# A Note on Some Identities of Frobenius-Euler Numbers and Polynomials

Authors: J. Choi; D. S. Kim; T. Kim; Y. H. Kim
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

The purpose of this paper is to give some identities on the Frobenius-Euler numbers and polynomials by using the fermionicp-adic q-integral equation
on ℤp.


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## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this paper, ℤp, ℚp, and ℂp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of ℚp, respectively. Let ℕ be the set of natural numbers and ℤ+=ℕ∪{0}. The p-adic absolute value on ℂp is normalized so that |p|p=1/p. Assume that q∈ℂp with |1-q|p<1.Letf be a continuous function on ℤp. Then the fermionic p-adic q-integral on ℤp for f is defined by Kim as follows:(1.1)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1+q1+qpN∑x=0pN-1f(x)(-q)x,(see[1]).
From (1.1), we note that(1.2)qnI-q(fn)=(-1)nI-q(f)+(1+q)∑l=0n-1(-1)n-1-lf(l)ql,
where n∈ℕ and fn(x)=f(x+n) (see [1]). The ordinary Euler polynomials En(x) are defined by(1.3)2et+1ext=eE(x)t=∑n=0∞En(x)tnn!,
with the usual convention about replacing En(x) by En(x) (see [1–10]). In the special case, x=0, En(0)=En is called the nth Euler number.As the extension of (1.3), the Frobenius-Euler polynomials are defined by(1.4)1-qet-qext=∑n=0∞Hn(q,x)tnn!,(see[2]).
In the special case, x=0, Hn(q,0)=Hn(q) is called the nth Frobenius-Euler number.By (1.3) and (1.4), we easily get Hn(-1,x)=En(x).From (1.4), we note that(1.5)Hn(q,x)=∑l=0n(nl)Hl(q)xn-l=(H(q)+x)n,(see[2]),
with the usual convention about replacing H(q)n by Hn(q).In this paper, we consider the fermionicp-adic q-integral on ℤp for the Frobenius-Euler numbers and polynomials. From these p-adic q-integrals on ℤp, we derive some new and interesting identities on the Frobenius-Euler numbers and polynomials.


## 2. Identities on the Frobenius-Euler Numbers

From (1.2) and (1.4), we can derive the following:(2.1)∫Zpe(x+y)tdμ-q(y)=1+q-1et+q-1ext=∑n=0∞Hn(-q-1,x)tnn!.
Thus, by (2.1), we get Witt's formula for Hn(-q-1,x) as follows:(2.2)∫Zp(x+y)ndμ-q(y)=Hn(-q-1,x),n∈Z+.
By (1.5) and (2.1), we get(2.3)(H(-q-1)+1)n+q-1Hn(-q-1)={1+q-1,ifn=0,0,ifn>0,
with the usual convention about replacing H(-q-1)n by Hn(-q-1).From (1.5) and (2.3), we note that(2.4)H0(-q-1)=1,Hn(-q-1,1)+q-1Hn(-q-1)=0,ifn>0.
By (2.1) and (2.2), we get(2.5)∫Zp(y+1-x)ndμ-q(y)=(-1)n∫Zp(y+x)ndμ-q-1(y).
Therefore, by (2.5), we obtain the following lemma.Lemma 2.1.
Forn∈ℤ+, one has
(2.6)Hn(-q-1,1-x)=(-1)nHn(-q,x).From (2.3), we can derive the following:(2.7)q2Hn(-q-1,2)-q2-q=q2∑l=0n(nl)(H(-q-1)+1)l-q2-q=q∑l=1n(nl)q(H(-q-1)+1)l-q=-q∑l=0n(nl)Hl(-q-1)=-(1+q)δ0,n+Hn(-q-1),
where δk,n is the Kronecker symbol.Therefore, by (2.7), we obtain the following theorem.Theorem 2.2.
Forn∈ℤ+, one has
(2.8)Hn(-q-1,2)=1+q-1-q-2(1+q)δ0,n+q-2Hn(-q-1).First we consider the fermionicp-adic q-integral on ℤp for the nth Frobenius-Euler polynomials as follows:(2.9)I1=∫ZpHn(-q-1,x)dμ-q(x)=∑l=0n(nl)Hl(-q-1)∫Zpxn-ldμ-q(x)=∑l=0n(nl)Hl(-q-1)Hn-l(-q-1),wheren∈Z+.
On the other hand, by (2.5) and Lemma 2.1, we get(2.10)I1=∫ZpHn(-q-1,x)dμ-q(x)=(-1)n∫ZpHn(-q,1-x)dμ-q(x)=(-1)n∑l=0n(nl)Hn-l(-q)∫Zp(1-x)ldμ-q(x)=∑l=0n(nl)(-1)n-lHn-l(-q)∫Zp(x-1)ldμ-q(x)=∑l=0n(nl)(-1)n-lHn-l(-q)Hl(-q-1,-1).
From Lemma 2.1, Theorem 2.2, and (2.10), we note that(2.11)I1=∑l=0n(nl)(-1)n-lHn-l(-q)Hl(-q-1,-1)=∑l=0n(nl)(-1)nHn-l(-q)Hl(-q,2)=∑l=0n(nl)(-1)nHn-l(-q){1+q-q2(1+q-1)δ0,l+q2Hl(-q)}=(-1)n(1+q)((1+q)δ0,n-qHn(-q))-Hn(-q)(q+q2)(-1)n+(-1)nq2∑l=0n(nl)Hn-l(-q)Hl(-q).
Therefore, by (2.10) and (2.11), we obtain the following theorem.Theorem 2.3.
Forn∈ℤ+, one has
(2.12)∑l=0n(nl)Hl(-q-1)Hn-l(-q-1)=(-1)n(1+q)((1+q)δ0,n-2qHn(-q))+(-1)nq2∑l=0n(nl)Hn-l(-q)Hl(-q).
In particular, for n∈ℕ, one has
(2.13)∑l=0n(nl)Hl(-q-1)Hn-l(-q-1)=2(-1)n+1q(1+q)Hn(-q)+(-1)nq2∑l=0n(nl)Hn-l(-q)Hl(-q).Let us consider the following fermionicp-adic q-integral on ℤp for the product of Bernoulli and Frobenius-Euler polynomials as follows:(2.14)I2=∫ZpBm(x)Hn(-q-1,x)dμ-q(x)=∑k=0m∑l=0n(mk)(nl)Bm-kHn-l(-q-1)∫Zpxk+ldμ-q(x)=∑k=0m∑l=0n(mk)(nl)Bm-kHn-l(-q-1)Hk+l(-q-1).
It is known that Bn(x)=(-1)nBn(1-x).On the other hand, by Lemma2.1, we get(2.15)I2=(-1)m+n∫ZpBm(1-x)Hn(-q,1-x)dμ-q(x)=(-1)m+n∑k=0m∑l=0n(mk)(nl)Bm-kHn-l(-q)∫Zp(1-x)k+ldμ-q(x)=(-1)m+n∑k=0m∑l=0n(mk)(nl)Bm-kHn-l(-q){(1+q)-q2(1+q-1)δ0,k+l+q2Hk+l(-q)}=(-1)m+n(1+q)Bm(1)Hn(-q,1)-(q2+q)(-1)m+nBmHn(-q)+q2(-1)m+n∑k=0m∑l=0n(mk)(nl)Bm-kHn-l(-q)Hk+l(-q).
Therefore, by (2.14) and (2.15), we obtain the following theorem.Theorem 2.4.
Form,n∈ℤ+, one has
(2.16)∑k=0m∑l=0n(mk)(nl)Bm-kHn-l(-q-1)Hk+l(-q-1)=(-1)m+n(1+q)Bm(1)Hn(-q,1)-(q2+q)(-1)m+nBmHn(-q)+q2(-1)m+n∑k=0m∑l=0n(mk)(nl)Bm-kHn-l(-q)Hk+l(-q).
In particular, for m∈ℕ-{1}, n∈ℕ, one has
(2.17)∑k=0m∑l=0n(mk)(nl)Bm-kHn-l(-q-1)Hk+l(-q-1)=2(-1)m+n+1(q2+q)BmHn(-q)+q2(-1)m+n∑k=0m∑l=0n(mk)(nl)Bm-kHn-l(-q)Hk+l(-q).It is known thatxn=(1/(n+1))∑l=0n(n+1l)Bl(x). Let us consider the following fermionic p-adic q-integral on ℤp:(2.18)∫Zpxndμ-q(x)=1n+1∑l=0n(n+1l)∫ZpBl(x)dμ-q(x).=1n+1∑l=0n(n+1l)∑k=0l(lk)Bl-k∫Zpxkdμ-q(x)=1n+1∑l=0n(n+1l)∑k=0l(lk)Bl-kHk(-q-1).
Therefore by (2.18), we obtain the following theorem.Theorem 2.5.
Forn∈ℤ+, one has
(2.19)Hn(-q-1)=1n+1∑l=0n(n+1l)∑k=0l(lk)Bl-kHk(-q-1).From (1.3), we can derive the following:(2.20)xn=En(x)+12∑l=0n-1(nl)El(x).
Let us take the fermionic p-adic q-integral on ℤp in (2.20) as follows:(2.21)∫Zpxndμ-q(x)=∫ZpEn(x)dμ-q(x)+12∑l=0n-1(nl)∫ZpEl(x)dμ-q(x)=∑l=0n(nl)En-lHl(-q-1)+12∑l=0n-1(nl)∑k=0l(lk)El-kHk(-q-1).
Therefore, by (2.21), we obtain the following theorem.Theorem 2.6.
Forn∈ℕ, one has
(2.22)Hn(-q-1)=∑l=0n(nl)En-lHl(-q-1)+12∑l=0n-1(nl)∑k=0l(lk)El-kHk(-q-1).By Theorems2.5 and 2.6, we obtain the following corollary.Corollary 2.7.
Forn∈ℕ, one has
(2.23)1n+1∑l=0n(n+1l)∑k=0l(lk)Bl-kHk(-q-1)=∑l=0n(nl)En-lHl(-q-1)+12∑l=0n-1(nl)∑k=0l(lk)El-kHk(-q-1).By (1.3), we easily get En(x)=(-1)nEn(1-x).Thus, we have(2.24)∫Zpxndμ-q(x)=(-1)n∫ZpEn(1-x)dμ-q(x)+12∑l=0n-1(nl)(-1)l∫ZpEl(1-x)dμ-q(x)=(-1)n∑l=0n(nl)En-l∫Zp(1-x)ldμ-q(x)+12∑l=0n-1(nl)(-1)l∑k=0l(lk)El-k∫Zp(1-x)kdμ-q(x)=∑l=0n(nl)En-l(-1)n-lHl(-q-1,-1)+12∑l=0n-1(nl)∑k=0l(lk)El-k(-1)l-kHk(-q-1,-1)=∑l=0n(nl)En-l(-1)nHl(-q,2)+12∑l=0n-1(nl)∑k=0l(lk)El-k(-1)lHk(-q,2).
Therefore, by (2.24), we obtain the following theorem.Theorem 2.8.
Forn∈ℕ, one has
(2.25)Hn(-q-1)=∑l=0n(nl)En-l(-1)nHl(-q,2)+12∑l=0n-1(nl)∑k=0l(lk)El-k(-1)lHk(-q,2).



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