# Symmetry Fermionicp-Adic q-Integral on ℤp for Eulerian Polynomials

Authors: Daeyeoul Kim; Min-Soo Kim
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

Kim et al. (2012) introduced an interestingp-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 ℤp, defined by
Kim (2008), we show a
symmetric relation between the q-extension of the alternating sum of integer
powers and the Eulerian polynomials.


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## Body

## 1. Introduction

The Eulerian polynomialsAn(t),n=0,1,…, which can be defined by the generating function
(1.1)1-te(t-1)x-t=∑n=0∞‍An(t)xnn!,
have numerous important applications in number theory, combinatorics, and numerical analysis, among other areas. From (1.1), we note that
(1.2)(A(t)+(t-1))n-tAn(t)=(1-t)δ0,n,
where δn,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].Letp be a fixed odd prime number. Let ℤp,ℚp, and ℂp be the ring of p-adic integers, the field of p-adic numbers, and the completion of the algebraic closure of ℚp, respectively. Let |·|p be the p-adic valuation on ℚ, where |p|p=p-1. The extended valuation on ℂp is denoted by the same symbol |·|p. Let q be an indeterminate, where |1-q|p<1. Then, the q-number is defined by
(1.3)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q.For a uniformly (or strictly) differentiable functionf:ℤp→ℂp (see [1, 3–6]), the fermionic p-adic q-integral on ℤp is defined by
(1.4)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1‍f(x)(-q)x.
Then, it is easy to see that
(1.5)1qI-1/q(f1)+I-1/q(f)=[2]1/qf(0),
where f1(x)=f(x+1).By using the same method as that described in [1], and applying (1.5) to f, where
(1.6)f(x)=q(1-ω)xe-x(1+q)ωt
for ω∈ℤ>0, we consider the generalized Eulerian polynomials on ℤp by using the fermionic p-adic q-integral on ℤp as follows:
(1.7)∫Zp‍q(1-ω)xe-x(1+q)ωtdμ-1/q(x)=1+qq1-ωe-(1+q)ωt+q=∑n=0∞‍An(-q,ω)tnn!.
By expanding the Taylor series on the left-hand side of (1.7) and comparing the coefficients of the terms tn/n!, we get
(1.8)∫Zpq(1-ω)xxndμ-1/q(x)=(-1)nωn(1+q)nAn(-q,ω).We note that, by substitutingω=1 into (1.8),
(1.9)An(-q,1)=An(-q)=(-1)n(1+q)n∫Zpxndμ-1/q(x)
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(-q) at q=1 associated with the Genocchi, Euler, and tangent numbers.LetTk,1/q(n) denote the q-extension of the alternating sum of integer powers, namely,
(1.10)Tk,1/q(n)=∑i=0n‍(-1)iikq-i=0kq0-1kq-1+⋯+(-1)nnkq-n,
where 00=1. If q→1, Tk,q(n)→Tk(n)=∑i=0n‍(-1)iik is the alternating sum of integer powers (see [4]). In particular, we have
(1.11)Tk,1/q(0)={1,fork=0,0,fork>0.Letω1,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.
Letω1,ω2 be any positive odd integers and n≥0. Then, one has
(1.12)∑i=0n‍(ni)Ai(-q,ω1)Tn-i,q-ω2(ω1-1)ω2n-i(-1-q)n-i=∑i=0n‍(ni)Ai(-q,ω2)Tn-i,q-ω1(ω2-1)ω1n-i(-1-q)n-i.Observe that Theorem1.1 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.
Letω1 be any positive odd integer and n≥0. Then, one has
(1.13)An(-q)=∑i=0n‍(ni)Ai(-q,ω1)Tn-i,q-1(ω1-1)(-1-q)n-i.


## 2. Proof of Theorem1.1

For the proof of Theorem1.1, 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.Letω1,ω2 be any positive odd integers. From (1.7), we obtain
(2.1)∫Zpq(1-ω1)xe-x(1+q)ω1tdμ-1/q(x)∫Zpq(1-ω1ω2)xe-x(1+q)ω1ω2tdμ-1/q(x)=1+(q-ω1e-(1+q)ω1t)ω21+q-ω1e-(1+q)ω1t.
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
(2.2)1+(q-ω1e-(1+q)ω1t)ω21+q-ω1e-(1+q)ω1t=∑i=0ω2-1‍(-1)iq-ω1ie-(1+q)ω1ti=∑k=0∞‍(∑i=0ω2-1‍(-1)iik(qω1)-iω1k(-1)k(1+q)k)tkk!.
In (1.10), let q=qω1. The left-hand, right-hand side, by definition, becomes
(2.3)1+(q-ω1e-(1+q)ω1t)ω21+q-ω1e-(1+q)ω1t=∑k=0∞‍(Tk,q-ω1(ω2-1)ω1k(-1)k(1+q)k)tkk!.
A comparison of (2.1) and (2.3) yields the identity
(2.4)∫Zpq(1-ω1)xe-x(1+q)ω1tdμ-1/q(x)∫Zpq(1-ω1ω2)xe-x(1+q)ω1ω2tdμ-1/q(x)=∑k=0∞‍(Tk,q-ω1(ω2-1)ω1k(-1)k(1+q)k)tkk!.
By slightly modifying the derivation of (2.4), we can obtain the following identity:
(2.5)∫Zpq(1-ω2)xe-x(1+q)ω2tdμ-1/q(x)∫Zpq(1-ω1ω2)xe-x(1+q)ω1ω2tdμ-1/q(x)=∑k=0∞‍(Tk,q-ω2(ω1-1)ω2k(-1)k(1+q)k)tkk!.Remark 2.1.
The derivations of identities are based on the fermionicp-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 theq-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 Theorem1.1.
Letω1,ω2 be any positive odd integers. Using the iterated fermionic p-adic q-integral on ℤp and (1.7), we have
(2.6)∫∫Zpq(1-ω1)x1+(1-ω2)x2e-(1+q)(ω1x1+ω2x2)tdμ-1/q(x1)dμ-1/q(x2)∫Zpq(1-ω1ω2)xe-x(1+q)ω1ω2tdμ-1/q(x)=[2]1/qq-ω1ω2e-(1+q)ω1ω2t+1(q-ω1e-(1+q)ω1t+1)(q-ω2e-(1+q)ω2t+1).
Now, we put
(2.7)I*=∫∫Zpq(1-ω1)x1+(1-ω2)x2e-(1+q)(ω1x1+ω2x2)tdμ-1/q(x1)dμ-1/q(x2)∫Zpq(1-ω1ω2)xe-x(1+q)ω1ω2tdμ-1/q(x).
From (1.7) and (2.5), we see that
(2.8)I*=(∫Zpq(1-ω1)x1e-(1+q)(ω1x1)tdμ-1/q(x1))×(∫Zpq(1-ω2)x2e-(1+q)(ω2x2)tdμ-1/q(x2)∫Zpq(1-ω1ω2)xe-x(1+q)ω1ω2tdμ-1/q(x))=(∑k=0∞‍Ak(-q,ω1)tkk!)×(∑l=0∞‍(Tl,q-ω2(ω1-1)ω2l(-1)l(1+q)l)tll!)=∑n=0∞‍(∑i=0n‍(-1)n-i(ni)Ai(-q,ω1)Tn-i,q-ω2(ω1-1)ω2n-i(1+q)n-i)tnn!.
On the other hand, from (1.7) and (2.4), we have
(2.9)I*=(∫Zp‍q(1-ω2)x2e-(1+q)(ω2x2)tdμ-1/q(x2))×(∫Zpq(1-ω1)x1e-(1+q)(ω1x1)tdμ-1/q(x1)∫Zpq(1-ω1ω2)xe-x(1+q)ω1ω2tdμ-1/q(x))=(∑k=0∞‍Ak(-q,ω2)tkk!)×(∑l=0∞‍(Tl,q-ω1(ω2-1)ω1l(-1)l(1+q)l)tll!)=∑n=0∞‍(∑i=0n‍(-1)n-i(ni)Ai(-q,ω2)Tn-i,q-ω1(ω2-1)ω1n-i(1+q)n-i)tnn!.
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)],
(3.1)An(-1,ω)=(-2ω)nEn(n≥0),
where En denotes the nth Euler number defined by En:=En(0), and the Euler polynomials are defined by the generating function
(3.2)2et+1ext=∑n=0∞‍En(x)tnn!,
and on putting q=1 in Theorem 1.1 and Corollary 1.2, we obtain
(3.3)∑i=0n‍(ni)ω1iEiTn-i(ω1-1)ω2n-i=∑i=0n‍(ni)ω2iEiTn-i(ω2-1)ω1n-i,(3.4)En=∑i=0n‍(ni)ω1iEiTn-i(ω1-1).
These formulae are valid for any positive odd integers ω1,ω2. The Genocchi numbers Gn may be defined by the generating function
(3.5)2tet+1=∑n=0∞‍Gntnn!,
which have several combinatorial interpretations in terms of certain surjective maps on finite sets. The well-known identity
(3.6)Gn=2(1-2n)Bn
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
(3.7)Gn=2nE2n-1(n≥1),
and from (3.2), (3.5) we deduce that
(3.8)En(x)=∑k=0n‍(nk)Gk+1k+1xn-k.
It is well known that the tangent coefficients (or numbers) Tn, defined by
(3.9)tant=∑n=1∞‍(-1)n-1T2nt2n-1(2n-1)!,
are closely related to the Bernoulli numbers, that is, (see [1])
(3.10)Tn=2n(2n-1)Bnn.
Ramanujan ([7, page 5]) observed that 2n(2n-1)Bn/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]).



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