# Explicit Formulas Involvingq-Euler Numbers and Polynomials

Authors: Serkan Araci; Mehmet Acikgoz; Jong Jin Seo
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We deal withq-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q-Euler numbers and polynomials by using their generating function and derivative operator. Also, we derive relations between the q-Euler numbers and q-Bernoulli numbers via the p-adic q-integral in the p-adic integer ring.


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## Body

## 1. Preliminaries

Imagine thatp is a fixed odd prime number. Throughout this paper we use the following notations, where ℤp denotes the ring of p-adic rational integers, ℚ denotes the field of rational numbers, ℚp denotes the field of p-adic rational numbers, and ℂp denotes the completion of algebraic closure of ℚp. Let ℕ be the set of natural numbers and ℕ*=ℕ∪{0}.Thep-adic absolute value is defined by
(1.1)|p|p=1p.In this paper, we will assume that|q-1|p<1 as an indeterminate.[
x
]
q is a q-extension of x, which is defined by
(1.2)[x]q=1-qx1-q.
We note that limq→1[x]q=x (see [1–12]).We say thatf is a uniformly differentiable function at a point a∈ℤp, if the difference quotient
(1.3)Ff(x,y)=f(x)-f(y)x-y,
has a limit f′(a) as (x,y)→(a,a) and denote this by f∈UD(ℤp).LetUD(ℤp) be the set of uniformly differentiable function on ℤp. For f∈UD(ℤp), let us start with the expression
(1.4)1[pN]q∑0≤ξ<pN‍f(ξ)qξ=∑0≤ξ<pN‍f(ξ)μq(ξ+pNℤp),
which represents p-adic q-analogue of Riemann sums for f. The integral of f on ℤp will be defined as the limit (N→∞) of these sums, when it exists. The p-adic q-integral of function f∈UD(ℤp) is defined by Kim
(1.5)Iq(f)=∫ℤp‍f(ξ)dμq(ξ)=limN→∞1[pN]q∑ξ=0pN-1‍f(ξ)qξ.The bosonic integral is considered as a bosonic limitq→1,  I1(f)=limq→1Iq(f). Similarly, the fermionic p-adic integral on ℤp is introduced by Kim as follows:
(1.6)I-q(f)=limq→-qIq(f)=∫ℤp‍f(ξ)dμ-q(ξ)
(for more details, see [9–12]).In [6], the q-Euler polynomials with weight 0 are introduced as
(1.7)E~n,q(x)=∫ℤp‍(x+y)ndμ-q(y).From (1.7), we have
(1.8)E~n,q(x)=∑l=0n‍(nl)xlE~n-l,q,
where E~n,q(0)=E~n,q are called q-Euler numbers with weight 0. Then, q-Euler numbers are defined as
(1.9)q(E~q+1)n+E~n,q={[2]q,ifn=0,0,ifn≠0,
where the usual convention about replacing (E~q)n by E~n,q is used.Similarly, theq-Bernoulli polynomials and numbers with weight 0 are defined, respectively, as
(1.10)B~n,q(x)=limn→∞1[pn]q∑y=0pn-1‍(x+y)nqy=∫ℤp‍(x+y)ndμq(y),B~n,q=∫ℤp‍yndμq(y)
(for more information, see [4]).We, by using the Kim et al. method in [2], will investigate some interesting identities on the q-Euler numbers and polynomials arising from their generating function and derivative operator. Consequently, we derive some properties on the q-Euler numbers and polynomials and q-Bernoulli numbers and polynomials by using q-Volkenborn integral and fermionic p-adic q-integral on ℤp.


## 2. On theq-Euler Numbers and Polynomials

Let us consider Kim’sq-Euler polynomials as follows:
(2.1)Fxq=Fxq(t)=[2]qqet+1ext=∑n=0∞‍E~n,q(x)tnn!.Herex is a fixed parameter. Thus, by expression of (2.1), we can readily see the following:
(2.2)qetFxq+Fxq=[2]qext.Last from equality, taking derivative operatorD as D=d/dt on the both sides of (2.2). Then, we easily see that
(2.3)qet(D+I)kFxq+DkFxq=[2]qxkext,
where k∈ℕ* and I is identity operator. By multiplying e-t on both sides of (2.3), we get
(2.4)q(D+I)kFxq+e-tDkFxq=[2]qxke(x-1)t.Let us take derivative operatorDm(m∈ℕ) on both sides of (2.4). Then we get
(2.5)qetDm(D+I)kFxq+Dk(D-I)mFxq=[2]qxk(x-1)mext.LetG[0] (not G(0)) be the constant term in a Laurent series of G(t). Then, from (2.5), we get
(2.6)∑j=0k‍(kj)(qetDk+m-jFxq(t))[0]+∑j=0m‍(mj)(-1)j(Dk+m-jFxq(t))[0]=[2]qxk(x-1)m.By (2.1), we see
(2.7)(DNFxq(t))[0]=E~N,q(x),(etDNFxq(t))[0]=E~N,q(x).By expressions of (2.6) and (2.7), we see that
(2.8)∑j=0max{k,m}‍[q(kj)+(-1)j(mj)]E~k+m-j,q(x)=[2]qxk(x-1)m.From (2.1), we note that
(2.9)ddx(E~n,q(x))=n∑l=0n-1‍(n-1l)E~l,qxn-1-l=nE~n-1,q(x).By (2.9), we easily see
(2.10)∫01‍E~n,q(x)dx=E~n+1,q(1)-E~n+1,qn+1=-[2]q-1n+1E~n+1,q.Now, let us consider definition of integral from0 to 1 in (2.8), then we have
(2.11)-[2]q-1∑j=0max{k,m}‍[q(kj)+(-1)j(mj)]E~k+m-j+1,qk+m-j+1=[2]q(-1)mB(k+1,m+1)=[2]q(-1)mΓ(k+1)Γ(m+1)Γ(k+m+2),
where B(m,n) is beta function which is defined by
(2.12)B(m,n)=∫01‍xm-1(1-x)n-1dx=Γ(m)Γ(n)Γ(m+n),m>0,n>0.As a result, we obtain the following theorem.Theorem 2.1.
Forn∈ℕ, one has
(2.13)∑j=1max{k,m}‍[q(kj)+(-1)j(mj)]E~k+m-j+1,qk+m-j+1=q(-1)m+1(k+m+1)(k+mk)-[2]qE~k+m+1,qk+m+1.Substitutingm=k+1 into Theorem 2.1, we readily get
(2.14)∑j=1k+1‍[q(kj)+(-1)j(k+1j)]E~2k+2-j,q2k+2-j=q(-1)k(2k+2)(2k+1k)-[2]qE~2k+2,q2k+2.By (2.1), it follows that
(2.15)∑j=0max{k,m}‍(k+m-j)[q(kj)+(-1)j(mj)]E~k+m-j-1,q(x)=[2]qxk-1(x-1)m-1((k+m)x-k).Letm=k in (2.1), we see that
(2.16)∑j=0k‍[q(kj)+(-1)j(kj)]E~2k-j,q(x)=[2]qxk(x-1)k.Last from equality, we discover the following:(2.17)[2]q∑j=0[k/2]‍(k2j)E~2k-2j,q(x)+(q-1)∑j=0[k/2]‍(k2j+1)E~2k-2j-1,q(x)=[2]qxk(x-1)k.Here[·] is Gauss’ symbol. Then, taking integral from 0to 1 in both sides of last equality, we get
(2.18)-[2]q-1[2]q∑j=0[k/2]‍(k2j)E~2k-2j+1,q2k-2j+1+[2]q-1(1-q)∑j=0[k/2]‍(k2j+1)E~2k-2j,q2k-2j=[2]q(-1)kB(k+1,k+1)=[2]q(-1)k(2k+1)(2kk).Consequently, we derive the following theorem.Theorem 2.2.
The following identity(2.19)[2]q∑j=0[k/2]‍(k2j)E~2k-2j+1,q2k-2j+1+(q-1)∑j=0[k/2]‍(k2j+1)E~2k-2j,q2k-2j=q(-1)k+1(2k+1)(2kk)
is true.In view of (2.1) and (2.17), we discover the following applications:
(2.20)=∑j=0k+1‍[q(kj)+(-1)j(k+1j)]E~2k+1-j,q(x)=[2]qE~2k+1,q(x)+∑j=1[(k+1)/2]‍[q(k2j)+(k2j)+(k2j-1)]E~2k+1-2j,q(x)+∑j=0[(k+1)/2]‍[q(k2j+1)-(k2j+1)-(k2j)]E~2k-2j,q(x)=-[∑j=0[k/2]‍(k2j)E~2k-2j,q(x)+q-11+q∑j=0[k/2]‍(k2j+1)E~2k-2j+1(x)]+[2]q∑j=0[k/2]‍(k2j)E~2k+1-2j,q(x)+∑j=1[k/2]‍(k2j-1)E~2k+1-2j,q(x)+(q-1)∑j=0[k/2]‍(k2j+1)E~2k-2j,q(x)+q-11+q∑j=0[k/2]‍(k2j+1)E~2k-2j+1(x).By expressions (2.17) and (2.20), we have the following theorem.Theorem 2.3.
Fork∈ℕ, one has
(2.21)[2]q∑j=0[k/2]‍(k2j)E~2k+1-2j,q(x)+∑j=1[k/2]‍(k2j-1)E~2k+1-2j,q(x)+(q-1)∑j=0[k/2]‍(k2j+1)[E~2k-2j,q(x)+11+qE~2k-2j+1(x)]=xk(x-1)k([2]qx-q).


## 3.p-adic Integral on ℤp Associated with Kim’s q-Euler Polynomials

In this section, we consider Kim’sq-Euler polynomials by means of p-adic q-integral on ℤp. Now we start with the following assertion.Letm,k∈ℕ. Then by (2.8),
(3.1)I1=[2]q∫ℤp‍xk(x-1)mdμ-q(x)=[2]q∑l=0m‍(ml)(-1)m-l∫ℤp‍xl+kdμ-q(x)=[2]q∑l=0m‍(ml)(-1)m-lE~l+k,q.On the other hand, in right hand side of (2.8),
(3.2)I2=∑j=0max{k,m}‍[q(kj)+(-1)j(mj)]∑l=0k+m-j‍(k+m-jl)E~k+m-j-l,q∫ℤp‍xldμ-q(x)=∑j=0max{k,m}‍[q(kj)+(-1)j(mj)]∑l=0k+m-j‍(k+m-jl)E~k+m-j-l,qE~l,q.EquatingI1 and I2, we get the following theorem.Theorem 3.1.
Form,k∈ℕ, one has
(3.3)∑j=0max{k,m}‍[q(kj)+(-1)j(mj)]∑l=0k+m-j‍(k+m-jl)E~k+m-j-l,qE~l,q=[2]q∑l=0m‍(ml)(-1)m-lE~l+k,q.Let us take fermionicp-adic q-integral on ℤp in left hand side of (2.21), we get
(3.4)I3=∫ℤp‍xk(x-1)k([2]qx-q)dμ-q(x)=[2]q∑l=0k‍(kl)(-1)k-l∫ℤp‍xk+l+1dμ-q(x)-q∑l=0k‍(kl)(-1)k-l∫ℤp‍xk+ldμ-q(x)=[2]q∑l=0k‍(kl)(-1)k-lE~k+l+1,q-q∑l=0k‍(kl)(-1)k-lE~k+l,q.In other words, we consider right hand side of (2.21) as follows:
(3.5)I4=[2]q∑j=0[k/2]‍(k2j)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,q∫ℤp‍xldμ-q(x)+∑j=1[k/2]‍(k2j-1)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,q∫ℤp‍xldμ-q(x)+∑j=0[k/2]‍(k2j+1)[(q-1)∑j=02k-2j(2k-2jl)E~2k-2j-l,q∫ℤp‍xldμ-q(x)+q-11+q∑l=02k-2j+1(2k-2j+1l)E~2k-2j-l+1∫ℤp‍xldμ-q(x)]=[2]q∑j=0[k/2]‍(k2j)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,qE~l,q+∑j=1[k/2]‍(k2j-1)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,qE~l,q+∑j=0[k/2]‍(k2j+1)[(q-1)∑j=02k-2j(2k-2jl)E~2k-2j-l,qE~l,q+q-11+q∑l=02k-2j+1(2k-2j+1l)E~2k-2j-l+1E~l,q].EquatingI3 and I4, we get the following theorem.Theorem 3.2.
Fork∈ℕ, one has
(3.6)∑l=0k‍(kl)(-1)k-l{[2]qE~k+l+1,q-qE~k+l,q}=[2]q∑j=0[k/2]‍(k2j)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,qE~l,q+∑j=1[k/2]‍(k2j-1)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,qE~l,q+∑j=0[k/2]‍(k2j+1){(q-1)∑j=02k-2j(2k-2jl)E~2k-2j-l,qE~l,q+q-11+q∑l=02k-2j+1(2k-2j+1l)E~2k-2j-l+1E~l,q}.Now, we consider (2.8) and (2.1) by means of q-Volkenborn integral. Then, by (2.8), we see
(3.7)[2]q∫ℤp‍xk(x-1)mdμq(x)=[2]q∑l=0m‍(ml)(-1)m-l∫ℤp‍xl+kdμq(x)=[2]q∑l=0m‍(ml)(-1)m-lB~l+k,q.On the other hand,(3.8)∑j=0max{k,m}‍[q(kj)+(-1)j(mj)]∑l=0k+m-j‍(k+m-jl)E~k+m-j-l,q∫ℤp‍xldμq(x)=∑j=0max{k,m}‍[q(kj)+(-1)j(mj)]∑l=0k+m-j‍(k+m-jl)E~k+m-j-l,qB~l,q.Therefore, we get the following theorem.Theorem 3.3.
Form,k∈ℕ, one has
(3.9)[2]q∑l=0m‍(ml)(-1)m-lB~l+k,q=∑j=0max{k,m}‍[q(kj)+(-1)j(mj)]∑l=0k+m-j‍(k+m-jl)E~k+m-j-l,qB~l,q.By using fermionicp-adic q-integral on ℤp in left hand side of (2.21), we get
(3.10)I5=[2]q∫ℤp‍xk(x-1)k([2]x-q)dμq(x)=[2]q∑l=0k‍(kl)(-1)k-l∫ℤp‍xk+l+1dμq(x)-q∑l=0k‍(kl)(-1)k-l∫ℤp‍xk+ldμq(x)=[2]q∑l=0k‍(kl)(-1)k-lB~k+l+1,q-q∑l=0k‍(kl)(-1)k-lB~k+l,q.Also, we consider right hand side of (2.21) as follows:
(3.11)I6=[2]q∑j=0[k/2]‍(k2j)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,q∫ℤp‍xldμq(x)+∑j=1[k/2]‍(k2j-1)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,q∫ℤp‍xldμq(x)+∑j=0[k/2]‍(k2j+1)[(q-1)∑j=02k-2j(2k-2jl)E~2k-2j-l,q∫ℤp‍xldμq(x)+q-11+q∑l=02k-2j+1(2k-2j+1l)E~2k-2j-l+1∫ℤp‍xldμq(x)]=[2]q∑j=0[k/2]‍(k2j)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,qB~l,q+∑j=1[k/2]‍(k2j-1)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,qB~l,q+∑j=0[k/2]‍(k2j+1)[(q-1)∑j=02k-2j(2k-2jl)E~2k-2j-l,qB~l,q+q-11+q∑l=02k-2j+1(2k-2j+1l)E~2k-2j-l+1B~l,q].EquatingI5 and I6, we get the following corollary.Corollary 3.4.
Fork∈ℕ, one gets
(3.12)∑l=0k‍(kl)(-1)k-l{[2]qB~k+l+1,q-qB~k+l,q}=[2]q∑j=0[k/2]‍(k2j)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,qB~l,q+∑j=1[k/2]‍(k2j-1)∑l=02k-2j+1‍(2k-2j+1l)E~2k+1-2j-l,qB~l,q+∑j=0[k/2]‍(k2j+1){(q-1)∑j=02k-2j(2k-2jl)E~2k-2j-l,qB~l,q+q-11+q∑l=02k-2j+1(2k-2j+1l)E~2k-2j-l+1B~l,q}.



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