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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 424189, 7 pages
Research Article

Symmetry Fermionic 𝑝-Adic π‘ž-Integral on ℀𝑝 for Eulerian Polynomials

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

Received 18 June 2012; Accepted 14 August 2012

Academic Editor: CheonΒ Ryoo

Copyright Β© 2012 Daeyeoul Kim and Min-Soo 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. (2012) introduced 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 on ℀𝑝, defined by Kim (2008), we show a symmetric relation between the q-extension of the alternating sum of integer powers and the Eulerian polynomials.

1. Introduction

The Eulerian polynomials 𝐴𝑛(𝑑),𝑛=0,1,…, which can be defined by the generating function 1βˆ’π‘‘π‘’(π‘‘βˆ’1)π‘₯=βˆ’π‘‘βˆžξ“π‘›=0𝐴𝑛(π‘₯𝑑)𝑛,𝑛!(1.1) have numerous important applications in number theory, combinatorics, and numerical analysis, among other areas. From (1.1), we note that (𝐴(𝑑)+(π‘‘βˆ’1))π‘›βˆ’π‘‘π΄π‘›(𝑑)=(1βˆ’π‘‘)𝛿0,𝑛,(1.2) where 𝛿𝑛,π‘˜ is the Kronecker symbol (see [1]). Thus far, few recurrences for the Eulerian polynomials other than (1.2) have been reported in the literature. Other recurrences are of importance as they might reveal new aspects and properties of the Eulerian polynomials, and they can help simplify the proofs of known properties. For more important properties, see, for instance, [1] or [2].

Let 𝑝 be a fixed odd prime number. Let ℀𝑝,β„šπ‘, and ℂ𝑝 be the ring of 𝑝-adic integers, the field of 𝑝-adic numbers, and the completion of the algebraic closure of β„šπ‘, respectively. Let |β‹…|𝑝 be the 𝑝-adic valuation on β„š, where |𝑝|𝑝=π‘βˆ’1. The extended valuation on ℂ𝑝 is denoted by the same symbol |β‹…|𝑝. Let π‘ž be an indeterminate, where |1βˆ’π‘ž|𝑝<1. Then, the π‘ž-number is defined by [π‘₯]π‘ž=1βˆ’π‘žπ‘₯,[π‘₯]1βˆ’π‘žβˆ’π‘ž=1βˆ’(βˆ’π‘ž)π‘₯.1+π‘ž(1.3)

For a uniformly (or strictly) differentiable function π‘“βˆΆβ„€π‘β†’β„‚π‘ (see [1, 3–6]), the fermionic 𝑝-adic π‘ž-integral on ℀𝑝 is defined by πΌβˆ’π‘ž(ξ€œπ‘“)=℀𝑝𝑓(π‘₯)π‘‘πœ‡βˆ’π‘ž(π‘₯)=limπ‘β†’βˆž1ξ€Ίπ‘π‘ξ€»π‘βˆ’π‘žπ‘βˆ’1π‘₯=0𝑓(π‘₯)(βˆ’π‘ž)π‘₯.(1.4) Then, it is easy to see that 1π‘žπΌβˆ’1/π‘žξ€·π‘“1ξ€Έ+πΌβˆ’1/π‘ž[2](𝑓)=1/π‘žπ‘“(0),(1.5) where 𝑓1(π‘₯)=𝑓(π‘₯+1).

By using the same method as that described in [1], and applying (1.5) to 𝑓, where 𝑓(π‘₯)=π‘ž(1βˆ’πœ”)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”π‘‘(1.6) for πœ”βˆˆβ„€>0, we consider the generalized Eulerian polynomials on ℀𝑝 by using the fermionic 𝑝-adic π‘ž-integral on ℀𝑝 as follows: ξ€œβ„€π‘π‘ž(1βˆ’πœ”)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”π‘‘π‘‘πœ‡βˆ’1/π‘ž(π‘₯)=1+π‘žπ‘ž1βˆ’πœ”π‘’βˆ’(1+π‘ž)πœ”π‘‘=+π‘žβˆžξ“π‘›=0𝐴𝑛𝑑(βˆ’π‘ž,πœ”)𝑛.𝑛!(1.7) By expanding the Taylor series on the left-hand side of (1.7) and comparing the coefficients of the terms 𝑑𝑛/𝑛!, we get ξ€œβ„€π‘π‘ž(1βˆ’πœ”)π‘₯π‘₯π‘›π‘‘πœ‡βˆ’1/π‘ž(π‘₯)=(βˆ’1)π‘›πœ”π‘›(1+π‘ž)𝑛𝐴𝑛(βˆ’π‘ž,πœ”).(1.8)

We note that, by substituting πœ”=1 into (1.8), 𝐴𝑛(βˆ’π‘ž,1)=𝐴𝑛(βˆ’π‘ž)=(βˆ’1)𝑛(1+π‘ž)π‘›ξ€œβ„€π‘π‘₯π‘›π‘‘πœ‡βˆ’1/π‘ž(π‘₯)(1.9) is the Witt's formula for the Eulerian polynomials in [1, Theorem 1]. Recently, Kim et al. [1] investigated new properties of the Eulerian polynomials 𝐴𝑛(βˆ’π‘ž) at π‘ž=1 associated with the Genocchi, Euler, and tangent numbers.

Let π‘‡π‘˜,1/π‘ž(𝑛) denote the π‘ž-extension of the alternating sum of integer powers, namely, π‘‡π‘˜,1/π‘ž(𝑛)=𝑛𝑖=0(βˆ’1)π‘–π‘–π‘˜π‘žβˆ’π‘–=0π‘˜π‘ž0βˆ’1π‘˜π‘žβˆ’1+β‹―+(βˆ’1)π‘›π‘›π‘˜π‘žβˆ’π‘›,(1.10) where 00=1. If π‘žβ†’1, π‘‡π‘˜,π‘ž(𝑛)β†’π‘‡π‘˜βˆ‘(𝑛)=𝑛𝑖=0(βˆ’1)π‘–π‘–π‘˜ is the alternating sum of integer powers (see [4]). In particular, we have π‘‡π‘˜,1/π‘žβŽ§βŽͺ⎨βŽͺ⎩(0)=1,forπ‘˜=0,0,forπ‘˜>0.(1.11)

Let πœ”1,πœ”2 be any positive odd integers. Our main result of symmetry between the π‘ž-extension of the alternating sum of integer powers and the Eulerian polynomials is given in the following theorem, which is symmetric in πœ”1 and πœ”2.

Theorem 1.1. Let πœ”1,πœ”2 be any positive odd integers and 𝑛β‰₯0. Then, one has 𝑛𝑖=0βŽ›βŽœβŽœβŽπ‘›π‘–βŽžβŽŸβŽŸβŽ π΄π‘–ξ€·βˆ’π‘ž,πœ”1ξ€Έπ‘‡π‘›βˆ’π‘–,π‘ž2βˆ’πœ”ξ€·πœ”1ξ€Έπœ”βˆ’12π‘›βˆ’π‘–(βˆ’1βˆ’π‘ž)π‘›βˆ’π‘–=𝑛𝑖=0βŽ›βŽœβŽœβŽπ‘›π‘–βŽžβŽŸβŽŸβŽ π΄π‘–ξ€·βˆ’π‘ž,πœ”2ξ€Έπ‘‡π‘›βˆ’π‘–,π‘ž1βˆ’πœ”ξ€·πœ”2ξ€Έπœ”βˆ’11π‘›βˆ’π‘–(βˆ’1βˆ’π‘ž)π‘›βˆ’π‘–.(1.12)

Observe that Theorem 1.1 can be obtained by the same method as that described in [4]. If π‘ž=1, Theorem 1.1 reduces to the form stated in the remark in [4, page 1275].

Using (1.11), if we take πœ”2=1 in Theorem 1.1, we obtain the following corollary.

Corollary 1.2. Let πœ”1 be any positive odd integer and 𝑛β‰₯0. Then, one has 𝐴𝑛(βˆ’π‘ž)=𝑛𝑖=0βŽ›βŽœβŽœβŽπ‘›π‘–βŽžβŽŸβŽŸβŽ π΄π‘–ξ€·βˆ’π‘ž,πœ”1ξ€Έπ‘‡π‘›βˆ’π‘–,π‘žβˆ’1ξ€·πœ”1ξ€Έβˆ’1(βˆ’1βˆ’π‘ž)π‘›βˆ’π‘–.(1.13)

2. Proof of Theorem 1.1

For the proof of Theorem 1.1, we will need the following two identities (see (2.4) and (2.5)) related to the Eulerian polynomials and the π‘ž-extension of the alternating sum of integer powers.

Let πœ”1,πœ”2 be any positive odd integers. From (1.7), we obtain βˆ«β„€π‘π‘ž(1βˆ’πœ”1)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”1π‘‘π‘‘πœ‡βˆ’1/π‘ž(π‘₯)βˆ«β„€π‘π‘ž(1βˆ’πœ”1πœ”2)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”1πœ”2π‘‘π‘‘πœ‡βˆ’1/π‘ž=ξ€·π‘ž(π‘₯)1+βˆ’πœ”1π‘’βˆ’(1+π‘ž)πœ”1π‘‘ξ€Έπœ”21+π‘žβˆ’πœ”1π‘’βˆ’(1+π‘ž)πœ”1𝑑.(2.1) This has an interesting 𝑝-adic analytic interpretation, which we shall discuss below (see Remark 2.1). It is easy to see that the right-hand side of (2.1) can be written as ξ€·π‘ž1+βˆ’πœ”1π‘’βˆ’(1+π‘ž)πœ”1π‘‘ξ€Έπœ”21+π‘žβˆ’πœ”1π‘’βˆ’(1+π‘ž)πœ”1𝑑=πœ”2βˆ’1𝑖=0(βˆ’1)π‘–π‘žβˆ’πœ”1π‘–π‘’βˆ’(1+π‘ž)πœ”1𝑑𝑖=βˆžξ“π‘˜=0βŽ›βŽœβŽœβŽπœ”2βˆ’1𝑖=0(βˆ’1)π‘–π‘–π‘˜(π‘žπœ”1)βˆ’π‘–πœ”π‘˜1(βˆ’1)π‘˜(1+π‘ž)π‘˜βŽžβŽŸβŽŸβŽ π‘‘π‘˜.π‘˜!(2.2) In (1.10), let π‘ž=π‘žπœ”1. The left-hand, right-hand side, by definition, becomes ξ€·π‘ž1+βˆ’πœ”1π‘’βˆ’(1+π‘ž)πœ”1π‘‘ξ€Έπœ”21+π‘žβˆ’πœ”1π‘’βˆ’(1+π‘ž)πœ”1𝑑=βˆžξ“π‘˜=0ξ€·π‘‡π‘˜,π‘ž1βˆ’πœ”ξ€·πœ”2ξ€Έπœ”βˆ’1π‘˜1(βˆ’1)π‘˜(1+π‘ž)π‘˜ξ€Έπ‘‘π‘˜.π‘˜!(2.3) A comparison of (2.1) and (2.3) yields the identity βˆ«β„€π‘π‘ž(1βˆ’πœ”1)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”1π‘‘π‘‘πœ‡βˆ’1/π‘ž(π‘₯)βˆ«β„€π‘π‘ž(1βˆ’πœ”1πœ”2)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”1πœ”2π‘‘π‘‘πœ‡βˆ’1/π‘ž=(π‘₯)βˆžξ“π‘˜=0ξ€·π‘‡π‘˜,π‘ž1βˆ’πœ”ξ€·πœ”2ξ€Έπœ”βˆ’1π‘˜1(βˆ’1)π‘˜(1+π‘ž)π‘˜ξ€Έπ‘‘π‘˜.π‘˜!(2.4) By slightly modifying the derivation of (2.4), we can obtain the following identity: βˆ«β„€π‘π‘ž(1βˆ’πœ”2)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”2π‘‘π‘‘πœ‡βˆ’1/π‘ž(π‘₯)βˆ«β„€π‘π‘ž(1βˆ’πœ”1πœ”2)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”1πœ”2π‘‘π‘‘πœ‡βˆ’1/π‘ž=(π‘₯)βˆžξ“π‘˜=0ξ€·π‘‡π‘˜,π‘ž2βˆ’πœ”ξ€·πœ”1ξ€Έπœ”βˆ’1π‘˜2(βˆ’1)π‘˜(1+π‘ž)π‘˜ξ€Έπ‘‘π‘˜.π‘˜!(2.5)

Remark 2.1. The derivations of identities are based on the fermionic 𝑝-adic π‘ž-integral expression of the generating function for the Eulerian polynomials in (1.7) and the quotient of integrals in (2.4), (2.5) that can be expressed as the exponential generating function for the π‘ž-extension of the alternating sum of integer powers.
Observe that similar identities related to the Eulerian polynomials and the π‘ž-extension of the alternating sum of integer powers in (2.4) and (2.5) can be found, for instance, in [3, (1.8)], [4, (21)], and [6, Theorem 4].

Proof of Theorem 1.1. Let πœ”1,πœ”2 be any positive odd integers. Using the iterated fermionic 𝑝-adic π‘ž-integral on ℀𝑝 and (1.7), we have βˆ«βˆ«β„€π‘π‘ž(1βˆ’πœ”1)π‘₯1+(1βˆ’πœ”2)π‘₯2π‘’βˆ’(1+π‘ž)(πœ”1π‘₯1+πœ”2π‘₯2)π‘‘π‘‘πœ‡βˆ’1/π‘žξ€·π‘₯1ξ€Έπ‘‘πœ‡βˆ’1/π‘žξ€·π‘₯2ξ€Έβˆ«β„€π‘π‘ž(1βˆ’πœ”1πœ”2)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”1πœ”2π‘‘π‘‘πœ‡βˆ’1/π‘ž=[2](π‘₯)1/π‘žπ‘žβˆ’πœ”1πœ”2π‘’βˆ’(1+π‘ž)πœ”1πœ”2𝑑+1ξ€·π‘žβˆ’πœ”1π‘’βˆ’(1+π‘ž)πœ”1π‘‘π‘ž+1ξ€Έξ€·βˆ’πœ”2π‘’βˆ’(1+π‘ž)πœ”2𝑑.+1(2.6) Now, we put πΌβˆ—=βˆ«βˆ«β„€π‘π‘ž(1βˆ’πœ”1)π‘₯1+(1βˆ’πœ”2)π‘₯2π‘’βˆ’(1+π‘ž)(πœ”1π‘₯1+πœ”2π‘₯2)π‘‘π‘‘πœ‡βˆ’1/π‘žξ€·π‘₯1ξ€Έπ‘‘πœ‡βˆ’1/π‘žξ€·π‘₯2ξ€Έβˆ«β„€π‘π‘ž(1βˆ’πœ”1πœ”2)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”1πœ”2π‘‘π‘‘πœ‡βˆ’1/π‘ž.(π‘₯)(2.7) From (1.7) and (2.5), we see that πΌβˆ—=ξƒ©ξ€œβ„€π‘π‘ž(1βˆ’πœ”1)π‘₯1π‘’βˆ’(1+π‘ž)(πœ”1π‘₯1)π‘‘π‘‘πœ‡βˆ’1/π‘žξ€·π‘₯1ξ€ΈξƒͺΓ—βŽ›βŽœβŽœβŽβˆ«β„€π‘π‘ž(1βˆ’πœ”2)π‘₯2π‘’βˆ’(1+π‘ž)(πœ”2π‘₯2)π‘‘π‘‘πœ‡βˆ’1/π‘žξ€·π‘₯2ξ€Έβˆ«β„€π‘π‘ž(1βˆ’πœ”1πœ”2)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”1πœ”2π‘‘π‘‘πœ‡βˆ’1/π‘žβŽžβŽŸβŽŸβŽ =(π‘₯)βˆžξ“π‘˜=0π΄π‘˜ξ€·βˆ’π‘ž,πœ”1ξ€Έπ‘‘π‘˜ξƒͺΓ—ξƒ©π‘˜!βˆžξ“π‘™=0𝑇𝑙,π‘ž2βˆ’πœ”ξ€·πœ”1ξ€Έπœ”βˆ’1𝑙2(βˆ’1)𝑙(1+π‘ž)𝑙𝑑𝑙ξƒͺ=𝑙!βˆžξ“π‘›=0βŽ›βŽœβŽœβŽπ‘›ξ“π‘–=0(βˆ’1)π‘›βˆ’π‘–βŽ›βŽœβŽœβŽπ‘›π‘–βŽžβŽŸβŽŸβŽ π΄π‘–ξ€·βˆ’π‘ž,πœ”1ξ€Έπ‘‡π‘›βˆ’π‘–,π‘ž2βˆ’πœ”ξ€·πœ”1ξ€Έπœ”βˆ’12π‘›βˆ’π‘–(1+π‘ž)π‘›βˆ’π‘–βŽžβŽŸβŽŸβŽ π‘‘π‘›.𝑛!(2.8) On the other hand, from (1.7) and (2.4), we have πΌβˆ—=ξƒ©ξ€œβ„€π‘π‘ž(1βˆ’πœ”2)π‘₯2π‘’βˆ’(1+π‘ž)(πœ”2π‘₯2)π‘‘π‘‘πœ‡βˆ’1/π‘žξ€·π‘₯2ξ€ΈξƒͺΓ—βŽ›βŽœβŽœβŽβˆ«β„€π‘π‘ž(1βˆ’πœ”1)π‘₯1π‘’βˆ’(1+π‘ž)(πœ”1π‘₯1)π‘‘π‘‘πœ‡βˆ’1/π‘žξ€·π‘₯1ξ€Έβˆ«β„€π‘π‘ž(1βˆ’πœ”1πœ”2)π‘₯π‘’βˆ’π‘₯(1+π‘ž)πœ”1πœ”2π‘‘π‘‘πœ‡βˆ’1/π‘žβŽžβŽŸβŽŸβŽ =(π‘₯)βˆžξ“π‘˜=0π΄π‘˜ξ€·βˆ’π‘ž,πœ”2ξ€Έπ‘‘π‘˜ξƒͺΓ—ξƒ©π‘˜!βˆžξ“π‘™=0𝑇𝑙,π‘ž1βˆ’πœ”ξ€·πœ”2ξ€Έπœ”βˆ’1𝑙1(βˆ’1)𝑙(1+π‘ž)𝑙𝑑𝑙ξƒͺ=𝑙!βˆžξ“π‘›=0βŽ›βŽœβŽœβŽπ‘›ξ“π‘–=0(βˆ’1)π‘›βˆ’π‘–βŽ›βŽœβŽœβŽπ‘›π‘–βŽžβŽŸβŽŸβŽ π΄π‘–ξ€·βˆ’π‘ž,πœ”2ξ€Έπ‘‡π‘›βˆ’π‘–,π‘ž1βˆ’πœ”ξ€·πœ”2ξ€Έπœ”βˆ’11π‘›βˆ’π‘–(1+π‘ž)π‘›βˆ’π‘–βŽžβŽŸβŽŸβŽ π‘‘π‘›.𝑛!(2.9) By comparing the coefficients on both sides of (2.8) and (2.9), we obtain the result in Theorem 1.1.

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(βˆ’1,πœ”)=(βˆ’2πœ”)𝑛𝐸𝑛(𝑛β‰₯0),(3.1) where 𝐸𝑛 denotes the 𝑛th Euler number defined by πΈπ‘›βˆΆ=𝐸𝑛(0), and the Euler polynomials are defined by the generating function 2𝑒𝑑𝑒+1π‘₯𝑑=βˆžξ“π‘›=0𝐸𝑛(𝑑π‘₯)𝑛,𝑛!(3.2) and on putting π‘ž=1 in Theorem 1.1 and Corollary 1.2, we obtain 𝑛𝑖=0βŽ›βŽœβŽœβŽπ‘›π‘–βŽžβŽŸβŽŸβŽ πœ”π‘–1πΈπ‘–π‘‡π‘›βˆ’π‘–ξ€·πœ”1ξ€Έπœ”βˆ’12π‘›βˆ’π‘–=𝑛𝑖=0βŽ›βŽœβŽœβŽπ‘›π‘–βŽžβŽŸβŽŸβŽ πœ”π‘–2πΈπ‘–π‘‡π‘›βˆ’π‘–ξ€·πœ”2ξ€Έπœ”βˆ’11π‘›βˆ’π‘–,(3.3)𝐸𝑛=𝑛𝑖=0βŽ›βŽœβŽœβŽπ‘›π‘–βŽžβŽŸβŽŸβŽ πœ”π‘–1πΈπ‘–π‘‡π‘›βˆ’π‘–ξ€·πœ”1ξ€Έ.βˆ’1(3.4) These formulae are valid for any positive odd integers πœ”1,πœ”2. The Genocchi numbers 𝐺𝑛 may be defined by the generating function 2𝑑𝑒𝑑=+1βˆžξ“π‘›=0𝐺𝑛𝑑𝑛,𝑛!(3.5) which have several combinatorial interpretations in terms of certain surjective maps on finite sets. The well-known identity 𝐺𝑛=2(1βˆ’2𝑛)𝐡𝑛(3.6) shows the relation between the Genocchi and the Bernoulli numbers. It follows from (3.6) and the Staudt-Clausen theorem that the Genocchi numbers are integers. It is easy to see that 𝐺𝑛=2𝑛𝐸2π‘›βˆ’1(𝑛β‰₯1),(3.7) and from (3.2), (3.5) we deduce that 𝐸𝑛(π‘₯)=π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ πΊπ‘˜+1π‘₯π‘˜+1π‘›βˆ’π‘˜.(3.8) It is well known that the tangent coefficients (or numbers) 𝑇𝑛, defined by tan𝑑=βˆžξ“π‘›=1(βˆ’1)π‘›βˆ’1𝑇2𝑛𝑑2π‘›βˆ’1,(2π‘›βˆ’1)!(3.9) are closely related to the Bernoulli numbers, that is, (see [1]) 𝑇𝑛=2𝑛(2π‘›π΅βˆ’1)𝑛𝑛.(3.10) Ramanujan ([7, page 5]) observed that 2𝑛(2π‘›βˆ’1)𝐡𝑛/𝑛 and, therefore, the tangent coefficients, are integers for 𝑛β‰₯1. From (3.3), (3.6), (3.7), and (3.10), the obtained symmetric formulae involve the Bernoulli, Genocchi, and tangent numbers (see [1]).


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


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