# Identities Involvingq-Bernoulli and q-Euler Numbers

Authors: D. S. Kim; T. Kim; J. Choi; Y. H. Kim
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We give some identities on theq-Bernoulli and q-Euler numbers by using p-adic integral equations 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 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 norm |·|p is normally defined by |p|p=1/p.As it is well known, the Euler polynomials are defined by(1.1)2et+1ext=eE(x)t=∑n=0∞En(x)tnn!,
with the usual convention about replacing En(x) by En(x)  (see [1–14]). In the special case, x=0, En(0)=En is called the nth Euler number.The ordinary Bernoulli polynomials are also defined by(1.2)tet-1ext=eB(x)t=∑n=0∞Bn(x)tnn!,
with the usual convention about replacing Bn(x) by Bn(x)h (see [1–14]). In the special case, x=0, Bn(0)=Bn is called the nth Bernoulli number.LetUD(ℤp) be the space of uniformly differentiable functions on ℤp. For f∈UD(ℤp), the bosonic p-adic integral on ℤp is defined by(1.3)I(f)=∫Zpf(x)dμ(x)=lim⁡N→∞∑x=0pN-1f(x)μ(x+pNZp)=lim⁡N→∞1pN∑x=0pN-1f(x),
(see [1, 7]). Let f1 be the translation of f with f1(x)=f(x+1). From (1.3) we have(1.4)I(f1)-I(f)=f′(0),
(see [1, 7]).The fermionicp-adic integral on ℤp is also defined by T. Kim as follows:(1.5)I-1(f)=∫Zpf(x)dμ-1(x)=lim⁡N→∞∑x=0pN-1f(x)μ-1(x+pNZp)=lim⁡N→∞∑x=0pN-1f(x)(-1)x,
(see [6, 15, 16]). By (1.5), we get(1.6)I-1(f1)+I-1(f)=2f(0),
(see [6, 8]).Letf(x)=ext∈UD(ℤp) with |t|p<|p|p1/(p-1) and |x|p≤1. From (1.4), we have(1.7)∫Zpe(x+y)tdμ(y)=tet-1ext=∑n=0∞Bn(x)tnn!.
Thus, by (1.7), we see that(1.8)∫Zp(x+y)ndμ(y)=Bn(x),n∈Z+.
By (1.8), we get(1.9)Bn(x)=∑l=0n(nl)xn-l∫Zpyldμ(y)=∑l=0n(nl)xn-lBl.
As an indeterminate, let us assume that q∈ℂp with |1-q|p<1.From (1.1) and (1.6), we note that the q-Euler polynomials are given by(1.10)∫Zpqye(x+y)tdμ-1(y)=2qet+1ext=∑n=0∞En,q(x)tnn!,
where En,q(x) are called the nth q-Euler polynomials (see [1, 3, 6, 8]).Thus, by (1.10), we get(1.11)En,q(x)=∫Zpqy(x+y)ndμ-1(y),n∈Z+.
In the special case, x=0, En,q(0)=En,q is called the nth q-Euler number (see [8, 9]). By (1.10) and (1.11), we get the recurrence formula for the q-Euler numbers as follows:(1.12)E0,q=2[2]q,q(Eq+1)n+En,q=2δ0,n,
with the usual convention about replacing Eqn by En,q. Here [x]q=(1-qx)/(1-q) is the q-number of x and δk,n is the Kronecker symbol (see [10, 11]).From (1.2), (1.7), and (1.8), we have(1.13)B0=1,(B+1)n-Bn=δ1,n,
with the usual convention about replacing Bn by Bn (see [1, 3, 14]).From (1.11), we easily see that(1.14)En,q(x)=∑l=0n(nl)xn-lEl,q,
(see [14]).In this paper we give some interesting properties ofp-adic integrals on ℤp associated with the q-Bernoulli and the q-Euler numbers. From those properties, we derive new identities involving the q-Bernoulli and the q-Euler numbers arising from p-adic integrals of polynomial identities.


## 2. Identities onq-Bernoulli and q-Euler Numbers

LetCpn be the cyclic group of order pn with Cpn={ζ∈ℂp∣ζpn=1}. Then Tp is defined by the direct limit as Tp=limn→∞Cpn=Cp∞. In this section, we assume that q(≠1)∈Tp, then |q-1|p<1. From (1.4), we can derive the following equation (2.1):(2.1)∫Zpqye(x+y)tdμ(y)=tqet-1ext=∑n=0∞Bn,q(x)tnn!,
where Bn,q(x) is called the nth q-Bernoulli polynomial (see [7]). In the special case, x=0, Bn,q(0)=Bn,q is called the nth q-Bernoulli number.By (2.1), we get(2.2)B0,q=0,q(Bq+1)n-Bn,q=δ1,n,
with the usual convention about replacing Bqn by Bn,q (see [7, 14]).From (1.3), we have(2.3)∫Zpf(-y)dμ(y)=∫Zpf(y+1)dμ(y).
By (2.3), we get(2.4)q∫Zp(1-x+y)nqydμ(y)=(-1)n∫Zpq-y(x+y)ndμ(y).
From (2.1) and (2.4), we have(2.5)qBn,q(1-x)=(-1)nBn,q-1(x),n∈Z+.
By using (1.4), we see that(2.6)q∫Zp(1+x+y)nqydμ(y)=∫Zpqy(x+y)ndμ(y)+nxn-1.
Thus, by (2.1) and (2.6), we get(2.7)qBn,q(1+x)=Bn,q(x)+nxn-1,n∈Z+.
Therefore, by (2.5) and (2.7), we obtain the following theorem.Theorem 2.1.
Forn∈ℤ+, one has
(2.8)(-1)nBn,q-1(-x)=qBn,q(1+x)=Bn,q(x)+nxn-1.From (1.5) and (1.6), we note that(2.9)∫Zp(1-x+y)nqydμ-1(y)=(-1)n∫Zp(x+y)nq-(y+1)dμ-1(y).
Therefore, by (1.11) and (2.9), we obtain the following theorem.Theorem 2.2.
Forn∈ℤ+, one has
(2.10)qEn,q(1-x)=(-1)nEn,q-1(x).By (1.6), we get(2.11)q∫Zpqy(x+1+y)ndμ-1(y)+∫Zpqy(x+y)ndμ-1(y)=2xn.
Thus, by (1.11) and (2.11), we have(2.12)qEn,q(x+1)=-En,q(x)+2xn,n∈Z+.
Therefore, by Theorem 2.2 and (2.12), we obtain the following theorem.Theorem 2.3.
Forn∈ℤ+, one has
(2.13)(-1)nEn,q-1(-x)=qEn,q(1+x)=-En,q(x)+2xn.By using thep-adic integrals on ℤp, we have the following equation (2.14):(2.14)∫Zp∫Zpq(x+y)e(x+y)tdμ(x)dμ-1(y)=∫Zpqxextdμ(x)∫Zpqyeytdμ-1(y)=(tqet-1)(2qet+1)=2tq2e2t-1=∫Zpq2xe2xtdμ(x).
By (2.14), we get(2.15)∫Zp∫Zpq(x+y)(x+y)ndμ(x)dμ-1(y)=2n∫Zpq2xxndμ(x).
It is not difficult to show that(2.16)∫Zp∫Zpq(x+y)(x+y)ndμ(x)dμ-1(y)=∑l=0n(nl)∫Zp∫Zpqxxn-lqyyldμ(x)dμ-1(y).
Therefore, by (2.15) and (2.16), we obtain the following theorem.Theorem 2.4.
Forn∈ℤ+, one has
(2.17)∑l=0n(nl)Bn-l,qEl,q=2nBn,q2.By (2.5), (2.7), (2.12), Theorems 2.1, and 2.3, we get(2.18)qBn,q(x)=(-1)nBn,q-1(1-x)=Bn,q(x-1)+n(x-1)n-1,(2.19)qEn,q(x)=(-1)nEn,q-1(1-x)=-En,q(x-1)+2(x-1)n.
From (2.1), we have(2.20)Bn,q(x)=∫Zpqy(x+y)ndμ(y)=∑l=0n(nl)Bl,qxn-l,n∈Z+.
Let(2.21)I1=q∫ZpqxBn,q(x)dμ(x)=q∑l=0n(nl)Bn-l,qBl,q.
From (2.18), (2.20), and (2.21), we note that(2.22)I1=(-1)n∑l=0n(nl)Bn-l,q∫Zpqx(x-1)ldμ(x)+n∫Zpqx(x-1)n-1dμ(x)=(-1)n∑l=0n(nl)Bn-l,qBl,q(-1)+nBn-1,q(-1)=(-1)n∑l=0n(nl)Bl,qBn-l,q(-1)+nBn-1,q(-1).
By (2.5), we get(2.23)qBn,q(-1)=(-1)nBn,q-1(2),n∈Z+.
By (2.3), we easily see that(2.24)q2Bn,q(2)=nq+qBn,q(1)=nq+Bn,q+δ1,n,
where δ1,n is the Kronecker symbol.Thus, by (2.23) and (2.24), we get(2.25)Bn,q(-1)=(-1)n(n+qBn,q-1+qδ1,n).
By (2.22) and (2.25), we get(2.26)I1=(-1)n∑l=0n(nl)Bl,q(-1)n-l(n-l+qBn-l,q-1+qδ1,n-l)+nBn-1,q(-1)=n∑l=0n-1(n-1l)Bl,q(-1)l+q∑l=0n(nl)(-1)lBl,qBn-l,q-1+q(-1)n-1nBn-1,q+nBn-1,q(-1)=q∑l=0n(nl)(-1)lBl,qBn-l,q-1+q(-1)n-1nBn-1,q+(1+(-1)n-1)nBn-1,q(-1).
Therefore, by (2.21) and (2.26), we obtain the following theorem.Theorem 2.5.
Forn∈ℤ+, one has
(2.27)∑l=02n(2nl)B2n-l,qBl,q-∑l=02n(2nl)(-1)lBl,qB2n-l,q-1=-2nB2n-1,q.Let us consider the following integral:(2.28)I2=∫Zpqx+1En,q(x)dμ-1(x)=q∑l=0n(nl)En-l,qEl,q.
By (2.19), we get(2.29)I2=-∫ZpqxEn,q(x-1)dμ-1(x)+2∫Zpqx(x-1)ndμ-1(x)=-∑l=0n(nl)El,qEn-l,q(-1)+2En,q(-1)=-∑l=0n(nl)En-l,qEl,q(-1)+2En,q(-1).
From Theorem 2.2, we note that(2.30)qEn,q(-1)=(-1)nEn,q-1(2),n∈Z+.
By (1.12), we get(2.31)q2En,q(2)=2q+En,q-2δ0,n.
Thus, by (2.30) and (2.31), we get(2.32)En,q(-1)=(-1)n(2+qEn,q-1-2qδ0,n).
From (2.29) and (2.32), we note that(2.33)I2=-∑l=0n(nl)El,q(-1)n-l(2+qEn-l,q-1-2qδ0,n-l)+2(-1)n(2+qEn,q-1-2qδ0,n)=-q∑l=0n(nl)(-1)n-lEl,qEn-l,q-1+2qEn,q.
Therefore, by (2.28) and (2.33), we obtain the following theorem.Theorem 2.6.
Forn∈ℤ+, one has
(2.34)∑l=0n(nl)En-l,qEl,q+∑l=0n(nl)(-1)n-lEl,qEn-l,q-1=2En,q.Now we consider the fermionicp-adic integral on ℤp for the nth q-Euler polynomials as follows:(2.35)I3=∫ZpqxEn,q(x)dμ-1(x)=∑l=0n(nl)El,q∫Zpqxxn-ldμ-1(x)=∑l=0n(nl)El,qEn-l,q,n∈Z+.
On the other hand, by Theorem 2.2, we get(2.36)I3=(-1)nq-1∫ZpEn,q-1(1-x)qxdμ-1(x)=(-1)nq-1∑l=0n(nl)En-l,q-1∫Zpqx(1-x)ldμ-1(x)=q-1∑l=0n(nl)En-l,q-1(-1)n-lEl,q(-1).
From (2.32) and (2.36), we note that(2.37)I3=(-1)nq-1∑l=0n(nl)En-l,q-1(2+qEl,q-1-2qδ0,l)=2(-1)nq-1En,q-1(1)+(-1)n∑l=0n(nl)En-l,q-1El,q-1-2(-1)nEn,q-1=-2(-1)nEn,q-1+4(-1)nδ0,n+(-1)n∑l=0n(nl)En-l,q-1El,q-1-2(-1)nEn,q-1.
Therefore, by (2.35) and (2.37), we obtain the following theorem.Theorem 2.7.
Forn∈ℤ+, one has
(2.38)∑l=02n+1(2n+1l)El,qE2n+1-l,q+∑l=02n+1(2n+1l)El,q-1E2n+1-l,q-1=4E2n+1,q-1.From (2.1) and (2.7), we note that(2.39)xn=qn+1∑l=0n(n+1l)Bl,q(x)+q-1n+1Bn+1,q(x).
Let us consider the following fermionic p-adic integral on ℤp:(2.40)∫Zpxnqxdμ-1(x)=qn+1∑l=0n(n+1l)∫ZpqxBl,q(x)dμ-1(x)+q-1n+1∫ZpqxBn+1,q(x)dμ-1(x)=qn+1∑l=0n(n+1l)∑k=0l(lk)Bl-k,qEk,q+q-1n+1∑l=0n+1(n+1l)Bn+1-l,qEl,q.
Therefore, by (2.40), we obtain the following theorem.Theorem 2.8.
Forn∈ℤ+, one has
(2.41)En,q=qn+1∑l=0n(n+1l)∑k=0l(lk)Bl-k,qEk,q+q-1n+1∑l=0n+1(n+1l)Bn+1-l,qEl,q.From (1.10) and (2.12), we note that(2.42)xn=[2]q2En,q(x)+q2∑l=0n-1(nl)El,q(x).
Thus, by (2.42), we get(2.43)∫Zpqxxndμ-1(x)=[2]q2∫ZpEn,q(x)qxdμ-1(x)+q2∑l=0n-1(nl)∫ZpqxEl,q(x)dμ-1(x)=[2]q2∑l=0n(nl)En-l,qEl,q+q2∑l=0n-1(nl)∑k=0l(lk)El-k,qEk,q.
Thus, by (2.43), we have(2.44)En,q=[2]q2∑l=0n(nl)En-l,qEl,q+q2∑l=0n-1(nl)∑k=0l(lk)El-k,qEk,q.



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