for Eulerian Polynomials

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

Research Article

Symmetry Fermionic p-Adic q-Integral on Z

p

for Eulerian Polynomials

Daeyeoul Kim

¹

and Min-Soo Kim

²

1National Institute for Mathematical Sciences, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon 305-811, Republic of Korea

2Division 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

e^t−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 importance as they might reveal new aspects and properties of the Eulerian polynomials,

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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

x_q 1−q^x

1−q, x_−q 1−

−q_x

1q . 1.3

For a uniformlyor strictlydiﬀerentiable functionf : Zp → Cp see1,3–6, the fermionicp-adicq-integral onZpis defined by

I_−q f

Zp

fxdμ_−qx lim

N→ ∞

1 p^N

−q p^N−1

x0

fx

−qx

. 1.4

Then, it is easy to see that 1 qI−1/q

f1

I−1/q f

2_1/qf0, 1.5

wheref1x fx1.

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

fx q^1−ωxe^−x1qωt 1.6

forω∈Z>0, we consider the generalized Eulerian polynomials onZpby using the fermionic p-adicq-integral onZpas follows:

Zp

q^1−ωxe^−x1qωtdμ−1/qx 1q q^1−ωe^−1qωtq ^∞

n0

An

−q, ωtⁿ 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

q^1−ωxxⁿdμ−1/qx −1ⁿ ωⁿ

1qnAn

−q, ω

. 1.8

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We note that, by substitutingω1 into1.8, An

−q,1 An

−q

−1ⁿ

1q_n

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

−1ⁱi^kq⁻ⁱ0^kq⁰−1^kq⁻¹· · · −1ⁿn^kq⁻ⁿ, 1.10

where 0⁰ 1. Ifq → 1,Tk,qn → Tkn _n

i0−1ⁱi^k 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 theq- 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, ω1

Tn−i,q^−ω²ω1−1ω₂ⁿ⁻ⁱ

−1−q_n−i

ⁿ

i0

n i

Ai

−q, ω2

Tn−i,q^−ω¹ω2−1ωⁿ⁻ⁱ₁

−1−q_n−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, ω1

Tn−i,q⁻¹ω1−1

−1−q_n−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 powers.

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

Zpq^1−ω¹^xe^−x1qω¹^tdμ_−1/qx

Zpq^1−ω¹^ω²^xe^−x1qω¹^ω²^tdμ_−1/qx 1

q^−ω¹e^−1qω¹^tω2

1q^−ω¹e^−1qω¹^t . 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

1

1q^−ω¹e^−1qω¹^t ^ω²⁻¹

i0

−1ⁱq^−ω¹ⁱe^−1qω¹^ti

^∞

k0

_ω

2−1

i0

−1ⁱi^k q^ω¹_−i

ω^k₁−1^k

1q_k t^k k!.

2.2

In1.10, letqq^ω¹. The left-hand, right-hand side, by definition, becomes

1

1q^−ω¹e^−1qω¹^t ^∞

k0

Tk,q^−ω¹ω2−1ω^k₁−1^k

1q_kt^k

k!. 2.3

A comparison of2.1and2.3yields the identity

Zpq^1−ω¹^xe^−x1qω¹^tdμ−1/qx

Zpq^1−ω¹^ω²^xe^−x1qω¹^ω²^tdμ−1/qx ^∞

k0

Tk,q^−ω¹ω2−1ω^k₁−1^k

1qkt^k

k!. 2.4

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

Zpq^1−ω²^xe^−x1qω²^tdμ−1/qx

Zpq^1−ω¹^ω²^xe^−x1qω¹^ω²^tdμ−1/qx ^∞

k0

Tk,q^−ω²ω1−1ω^k₂−1^k

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, 1.8,4,21, and6, Theorem 4.

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Proof of Theorem1.1. Letω1, ω2be any positive odd integers. Using the iterated fermionicp- adicq-integral onZpand1.7, we have

Zpq^1−ω¹^x¹^1−ω²^x²e^−1qω¹^x¹^ω²^x²^tdμ−1/qx1dμ−1/qx2

Zpq^1−ω¹^ω²^xe^−x1qω¹^ω²^tdμ−1/qx 2_1/q q^−ω¹^ω²e^−1qω¹^ω²^t1

q^−ω¹e^−1qω¹^t1

q^−ω²e^−1qω²^t1.

2.6

Now, we put

I^∗ ^Z^pq^1−ω¹^x¹^1−ω²^x²e^−1qω¹^x¹^ω²^x²^tdμ_−1/qx1dμ_−1/qx2

Zpq^1−ω¹^ω²^xe^−x1qω¹^ω²^tdμ−1/qx . 2.7

From1.7and2.5, we see that

I^∗

Zp

q^1−ω¹^x¹e^−1qω¹^x¹^tdμ−1/qx1

×

⎛

⎝

Zpq^1−ω²^x²e^−1qω²^x²^tdμ−1/qx2

Zpq^1−ω¹^ω²^xe^−x1qω¹^ω²^tdμ−1/qx

⎞

⎠

_∞

k0

Ak

−q, ω1

t^k k!

× _∞

l0

Tl,q^−ω²ω1−1ω^l₂−1^l

1q_lt^l l!

^∞

n0

_n

i0

−1ⁿ⁻ⁱ n

i

Ai

−q, ω1

T_n−i,q^−ω2ω1−1ω₂ⁿ⁻ⁱ

1q_n−i tⁿ n!.

2.8

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

I^∗

Zp

q^1−ω²^x²e^−1qω²^x²^tdμ−1/qx2

×

⎛

⎝

Zpq^1−ω¹^x¹e^−1qω¹^x¹^tdμ_−1/qx1

Zpq^1−ω¹^ω²^xe^−x1qω¹^ω²^tdμ_−1/qx

⎞

⎠

_∞

k0

Ak

−q, ω2

t^k k!

× _∞

l0

Tl,q^−ω¹ω2−1ω^l₁−1^l

1qlt^l l!

^∞

n0

_n

i0

−1ⁿ⁻ⁱ n

i

Ai

−q, ω2

Tn−i,q^−ω¹ω2−1ω₁ⁿ⁻ⁱ

1q_n−i tⁿ n!.

2.9

By comparing the coeﬃcients on both sides of 2.8 and 2.9, we obtain the result in Theorem1.1.

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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 −2ωⁿEn n≥0, 3.1 whereEn denotes thenth Euler number defined byEn : En0, and the Euler polynomials are defined by the generating function

2

e^t1e^xt^∞

n0

Enxtⁿ

n!, 3.2

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

i0

n i

ωⁱ₁EiT_n−iω1−1ωⁿ⁻ⁱ₂ ⁿ

i0

n i

ωⁱ₂EiT_n−iω2−1ω₁ⁿ⁻ⁱ, 3.3

Enⁿ

i0

n i

ωⁱ₁EiT_n−iω1−1. 3.4

These formulae are valid for any positive odd integersω1, ω2. The Genocchi numbersGnmay be defined by the generating function

2t e^t1 ^∞

n0

Gntⁿ

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

G_k1

k1x^n−k. 3.8

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

tant^∞

n1

−1ⁿ⁻¹T2n t²ⁿ⁻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/nand, 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.

7 S. Ramanujan, Collected Papers of Srinivasa Ramanujan, AMS Chelsea Publishing, Providence, RI, USA, 2000.

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for Eulerian Polynomials

Research Article

Symmetry Fermionic p-Adic q-Integral on Z

for Eulerian Polynomials

Daeyeoul Kim

and Min-Soo Kim

1. Introduction

2. Proof of Theorem 1.1

3. Concluding Remarks

Acknowledgment

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Discrete Mathematics

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