# Some Identities on the Generalizedq-Bernoulli, q-Euler, and q-Genocchi Polynomials

Authors: Daeyeoul Kim; Burak Kurt; Veli Kurt
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Mahmudov (2012, 2013) introduced and investigated someq-extensions of the q-Bernoulli polynomials ℬn,qαx,y of order α, the q-Euler polynomials ℰn,qαx,y of order α, and the q-Genocchi polynomials 𝒢n,qαx,y of order α. In this paper, we give some identities
for ℬn,qαx,y, 𝒢n,qαx,y, and ℰn,qαx,y and the recurrence relations between these polynomials. This is an analogous result to the q-extension of the Srivastava-Pintér addition theorem in Mahmudov (2013).


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## Body

## 1. Introduction, Definitions, and Notations

Throughout this paper, we always make use of the following notation:ℕ denotes the set of natural numbers and ℂ denotes the set of complex numbers. The q-numbers and q-factorial are defined by
(1)[a]q=1-qa1-q,q≠1,[n]q!=[n]q[n-1]q⋯[2]q[1]q,
respectively, where [0]q!=1, n∈ℕ, and a∈ℂ. The q-binomial coefficient is defined by
(2)[nk]q=(q:q)n(q:q)n-k(q:q)k,
where (q:q)n=(1-q)⋯(1-qn). The q-analogue of the function (x+y)qn is defined by
(3)(x+y)qn=∑k=0n[nk]qq(k(k-1))/2xn-kyk.
The q-binomial formula is known as
(4)(n;q)a=(1-a)qn=∏j=0n-1(1-qja)=∑k=0n[nk]qq(k(k-1))/2(-1)kak.Theq-exponential functions are given by
(5)eq(z)=∑n=0∞zn[n]q!=∏k=0∞1(1-(1-q)qkz),0<|q|<1,∏k=0∞11(1-(1-q)qkz),|z|<1|1-q|,Eq(z)=∑n=0∞q(n(n-1))/2zn[n]q!=∏k=0∞(1+(1-q)qkz),0<|q|<1,z∈ℂ.
From these forms, we easily see that eq(z)Eq(-z)=1. Moreover, Dqeq(z)=eq(z) and DqEq(z)=Eq(qz), where Dq is defined by
(6)Dqf(z)=f(qz)-f(z)qz-z,0<|q|<1,0≠z∈ℂ.The previousq-standard notation can be found in [1, 2]. Carlitz firstly extended the classical Bernoulli numbers and polynomials and Euler numbers and polynomials [3, 4]. There are numerous recent investigations on this subject by many other authors. Among them are Cenkci et al. [5, 6], Choi et al. [1], Cheon [7], Kim [8], Kurt [9], Kurt [10], Luo and Srivastava [11–13], Srivastava et al. [14, 15], Natalini and Bernardini [16], Tremblay et al. [17, 18], Gaboury and Kurt [19], Mahmudov [2, 20, 21], Araci et al. [22], and Kupershmidt [23].Mahmudov defined and studied the properties of the following generalizedq-Bernoulli polynomials ℬn,q(α)(x,y) of order α and  q-Euler polynomials ℰn,q(α)(x,y) of order α as follows [2].Letq∈ℂ, α∈ℕ, and 0<|q|<1. The q-Bernoulli numbers ℬn,q(α) and polynomials ℬn,q(α)(x,y) in x and  y of order α are defined by means of the generating functions:
(7)∑n=0∞ℬn,q(α)tn[n]q!=(teq(t)-1)α,|t|<2π,(8)∑n=0∞ℬn,q(α)(x,y)tn[n]q!=(teq(t)-1)αeq(tx)Eq(ty),=(teq(t)-1)αe(tx)|t|<2π.
The q-Euler numbers ℰn,q(α) and polynomials ℰn,q(α)(x,y) in x and y of order α are defined by means of the generating functions:
(9)∑n=0∞ℰn,q(α)tn[n]q!=(2eq(t)+1)α,|t|<π,(10)∑n=0∞ℰn,q(α)(x,y)tn[n]q!=(2eq(t)+1)αeq(tx)Eq(ty),|t|<π.Theq-Genocchi numbers 𝒢n,q(α) and polynomials 𝒢n,q(α)(x,y) in x and y of order α are defined by means of the generating functions:
(11)∑n=0∞𝒢n,q(α)tn[n]q!=(2teq(t)+1)α,|t|<π,(12)∑n=0∞𝒢n,q(α)(x,y)tn[n]q!=(2teq(t)+1)αeq(tx)Eq(ty),|t|<π.The familiarq-Stirling numbers S2,q(n,k) of the second kind are defined by
(13)(eq(t)-1)k[k]q!=∑n=0∞S2,q(n,k)tn[n]q!.It is obvious that(14)ℬn,q(1)(x,y)∶=ℬn,q(x,y),ℰn,q(1)(x,y)∶=ℰn,q(x,y),𝒢n,q(1)(x,y)∶=𝒢n,q(x,y),ℬn,q(0,0)∶=ℬn,q,ℰn,q(0,0)∶=ℰn,q,𝒢n,q(0,0)∶=𝒢n,q,ℬn,q(α)=ℬn,q(α)(0,0),limq→1-ℬn,q(α)(x,y)=ℬn(α)(x+y),limq→1-ℬn,q(α)=ℬn(α),ℰn,q(α)=ℰn,q(α)(0,0),limq→1-ℰn,q(α)(x,y)=ℰn(α)(x+y),limq→1-ℰn,q(α)=ℰn(α),𝒢n,q(α)=𝒢n,q(α)(0,0),limq→1-𝒢n,q(α)(x,y)=𝒢n(α)(x+y),limq→1-𝒢n,q(α)=𝒢n(α).
From (8) and (10), it is easy to check that
(15)ℬn,q(α)(x,y)=∑k=0n[nk]qℬn-k,q(x,0)ℬk,q(α-1)(0,y),ℰn,q(α)(x,y)=∑k=0n[nk]qℰn-k,q(α-1)(x,0)ℰk,q(0,y).In this work, we give some identities for theq-Bernoulli polynomials. Also, we give some relations between the q-Bernoulli polynomials and q-Euler polynomials and the q-Genocchi polynomials and q-Bernoulli polynomials. Furthermore, we give a different form of the analogue of the Srivastava-Pintér addition theorem. More precisely, we prove the following theorems.Theorem 1.
There are the following relations between theq-Bernoulli polynomials and q-Stirling numbers of the second kind:
(16)ℬn,q(α)(x,y)=[k]q![n]q![n+k]q!×∑l=0n+k[n+kl]qℬl,q(α+k)(x,y)×S2,q(n+k-l,k),(17)∑k=0n[nk]qℬn-k,q(α)(x,y)[α]q!S2,q(k,α)=∑l=0n-α[n-αl]q[n]q![n-α]q!xn-α-lylq(l2),
where q∈ℂ, α,n∈ℕ, and 0<|q|<1.Theorem 2.
Theq-Stirling numbers of the second kind satisfy the following relations:
(18)ℰn,q(α)(x,y)=∑j=0∞(-αj)12j[j]q!×∑p=0n[np]qS2,q(n-p,j)×∑l=0p[pl]qxp-lylq(l2),(19)ℬn,q(α)=[α]q!∑j=0∞(-αj)×∑k=0j(jk)[k]q!S2,q(n+k,k)[n+k]q![k]q!(-1)j-k,ℬn,q(-α)(x,y)=[α]q!∑m=0n+α[n+αm]qS2,q(m,α)×(x+y)qn+α-m[n]q![n+α]q!,
where q∈ℂ, α,n∈ℕ, and 0<|q|<1.Theorem 3.
The generalizedq-Euler polynomials satisfy the following relation:
(20)∑k=0n[nk]qℰk,q(x,y)=2(x+y)qn-ℰn,q(x,y),
where q∈ℂ, α,n∈ℕ, and 0<|q|<1.Theorem 4.
The polynomialsBn,q(x,y) and 𝒢n,q(x,y) satisfy the following difference relationships:
(21)ℬn,q(x,y)=∑n+1l=0l≠n[n+1l]q1[n+1]q𝒢l,q(x,y)ℬn+1-l,q,(22)𝒢n,q(x,y)=-2∑nl=0l≠n[nl]q1[l+1]q𝒢l+1,qℬn-l,q(x,y),
where q∈ℂ, α,n∈ℕ, and 0<|q|<1.Theorem 5.
There is the following relation between the generalizedq-Euler polynomials and generalized q-Bernoulli polynomials:
(23)ℰn,q(α)(x,y)={∑s=0n+1[n+1s]q∑l=0s[sl]qℬs-l,q(mx,0)-∑l=0n+1[n+1l]qℬn+1-l,q(mx,0)}×m[n+1]q!ℰl,q(α)(0,y)ml-n-1,
where q∈ℂ, α,n∈ℕ, and 0<|q|<1.


## 2. Proof of the Theorems

Lemma 6.
The generalizedq-Bernoulli polynomials, q-Euler polynomials, and q-Genocchi polynomials satisfy the following relations:
(24)∑k=0n[nk]qℬk,q(α)(x,y)ℬn-k,q(-α)=(x+y)qn,∑k=0n[nk]qℬk,q(α)(0,y)ℬn-k,q(-α)=q(n(n-1))/2yn,ℬn,q(α)(x,y)=∑l=0n[nl]qℬn-l,q(α)(0,y)×∑k=0l[lk]qℰk,q(α)(x,0)ℰl-k,q(-α),ℰn,q(α)(x,y)=∑l=0n[nl]qℰn-l,q(α)(0,y)×∑k=0l[lk]qℬk,q(α)(x,0)ℬl-k,q(-α),∑k=0n[nk]q𝒢k,q(x,y)+𝒢n,q(x,y)=2[n]q(x+y)qn-1,𝒢n,q(α-β)(x,y)=∑k=0n[nk]q𝒢k,q(α)(x,0)𝒢n-k,q(-β)(0,y).Proof.
The proof of this lemma can be found from (7)–(12).Proof of Theorem1.
By (8) and (13) we have
(25)∑n=0∞ℬn,q(α)(x,y)tn[n]q!=(teq(t)-1)αeq(tx)Eq(ty)×[k]q!(eq(t)-1)k(eq(t)-1)k[k]q!=[k]q!tα(eq(t)-1)α+keq(tx)Eq(ty)×∑m=0∞S2,q(m,k)tm[m]q!=[k]q!t-k∑n=0∞∑l=0n[nl]qℬl,q(α+k)=[k]q!t∑n=0∞∑l=0n×(x,y)S2,q(n-l,k)tn[n]q!=[k]q!∑n=0∞∑l=0n[nl]qℬl,q(α+k)=!∑n=0∞∑l=0n111×(x,y)S2,q(n-l,k)tn-k[n]q!=[k]q!∑n=-k∞∑l=0n+k[n+kl]qℬl,q(α+k)=!∑n=0∞∑l=0n11111×(x,y)S2,q(n+k-l,k)tn-k[n]q!.
Equating the coefficients of (tn/[n]q!), we obtain (16).
Similarly, we have (17).Proof of Theorem2.
Combining (10) and (13), we obtain
(26)(2eq(t)+1)α=(1+eq(t)-12)(-α)=∑j=0∞(-αj)(eq(t)-12)(j),∑n=0∞ℰn,q(α)(x,y)tn[n]q!=∑j=0∞(-αj)(eq(t)-12)(j)eq(tx)Eq(ty)=∑j=0∞(-αj)12j[j]q!∑n=0∞S2,q(n,j)tn[n]q!×∑n=0∞∑k=0n[nk]qxn-kykq(k2)tn[n]q!=∑j=0∞(-αj)12j×∑n=0∞∑p=0n[np]q[j]q!S2,q(n-p,j)×∑l=0p[pl]qxp-lylq(l2)tn[n]q!.
Comparing the coefficients of(tn/[n]q!), we find (18). Similarly, we have (19).Proof of Theorem3.
It is obvious that(27)-2(eq(t)+1)eq(t)=2(eq(t)+1)-2eq(t).
We write it as
(28)-2eq(t)+1eq(tx)Eq(ty)eq(t)=2eq(t)+1eq(tx)Eq(ty)-2eq(t)eq(tx)Eq(ty),-2eq(t)+1eq(tx)Eq(ty)=2eq(t)+1eq(tx)Eq(ty)-2eq(tx)Eq(ty)-∑n=0∞ℰn,q(x,y)tn[n]q!=∑n=0∞ℰn,q(x,y)tn[n]q!×∑n=0∞tn[n]q!-2∑n=0∞(x+y)qntn[n]q!.
Using the Cauchy product and comparing the coefficients of (tn/[n]q!), we have
(29)∑k=0n[nk]qℰk,q(x,y)=2(x+y)qn-ℰk,q(x,y).Finally, we consider the interesting relationships between theq-Bernoulli polynomials and q-Genocchi polynomials and the q-Euler polynomials and q-Bernoulli polynomials. These relations are q-analogues to the  Srivastava-Pintér addition theorems.Proof of Theorem4.
It follows immediately that(30)∑n=0∞ℬn,q(x,y)tn[n]q!=122teq(tx)Eq(ty)eq(t)+1+1t(teq(t)-1)2teq(t)+1eq(tx)Eq(ty)=12∑n=0∞𝒢n,q(x,y)tn[n]q!+1t×∑n=0∞ℬn,qtn[n]q!∑n=0∞𝒢n,q(x,y)tn[n]q!=12∑n=0∞𝒢n,q(x,y)tn[n]q!+∑n=0∞∑l=0n[nl]q1[n]q𝒢l,q(x,y)+∑n=0∞∑l=0n×ℬn-l,qtn-1[n-1]q!=12∑n=0∞𝒢n,q(x,y)tn[n]q!+∑n=0∞(∑l=0n+1[n+1l]q-12𝒢n,q(x,y)+∑l=0n+1[n+1l]q1[n+1]q×𝒢l,q(x,y)ℬn+1-l,q+∑l=0n+1[n+1l]q)tn[n]q!=∑n=0∞(∑n+1l=0l≠n[n+1l]q1[n+1]q×𝒢l,q(x,y)ℬn+1-l,q∑n+1l=0l≠n)tn[n]q!.Equating the coefficients of (tn/[n]q!), we have (21).
In a similar fashion, (12) yields
(31)∑n=0∞𝒢n,q(x,y)tn[n]q!=1t(2teq(t)+1(eq(t)-1))(teq(tx)Eq(ty)eq(t)-1)=1t(2t-22teq(t)+1)(teq(t)-1eq(tx)Eq(ty))=1t(2t-2∑n=0∞𝒢n,qtn[n]q!)(∑n=0∞ℬn,q(x,y)tn[n]q!)=1t(-2∑l=0∞1[l+1]q!𝒢l+1,qtl+1[l]q!)(∑n=0∞ℬn,q(x,y)tn[n]q!)=∑n=1∞(-2∑l=0l≠nn[nl]q𝒢l+1,q[l+1]qℬn-l,q(x,y))tn[n]q!.
Comparing the coefficients of (tn/[n]q!), we have (22).Proof of Theorem5.
By (10), we write
(32)∑n=0∞ℰn,q(α)(x,y)tn[n]q!=(2eq(t)+1)α×Eq(ty)eq(t/m)-1(t/m)(t/m)eq(t/m)-1eq((t/m)mx)=mt{∑n=0∞ℰn,q(α)(x,y)tn[n]q!×∑n=0∞ℬn,q(mx,0)tnmn[n]q!×∑n=0∞tnmn[n]q!-∑n=0∞ℰn,q(α)(0,y)tn[n]q!×∑n=0∞ℬn,q(mx,0)tnmn[n]q!}=m∑n=-1∞1[n+1]q×{∑s=0n+1[n+1s]q∑l=0s[sl]qℬs-l,q(mx,0)-∑l=0n+1[n+1l]qℬn+1-l,q(mx,0)}×m[n+1]q!ℰl,q(α)(0,y)ml-n-1tn[n]q!.
By equating the coefficients of (tn/[n]q!), we get the theorem.Remark 7.
There are many different relationships which are analogues to the Srivastava-Pintér addition theorems at these polynomials.



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