1dμ−1/qx2

Zpq

21/q

q−ω1ω2_e−1qω1ω2t₁

q−ω1e−1qω1t₁_q−ω2e−1qω2t₁.

2.6

Now, we put

I∗ Zpq

1−ω1x11−ω2x2_e−1qω1x1ω2x2t_dμ−1/qx1dμ−1/qx2

Zpq

1−ω1ω2x_e−x1qω1ω2tdμ−1/qx . 2.7

From1.7and2.5, we see that

I∗

Zp

q1−ω1x1_e−1qω1x1tdμ−1/qx1

×

⎛ ⎝

Zpq

1−ω2x2_e−1qω2x2tdμ−1/qx2

Zpq

⎞ ⎠

_∞

k0

Ak−q, ω1t

k k! × _∞ l0

Tl,q−ω2ω1−1ωl₂−1l

1qlt

l l! ∞ n0 _n i0

−1n−in

i

Ai−q, ω1Tn−i,q−ω2ω1−1ω₂n−i

1qn−i

tn

n!.

2.8

On the other hand, from1.7and2.4, we have

I∗

Zp

q1−ω2x2_e−1qω2x2tdμ−1/qx2

×

⎛ ⎝

Zpq

1−ω1x1_e−1qω1x1t_dμ−1/qx1

Zpq

⎞ ⎠ _∞ k0

Ak−q, ω2t

k k! × _∞ l0

Tl,q−ω1ω2−1ωl₁−1l

1qlt

l l! ∞ n0 _n i0 −1n−i n i

Ai−q, ω2Tn−i,q−ω1ω2−1ω₁n−i

1qn−i

tn

n!.

2.9

By comparing the coeﬃcients on both sides of 2.8 and 2.9, we obtain the result in

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3. Concluding Remarks

Note that many other interesting symmetric properties for the Euler, Genocchi, and tangent numbers are derivable as corollaries of the results presented herein. For instance, considering

1,5,

An−1, ω −2ωn

En n≥0, 3.1

whereEn denotes thenth Euler number defined byEn : En0, and the Euler polynomials

are defined by the generating function

2

et₁e

xt∞

n0

Enxtn

n!, 3.2

and on puttingq1 in Theorem1.1and Corollary1.2, we obtain

n

i0

n i

ωi₁EiTn−iω1−1ωn−i₂

n

i0

n i

ωi₂EiTn−iω2−1ω₁n−i, 3.3

En

n

i0

n i

ωi

1EiTn−iω1−1. 3.4

These formulae are valid for any positive odd integersω1, ω2. The Genocchi numbersGnmay

be defined by the generating function

2t

et₁

∞

n0

Gnt

n

n!, 3.5

which have several combinatorial interpretations in terms of certain surjective maps on finite sets. The well-known identity

Gn21−2nBn 3.6

shows the relation between the Genocchi and the Bernoulli numbers. It follows from3.6

and the Staudt-Clausen theorem that the Genocchi numbers are integers. It is easy to see that

Gn2nE2n−1 n≥1, 3.7

and from3.2,3.5we deduce that

Enx n

k0

n k

Gk1

k1x

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It is well known that the tangent coeﬃcientsor numbersTn, defined by

tant

∞

n1

−1n−1T2n t

2n−1

2n−1!, 3.9

are closely related to the Bernoulli numbers, that is,see1

Tn2n2n−1Bn

n. 3.10

Ramanujan7, page 5observed that 2n₂n₋_1Bn/n_{and, therefore, the tangent coeﬃcients,}

are integers forn ≥ 1. From3.3,3.6,3.7, and3.10, the obtained symmetric formulae

involve the Bernoulli, Genocchi, and tangent numberssee1.

Acknowledgment

This work was supported by the Kyungnam University Foundation grant, 2012.

References

1 D. S. Kim, T. Kim, W. J. Kim, and D. V. Dolgy, “A note on Eulerian polynomials,” Abstract and Applied

Analysis, vol. 2012, Article ID 269640, 10 pages, 2012.

2 P. Luschny, “Eulerian polynomials,”http://www.luschny.de/math/euler/EulerianPolynomials.html, 2011.

3 D. S. Kim and K. H. Park, “Identities of symmetry for Euler polynomials arising from quotients of fermionic integrals invariant underS3,” Journal of Inequalities and Applications, vol. 2010, Article ID

851521, 16 pages, 2010.

4 T. Kim, “Symmetryp-adic invariant integral onZp for Bernoulli and Euler polynomials,” Journal of Difference Equations and Applications, vol. 14, no. 12, pp. 1267–1277, 2008.

5 C. S. Ryoo, “A note onq-Bernoulli numbers and polynomials,” Applied Mathematics Letters, vol. 20, no. 5, pp. 524–531, 2007.

6 Y. Simsek, “Complete sum of products ofh, q-extension of Euler polynomials and numbers,” Journal

of Diﬀerence Equations and Applications, vol. 16, no. 11, pp. 1331–1348, 2010.

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