Symmetry Fermionic 𝑝-Adic 𝑞-Integral on ℤ_𝑝 for Eulerian Polynomials

Abstract

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 ℤ_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 𝐴_𝑛(𝑡),𝑛=0,1,…, which can be defined by the generating function 1−𝑡𝑒^{(𝑡−1)𝑥}−𝑡=^∞_𝑛=0𝐴_𝑛(𝑡)𝑥^𝑛𝑛!,(1.1) have numerous important applications in number theory, combinatorics, and numerical analysis, among other areas. From (1.1), we note that (𝐴(𝑡)+(𝑡−1))^𝑛−𝑡𝐴_𝑛(𝑡)=(1−𝑡)𝛿_0,𝑛,(1.2) where 𝛿_𝑛,𝑘 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 ℤ_𝑝,ℚ_𝑝, and ℂ_𝑝 be the ring of 𝑝-adic integers, the field of 𝑝-adic numbers, and the completion of the algebraic closure of ℚ_𝑝, respectively. Let |⋅|_𝑝 be the 𝑝-adic valuation on ℚ, where |𝑝|_𝑝=𝑝⁻¹. The extended valuation on ℂ_𝑝 is denoted by the same symbol |⋅|_𝑝. Let 𝑞 be an indeterminate, where |1−𝑞|_𝑝<1. Then, the 𝑞-number is defined by [𝑥]_𝑞=1−𝑞^𝑥1−𝑞,[𝑥]^−𝑞=1−(−𝑞)^𝑥1+𝑞.(1.3)

For a uniformly (or strictly) differentiable function 𝑓∶ℤ_𝑝→ℂ_𝑝 (see [1, 3–6]), the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 is defined by 𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁_𝑝−𝑞^𝑁−1_𝑥=0𝑓(𝑥)(−𝑞)^𝑥.(1.4) Then, it is easy to see that 1𝑞𝐼^−1/𝑞𝑓₁+𝐼_−1/𝑞(𝑓)=[2]_1/𝑞𝑓(0),(1.5) where 𝑓₁(𝑥)=𝑓(𝑥+1).

By using the same method as that described in [1], and applying (1.5) to 𝑓, where 𝑓(𝑥)=𝑞^{(1−𝜔)𝑥}𝑒^{−𝑥(1+𝑞)𝜔𝑡}(1.6) for 𝜔∈ℤ_>0, we consider the generalized Eulerian polynomials on ℤ_𝑝 by using the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 as follows: _ℤ𝑝𝑞^{(1−𝜔)𝑥}𝑒^{−𝑥(1+𝑞)𝜔𝑡}𝑑𝜇_−1/𝑞(𝑥)=1+𝑞=𝑞^1−𝜔𝑒^{−(1+𝑞)𝜔𝑡}+𝑞_∞_𝑛=0𝐴_𝑛(−𝑞,𝜔)𝑡^𝑛𝑛!.(1.7) By expanding the Taylor series on the left-hand side of (1.7) and comparing the coefficients of the terms 𝑡^𝑛/𝑛!, we get _ℤ𝑝𝑞^{(1−𝜔)𝑥}𝑥^𝑛𝑑𝜇_−1/𝑞(𝑥)=(−1)^𝑛𝜔^𝑛(1+𝑞)^𝑛𝐴_𝑛(−𝑞,𝜔).(1.8)

We note that, by substituting 𝜔=1 into (1.8), 𝐴_𝑛(−𝑞,1)=𝐴_𝑛(−𝑞)=(−1)^𝑛(1+𝑞)^𝑛_ℤ𝑝𝑥^𝑛𝑑𝜇_−1/𝑞(𝑥)(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 𝐴_𝑛(−𝑞) at 𝑞=1 associated with the Genocchi, Euler, and tangent numbers.

Let 𝑇_𝑘,1/𝑞(𝑛) denote the 𝑞-extension of the alternating sum of integer powers, namely, 𝑇_𝑘,1/𝑞(𝑛)=^𝑛_𝑖=0(−1)^𝑖𝑖^𝑘𝑞^−𝑖=0^𝑘𝑞⁰−1^𝑘𝑞⁻¹+⋯+(−1)^𝑛𝑛^𝑘𝑞^−𝑛,(1.10) where 0⁰=1. If 𝑞→1, 𝑇_𝑘,𝑞(𝑛)→𝑇_𝑘(𝑛)=∑^𝑛𝑖=0(−1)^𝑖𝑖^𝑘 is the alternating sum of integer powers (see [4]). In particular, we have 𝑇_𝑘,1/𝑞(0)=⎧⎪⎨⎪⎩1,0,for𝑘=0,for𝑘>0.(1.11)

Let 𝜔₁,𝜔₂ be 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 in 𝜔₁ and 𝜔₂.

Theorem 1.1. Let 𝜔₁,𝜔₂ be any positive odd integers and 𝑛≥0. Then, one has ^𝑛_𝑖=0⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠𝐴_𝑖−𝑞,𝜔₁𝑇_{𝑛−𝑖,𝑞}^2−𝜔𝜔₁−1𝜔^2𝑛−𝑖(−1−𝑞)^𝑛−𝑖=^𝑛_𝑖=0⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠𝐴_𝑖−𝑞,𝜔₂𝑇_{𝑛−𝑖,𝑞}^1−𝜔𝜔₂−1𝜔^1𝑛−𝑖(−1−𝑞)^𝑛−𝑖.(1.12)

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

Using (1.11), if we take 𝜔₂=1 in Theorem 1.1, we obtain the following corollary.

Corollary 1.2. Let 𝜔₁ be any positive odd integer and 𝑛≥0. Then, one has 𝐴_𝑛(−𝑞)=^𝑛_𝑖=0⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠𝐴_𝑖−𝑞,𝜔₁𝑇_{𝑛−𝑖,𝑞}⁻¹𝜔₁−1(−1−𝑞)^𝑛−𝑖.(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 𝑞-extension of the alternating sum of integer powers.

Let 𝜔₁,𝜔₂ be any positive odd integers. From (1.7), we obtain ∫_ℤ𝑝𝑞^{(1−𝜔1)𝑥}𝑒^{−𝑥(1+𝑞)𝜔1𝑡}𝑑𝜇_−1/𝑞(𝑥)∫_ℤ𝑝𝑞^{(1−𝜔1𝜔2)𝑥}𝑒^{−𝑥(1+𝑞)𝜔1𝜔2𝑡}𝑑𝜇_−1/𝑞(𝑥)=1+𝑞^−𝜔1𝑒^{−(1+𝑞)𝜔1𝑡}^𝜔21+𝑞^−𝜔1𝑒^{−(1+𝑞)𝜔1𝑡}.(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 1+𝑞^−𝜔1𝑒^{−(1+𝑞)𝜔1𝑡}^𝜔21+𝑞^−𝜔1𝑒^{−(1+𝑞)𝜔1𝑡}=^𝜔2−1_𝑖=0(−1)^𝑖𝑞^−𝜔1𝑖𝑒^{−(1+𝑞)𝜔1𝑡𝑖}=^∞_𝑘=0⎛⎜⎜⎝^𝜔2−1_𝑖=0(−1)^𝑖𝑖^𝑘(𝑞^𝜔1)^−𝑖𝜔^𝑘1(−1)^𝑘(1+𝑞)^𝑘⎞⎟⎟⎠𝑡_𝑘𝑘!.(2.2) In (1.10), let 𝑞=𝑞^𝜔1. The left-hand, right-hand side, by definition, becomes 1+𝑞^−𝜔1𝑒^{−(1+𝑞)𝜔1𝑡}^𝜔21+𝑞^−𝜔1𝑒^{−(1+𝑞)𝜔1𝑡}=^∞_𝑘=0𝑇_𝑘,𝑞^1−𝜔𝜔₂−1𝜔^𝑘1(−1)^𝑘(1+𝑞)^𝑘𝑡^𝑘𝑘!.(2.3) A comparison of (2.1) and (2.3) yields the identity ∫_ℤ𝑝𝑞^{(1−𝜔1)𝑥}𝑒^{−𝑥(1+𝑞)𝜔1𝑡}𝑑𝜇_−1/𝑞(𝑥)∫_ℤ𝑝𝑞^{(1−𝜔1𝜔2)𝑥}𝑒^{−𝑥(1+𝑞)𝜔1𝜔2𝑡}𝑑𝜇_−1/𝑞(𝑥)=^∞_𝑘=0𝑇_𝑘,𝑞^1−𝜔𝜔₂−1𝜔^𝑘1(−1)^𝑘(1+𝑞)^𝑘𝑘𝑡^𝑘!.(2.4) By slightly modifying the derivation of (2.4), we can obtain the following identity: ∫_ℤ𝑝𝑞^{(1−𝜔2)𝑥}𝑒^{−𝑥(1+𝑞)𝜔2𝑡}𝑑𝜇_−1/𝑞(𝑥)∫_ℤ𝑝𝑞^{(1−𝜔1𝜔2)𝑥}𝑒^{−𝑥(1+𝑞)𝜔1𝜔2𝑡}𝑑𝜇_−1/𝑞(𝑥)=^∞_𝑘=0𝑇_𝑘,𝑞^2−𝜔𝜔₁−1𝜔^𝑘2(−1)^𝑘(1+𝑞)^𝑘𝑘𝑡^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 in (1.7) and the quotient of integrals in (2.4), (2.5) that 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 the 𝑞-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 𝜔₁,𝜔₂ be any positive odd integers. Using the iterated fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 and (1.7), we have ∫∫_ℤ𝑝𝑞^{(1−𝜔1)𝑥1+(1−𝜔2)𝑥2}𝑒^{−(1+𝑞)(𝜔1𝑥1+𝜔2𝑥2)𝑡}𝑑𝜇_−1/𝑞𝑥₁𝑑𝜇_−1/𝑞𝑥₂=[2]∫_ℤ𝑝𝑞^{(1−𝜔1𝜔2)𝑥}𝑒^{−𝑥(1+𝑞)𝜔1𝜔2𝑡}𝑑𝜇_−1/𝑞(𝑥)_1/𝑞𝑞^{−𝜔1𝜔2}𝑒^{−(1+𝑞)𝜔1𝜔2𝑡}+1𝑞^−𝜔1𝑒^{−(1+𝑞)𝜔1𝑡}+1𝑞^−𝜔2𝑒^{−(1+𝑞)𝜔2𝑡}+1.(2.6) Now, we put 𝐼^∗=∫∫^ℤ𝑝𝑞^{(1−𝜔1)𝑥1+(1−𝜔2)𝑥2}𝑒^{−(1+𝑞)(𝜔1𝑥1+𝜔2𝑥2)𝑡}𝑑𝜇_−1/𝑞𝑥₁𝑑𝜇_−1/𝑞𝑥₂∫_ℤ𝑝𝑞^{(1−𝜔1𝜔2)𝑥}𝑒^{−𝑥(1+𝑞)𝜔1𝜔2𝑡}𝑑𝜇_−1/𝑞(𝑥).(2.7) From (1.7) and (2.5), we see that 𝐼^∗=_ℤ𝑝𝑞^{(1−𝜔1)𝑥1}𝑒^{−(1+𝑞)(𝜔1𝑥1)𝑡}𝑑𝜇_−1/𝑞𝑥₁×⎛⎜⎜⎝∫^ℤ𝑝𝑞^{(1−𝜔2)𝑥2}𝑒^{−(1+𝑞)(𝜔2𝑥2)𝑡}𝑑𝜇_−1/𝑞𝑥₂=∫_ℤ𝑝𝑞^{(1−𝜔1𝜔2)𝑥}𝑒^{−𝑥(1+𝑞)𝜔1𝜔2𝑡}𝑑𝜇_−1/𝑞(𝑥)_∞⎞⎟⎟⎠_𝑘=0𝐴_𝑘−𝑞,𝜔₁𝑡^𝑘𝑘!×^∞_𝑙=0𝑇_𝑙,𝑞^2−𝜔𝜔₁−1𝜔^𝑙2(−1)^𝑙(1+𝑞)^𝑙𝑡^𝑙=𝑙!_∞_𝑛=0⎛⎜⎜⎝^𝑛_𝑖=0(−1)^𝑛−𝑖⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠𝐴_𝑖−𝑞,𝜔₁𝑇_{𝑛−𝑖,𝑞}^2−𝜔𝜔₁−1𝜔^2𝑛−𝑖(1+𝑞)^𝑛−𝑖⎞⎟⎟⎠𝑡_𝑛𝑛!.(2.8) On the other hand, from (1.7) and (2.4), we have 𝐼^∗=_ℤ𝑝𝑞^{(1−𝜔2)𝑥2}𝑒^{−(1+𝑞)(𝜔2𝑥2)𝑡}𝑑𝜇_−1/𝑞𝑥₂×⎛⎜⎜⎝∫^ℤ𝑝𝑞^{(1−𝜔1)𝑥1}𝑒^{−(1+𝑞)(𝜔1𝑥1)𝑡}𝑑𝜇_−1/𝑞𝑥₁=∫_ℤ𝑝𝑞^{(1−𝜔1𝜔2)𝑥}𝑒^{−𝑥(1+𝑞)𝜔1𝜔2𝑡}𝑑𝜇_−1/𝑞(𝑥)_∞⎞⎟⎟⎠_𝑘=0𝐴_𝑘−𝑞,𝜔₂𝑡^𝑘𝑘!×^∞_𝑙=0𝑇_𝑙,𝑞^1−𝜔𝜔₂−1𝜔^𝑙1(−1)^𝑙(1+𝑞)^𝑙𝑡^𝑙=𝑙!_∞_𝑛=0⎛⎜⎜⎝^𝑛_𝑖=0(−1)^𝑛−𝑖⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠𝐴_𝑖−𝑞,𝜔₂𝑇_{𝑛−𝑖,𝑞}^1−𝜔𝜔₂−1𝜔^1𝑛−𝑖(1+𝑞)^𝑛−𝑖⎞⎟⎟⎠𝑡_𝑛𝑛!.(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)], 𝐴_𝑛(−1,𝜔)=(−2𝜔)^𝑛𝐸_𝑛(𝑛≥0),(3.1) where 𝐸_𝑛 denotes the 𝑛th Euler number defined by 𝐸_𝑛∶=𝐸_𝑛(0), and the Euler polynomials are defined by the generating function 2𝑒^𝑡+1𝑒^𝑥𝑡=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛𝑛!,(3.2) and on putting 𝑞=1 in Theorem 1.1 and Corollary 1.2, we obtain ^𝑛_𝑖=0⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠𝜔_𝑖1𝐸_𝑖𝑇_𝑛−𝑖𝜔₁−1𝜔^2𝑛−𝑖=^𝑛_𝑖=0⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠𝜔_𝑖2𝐸_𝑖𝑇_𝑛−𝑖𝜔₂−1𝜔^1𝑛−𝑖,(3.3)𝐸_𝑛=^𝑛_𝑖=0⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠𝜔_𝑖1𝐸_𝑖𝑇_𝑛−𝑖𝜔₁−1.(3.4) These formulae are valid for any positive odd integers 𝜔₁,𝜔₂. The Genocchi numbers 𝐺_𝑛 may be defined by the generating function 2𝑡𝑒^𝑡+1=^∞_𝑛=0𝐺_𝑛𝑡^𝑛𝑛!,(3.5) which have several combinatorial interpretations in terms of certain surjective maps on finite sets. The well-known identity 𝐺_𝑛=2(1−2^𝑛)𝐵_𝑛(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 𝐺_𝑛=2𝑛𝐸_2𝑛−1(𝑛≥1),(3.7) and from (3.2), (3.5) we deduce that 𝐸_𝑛(𝑥)=^𝑛_𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝐺_𝑘+1𝑘+1𝑥^𝑛−𝑘.(3.8) It is well known that the tangent coefficients (or numbers) 𝑇_𝑛, defined by tan𝑡=^∞_𝑛=1(−1)^𝑛−1𝑇_2𝑛𝑡^2𝑛−1(2𝑛−1)!,(3.9) are closely related to the Bernoulli numbers, that is, (see [1]) 𝑇_𝑛=2^𝑛(2^𝑛−1)𝐵^𝑛𝑛.(3.10) Ramanujan ([7, page 5]) observed that 2^𝑛(2^𝑛−1)𝐵_𝑛/𝑛 and, therefore, the tangent coefficients, are integers for 𝑛≥1. From (3.3), (3.6), (3.7), and (3.10), the obtained symmetric formulae involve the Bernoulli, Genocchi, and tangent numbers (see [1]).

Acknowledgment

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