# Some Identities of the Frobenius-Euler Polynomials

Authors: Taekyun Kim; Byungje Lee
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

By using the ordinary fermionicp-adic invariant integral on ℤp, we derive
some interesting identities related to the Frobenius-Euler polynomials.


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## Body

## 1. Introduction

Letp be a fixed
prime. Throughout  this paper ℤp,ℚp,ℂ, and ℂp will,
respectively, denote the ring of p-adic rational
integers, the field of p-adic rational
numbers, the complex number field, and the completion of algebraic closure of ℚp. When one talks
about q-extension, q is variously
considered as an indeterminate, a complex q∈ℂ, or a p-adic number q∈ℂp; see [1–14]. If q∈ℂ, then we assume |q|<1. If q∈ℂp, then we assume |1−q|p<1. For x∈ℚp, we use the
notation [x]q=(1−qx)/(1−q), and [x]−q=(1−(−q)x)/(1+q); see [15, 16]. The
normalized valuation in ℂp is denoted by |⋅|p with |p|p=1/p. We say that f is a uniformly
differentiable function at a point a∈ℤp and denote this
property by f∈UD(ℤp), if the difference quotients Ff(x,y)=(f(x)−f(y))/(x−y) have a limit l=f′(a) as (x,y)→(a,a). For f∈UD(ℤp), let us start with the expression(1.1)1[pN]q∑0≤j<pNqjf(j)=∑0≤j<pNf(j)μq(j+pNℤp),representing a q-analogue of
Riemann sums for f; see [15, 16]. The integral of f on ℤp will be defined
as a limit (n→∞) of those sums, when it exists. The q-deformed
bosonic p-adic integral
of the  function f∈UD(ℤp) is defined by(1.2)Iq(f)=∫ℤpf(x)dμq(x)=limN→∞1[dpN]q∑0≤x<dpNf(x)qx, see [15]. Thus, we  note that(1.3)qIq(f1)=Iq(f)+(q−1)f(0)+q−1logqf′(0),where f1(x)=f(x+1),f′(0)=df(0)/dx.The fermionicp-adic invariant
integral on ℤp is defined as(1.4)I−1(f)=∫ℤpf(x)dμ−1(x)=limN→∞1[pN]−1∑x=0pN−1f(x)(−1)x, see [15].In this paper, we prove an identity of symmetry for the Frobenius-Euler polynomials.
Finally we investigate the several further interesting properties of the symmetry
for the fermionicp-adic invariant integral on ℤp related to the Frobenius-Euler polynomials
and numbers.


## 2. Some Identities of the Frobenius-Euler Polynomials

Letu(≠1)∈ℂp (or ℂ) be algebraic.
Then the nth
Frobenius-Euler numbers Hn(u) are defined as (2.1)H0(u)=1,(H(u)+1)n−uHn(u)=0,ifn≥1, with the usual convention about replacing Hn(u) by Hn(u).Thenth
Frobenius-Euler polynomials Hn(u,x) are also
defined as(2.2)Hn(u,x)=∑l=0n(nl)xn−lHl(u).From (1.4), we can easily derive(2.3)I−1(f1)+I−1(f)=2f(0),wheref1(x)=f(x+1). By continuing this process, we see that(2.4)I−1(fn)+(−1)n−1I−1(f)=2∑l=0n−1(−1)n−1−lf(l),wherefn(x)=f(x+n).When n is an odd
positive integer, we obtain(2.5)I−1(fn)+I−1(f)=2∑l=0n−1(−1)lf(l).If n∈ℕ with n≡0(mod2), then we have(2.6)I−1(fn)−I−1(f)=2∑l=0n−1(−1)l−1f(l).From (1.4) and (2.3), we derive(2.7)∫ℤpextqxdμ−1(x)=2[2]q1−(−q)−1et−(−q)−1=2[2]q∑n=0∞Hn(−q−1)tnn!.Thus, we note that(2.8)∫ℤpxnqxdμ−1(x)=2[2]qHn(−q−1),∫ℤp(y+x)nqydμ−1(x)=2[2]qHn(−q−1,x). Let n∈ℕ with n≡1(mod2). Then we obtain(2.9)[2]q∑l=0n−1(−1)lqllm=qnHm(−q−1,n)+Hm(−q−1). For n∈ℕ with n≡0(mod2), we have(2.10)qnHm(−q−1,n)−Hm(−q−1)=[2]q∑l=0n−1(−1)l−1qllm. By substituting f(x)=qxext into (2.5), we
can easily see that(2.11)∫ℤpqn+xe(x+n)tdμ−1(x)+∫ℤpqxextdμ−1(x)=2qnent+1qet+1=2∑l=0n−1(−1)lqlelt.Let Sk,q(n)=∑l=0n(−1)llkql. Then Sk,q(n) is called the
alternating sums of powers of consecutive q-integers. From
the definition of the fermionic p-adic invariant
integral on ℤp, we can derive (2.12)∫ℤpqx+ne(x+n)tdμ−1(x)+∫ℤpqxextdμ−1(x)=2∫ℤpqxextdμ−1(x)∫ℤpenxtqnxdμ−1(x).By (2.12), we easily see that(2.13)∫ℤpqnxenxtdμ−1(x)=2qnent+1.Letw1,w2(∈ℕ) be odd. By
using double fermionic p-adic invariant
integral on ℤp, we obtain(2.14)∫ℤp∫ℤpe(w1x1+w2x2)tqw1x1+w2x2dμ−1(x1)dμ−1(x2)∫ℤpew1w2xtqw1w2xdμ−1(x)=2(qw1w2ew1w2t+1)(qw1ew1t+1)(qw2ew2t+1).Now we also consider the following fermionicp-adic invariant
integral on ℤp associated with
Frobenius-Euler polynomials: (2.15)∫ℤp∫ℤpe(w1x1+w2x2+w1w2x)tqw1x1+w2x2dμ−1(x1)dμ−1(x2)∫ℤpew1w2xtqw1w2xdμ−1(x)=2ew1w2xt(qw1w2ew1w2t+1)(qw1ew1t+1)(qw2ew2t+1).From (2.15) and (2.12), we can derive (2.16)2∫ℤpqxextdμ−1(x)∫ℤpew1xtqw1xdμ−1(x)=2∑l=0w1−1(−1)lqlelt=∑k=0∞(2∑l=0w1−1(−1)lqllk)tkk!=∑k=0∞2Sk,q(w1−1)tkk!.Let(2.17)M(w1,w2)(t,x)=∫ℤp∫ℤpqw1x1+w2x2e(w1x1+w2x2+w1w2x)tdμ−1(x1)dμ−1(x2)∫ℤpew1w2x3tqw1w2x3dμ−1(x3).By (2.15), (2.16), and (2.17), we see that(2.18)M(w1,w2)(t,x)=ew1w2xt(qw1w2ew1w2t+1)(qw1ew1t+1)(qw2ew2t+1). From (2.17) we derive(2.19)M(w1,w2)(t,x)=(12∫ℤpew1(x1+w2x)tqw1x1dμ−1(x1))(2∫ℤpew2x2tqw2x2dμ−1(x2)∫ℤpew1w2xtqw1w2xdμ−1(x)). By (2.16) and (2.19), we see that (2.20)M(w1,w2)(t,x)=(11+qw1∑i=0∞Hi(−q−w1,w2x)w1ii!ti)(∑l=0∞Sl,qw2(w1−1)w2ll!tl)=∑n=0∞(∑i=0n(ni)Hi(−q−w1,w2x)1+qw1Sn−i,qw2(w1−1)w1iw2n−i)tnn!.By the symmetry of p-adic invariant
integral on ℤp, we also see that(2.21)M(w1,w2)(t,x)=∑n=0∞(∑i=0n(ni)Hi(−q−w2,w1x)1+qw2Sn−i,qw1(w2−1)w2iw1n−i)tnn!,where Hn(−q−1,x) are the nth
Frobenius-Euler polynomials.By comparing the coefficients on the both sides of
(2.20) and (2.21), we obtain the following
theorem.Theorem 2.1.
Forw1,w2,n∈ℕ with n≡1(mod2), w1≡1(mod2), w2≡1(mod2), one has (2.22)∑i=0n(ni)Hi(−q−w1,w2x)1+qw1Sn−i,qw2(w1−1)w1iw2n−i=∑i=0n(ni)Hi(−q−w2,w1x)1+qw2Sn−i,qw1(w2−1)w2iw1n−i,where Hn(q,x) are  the nth
Frobenius-Euler polynomials.If we takew2=1 in Theorem 2.1,
then we have(2.23)Hn(−q−1,w1x)1+q=∑i=0n(ni)Hi(−q−w1,x)1+qw1Sn−i,q(w1−1)w1i.From (2.11) and (2.12), we derive(2.24)M(w1,w2)(t,x)=(ew1w2xt2∫ℤpew1x1tqw1x1dμ−1(x1))(2∫ℤpew2x2tqw2x2dμ−1(x2)∫ℤpew1w2xtqw1w2xdμ−1(x))=(ew1w2xt2∫ℤpew1x1tqw1x1dμ−1(x1))(2∑l=0w1−1(−1)lqw2lew2lt)=∑l=0w1−1(−1)lqw2l∫ℤpe(x1+w2x+(w2/w1)l)tw1qx1w1dμ−1(x1)=∑n=0∞(2∑l=0w1−1(−1)lHn(−q−w1,w2x+(w2/w1)l)1+qw1qw2l)tnn!.From the symmetry of M(w1,w2)(t,x), we note that (2.25)M(w1,w2)(t,x)=∑n=0∞(2∑l=0w2−1(−1)lHn(−q−w2,w1x+(w1/w2)l)1+qw2qw1l)tnn!.By comparing the coefficients on
the both sides of (2.24) and (2.25), we obtain the
following theorem.Theorem 2.2.
Letw1,w2(∈ℕ) be odd, and let n∈ℤ+ with n≡1(mod2). Then, one has(2.26)∑l=0w1−1(−1)lHn(−q−w1,w2x+(w2/w1)l)1+qw1qw2l=∑l=0w2−1(−1)lHn(−q−w2,w1x+(w1/w2)l)1+qw2qw1l.By settingw2=1 in Theorem 2.2,
we get the multiplication theorem for the Frobenius-Euler polynomials as
follows:(2.27)Hn(−q−1,w1x)1+q=∑l=0w1−1(−1)lqlHn(−q−w1,x+lw1).Remark 2.3.
By using the fermionicp-adic invariant q-integral on ℤp, the symmetric properties related to Frobenius-Euler
polynomials are studied in [17]. In this paper, we  have studied the symmetric
properties of Frobenius-Euler polynomials, which are different from the symmetric
properties  treated  in a previous paper [17]. To derive the symmetric properties
of Frobenius-Euler polynomials, we used the ordinary fermionic p-adic invariant
integrals on ℤp in this paper.



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