# q-Analogues of the Bernoulli and Genocchi Polynomials and the Srivastava-Pintér Addition Theorems

Authors: N. I. Mahmudov
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli and Genocchi
polynomials based on theq-integers. The q-analogues of well-known formulas are derived. The q-analogue
of the Srivastava-Pintér addition theorem is obtained.


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## Body

## 1. Introduction

Throughout this paper, we always make use of the following notation:ℕ denotes the set of natural numbers, ℕ0 denotes the set of nonnegative integers, ℝ denotes the set of real numbers, and ℂ denotes the set of complex numbers.Theq-shifted factorial is defined by
(1.1)(a;q)0=1,(a;q)n=∏j=0n-1‍(1-qja),n∈ℕ,(a;q)∞=∏j=0∞‍(1-qja),|q|<1,a∈ℂ.
The q-numbers and q-numbers factorial is defined by
(1.2)[a]q=1-qa1-q(q≠1);[0]q!=1;[n]q!=[1]q[2]q⋯[n]qn∈ℕ,a∈ℂ,
respectively. The q-polynomial coefficient is defined by
(1.3)[nk]q=(q;q)n(q;q)n-k(q;q)k.
The q-analogue of the function (x+y)n is defined by
(1.4)(x+y)qn:=∑k=0n[nk]qq(1/2)k(k-1)xn-kyk,n∈ℕ0.
In the standard approach to the q-calculus two exponential function are used:
(1.5)eq(z)=∑n=0∞‍zn[n]q!=∏k=0∞‍1(1-(1-q)qkz),0<|q|<1,|z|<1|1-q|,Eq(z)=∑n=0∞‍q(1/2)n(n-1)zn[n]q!=∏k=0∞‍(1+(1-q)qkz),0<|q|<1,z∈ℂ.
From this form we easily see that eq(z)Eq(-z)=1. Moreover,
(1.6)Dqeq(z)=eq(z),DqEq(z)=Eq(qz),
where Dq is defined by
(1.7)Dqf(z):=f(qz)-f(z)qz-z.
The previous q-standard notation can be found in [1].Carlitz has introduced theq-Bernoulli numbers and polynomials in [2]. Srivastava and Pintér proved some relations and theorems between the Bernoulli polynomials and Euler polynomials in [3]. They also gave some generalizations of these polynomials. In [4–6], Kim et al. investigated some properties of the q-Euler polynomials and Genocchi polynomials. They gave some recurrence relations. In [7], Cenkci et al. gave the q-extension of Genocchi numbers in a different manner. In [5], Kim gave a new concept for the q-Genocchi numbers and polynomials. In [8], Simsek et al. investigated the q-Genocchi zeta function and l-function by using generating functions and Mellin transformation. We also recall the definitions of the q-Bernoulli and the q-Genocchi polynomials of higher order (see [2, 9–12]):
(1.8)(-t)α∑n=0∞‍([α]q)n[n]q!qn+xet[n+x]q=∑n=0∞‍Bn,q(α)(x)tnn!,(2t)α∑n=0∞‍([α]q)n[n]q!(-1)nqn+xet[n+x]q=∑n=0∞‍Gn,q(α)(x)tnn!.
We propose the following definitions. We define the q-Bernoulli and the q-Genocchi polynomials of higher order in two variables x and y, using two q-exponential functions, which helps us easily prove some properties of these polynomials and q-analogue of the Srivastava and Pintér addition theorem.Definition 1.1.
Theq-Bernoulli numbers 𝔅n,q(α) and polynomials 𝔅n,q(α)(x,y) in x,y of order α are defined by means of the generating function functions:
(1.9)(teq(t)-1)α=∑n=0∞‍𝔅n,q(α)tn[n]q!,|t|<2π,(teq(t)-1)αeq(tx)Eq(ty)=∑n=0∞‍𝔅n,q(α)(x,y)tn[n]q!,|t|<2π.Definition 1.2.
Theq-Genocchi numbers 𝔊n,q(α) and polynomials 𝔊n,q(α)(x,y) in x,y are defined by means of the generating functions:
(1.10)(2teq(t)+1)α=∑n=0∞‍𝔊n,q(α)tn[n]q!,|t|<π,(2teq(t)+1)αeq(tx)Eq(ty)=∑n=0∞‍𝔊n,q(α)(x,y)tn[n]q!,|t|<π.It is obvious that(1.11)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),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).
Here Bn(α)(x) and En(α)(x) denote the classical Bernoulli, and Genocchi polynomials of order α are defined by
(1.12)(tet-1)αetx=∑n=0∞‍Bn(α)(x)tnn!,(2et+1)αetx=∑n=0∞‍Gn(α)(x)tnn!.The aim of the present paper is to obtain some results for theq-Genocchi polynomials (properties of the q-Bernoulli polynomials are studied in [13]). The q-analogues of well-known results, for example, Srivastava and Pintér [3], can be derived from these q-identities. It should be mentioned that probabilistic proofs the Srivastava-Pintér addition theorems were given recently in [14]. The formulas involving the q-Stirling numbers of the second kind, q-Bernoulli polynomials and q-Bernstein polynomials, are also given. Furthermore some special cases are also considered.The following elementary properties of theq-Genocchi polynomials 𝔈n,q(α)(x,y) of order α are readily derived from Definition 1.2. We choose to omit the details involved.Property 1.3.
Special values of theq-Genocchi polynomials of order α:
(1.13)𝔈n,q(0)(x,0)=xn,𝔈n,q(0)(0,y)=q(1/2)n(n-1)yn.Property 1.4.
Summation formulas for theq-Genocchi polynomials of order α:
(1.14)𝔈n,q(α)(x,y)=∑k=0n‍[nk]q𝔈k,q(α)(x+y)qn-k,𝔈n,q(α)(x,y)=∑k=0n‍[nk]q𝔈n-k,q(α-1)𝔈k,q(x,y),𝔊n,q(α)(x,y)=∑k=0n‍[nk]qq(n-k)(n-k-1)/2𝔊k,q(α)(x,0)yn-k=∑k=0n‍[nk]q𝔊k,q(α)(0,y)xn-k,𝔊n,q(α)(x,0)=∑k=0n‍[nk]q𝔊k,q(α)xn-k,𝔊n,q(α)(0,y)=∑k=0n‍[nk]qq(n-k)(n-k-1)/2𝔊k,q(α)yn-k.Property 1.5.
Difference equations:(1.15)𝔊n,q(α)(1,y)+𝔊n,q(α)(0,y)=2[n]q𝔊n-1,q(α-1)(0,y),𝔊n,q(α)(x,0)+𝔊n,q(α)(x,-1)=2[n]q𝔊n-1,q(α-1)(x,-1).Property 1.6.
Differential relations:(1.16)Dq,x𝔊n,q(α)(x,y)=[n]q𝔊n-1,q(α)(x,y),Dq,y𝔊n,q(α)(x,y)=[n]q𝔊n-1,q(α)(x,qy).Property 1.7.
Addition theorem of the argument:(1.17)𝔈n,q(α+β)(x,y)=∑k=0n‍[nk]q𝔈n-k,q(α)(x,0)𝔈k,q(β)(0,y).Property 1.8.
Recurrence relationships:(1.18)𝔊n,q(α)(1m,y)+∑k=0n‍[nk]q(1m-1)qn-k𝔊k,q(α)(0,y)=2[n]q∑k=0n-1‍[n-1k]q(1m-1)qn-1-k𝔊k,q(α-1)(0,y).


## 2. Explicit Relationship between theq-Genocchi and the q-Bernoulli Polynomials

In this section we prove an interesting relationship between theq-Genocchi polynomials 𝔊n,q(α)(x,y) of order α and the q-Bernoulli polynomials. Here some q-analogues of known results will be given. We also obtain new formulas and their some special cases in the following.Theorem 2.1.
Forn∈ℕ0, the following relationship
(2.1)𝔊n,q(α)(x,y)=∑k=0n‍1mn-k-1[k+1]q[2[k+1]q∑j=0k‍[kj]q1mk-j𝔊j,q(α-1)(x,-1)-∑j=0k+1‍[k+1j]q1mk+1-j𝔊j,q(α)(x,-1)-𝔊k+1,q(α)(x,0)]𝔅n-k,q(0,my)
holds true between the q-Genocchi and the q-Bernoulli polynomials.Proof.
Using the following identity:(2.2)(2teq(t)+1)αeq(tx)Eq(ty)=(2teq(t)+1)αeq(tx)·eq(t/m)-1t·teq(t/m)-1·Eq(tmmy),
we have
(2.3)∑n=0∞‍𝔊n,q(α)(x,y)tn[n]q!=mt∑n=0∞‍(∑k=0n‍[nk]q1mn-k𝔊k,q(α)(x,0)-𝔊n,q(α)(x,0))tn[n]q!∑n=0∞‍𝔅n,q(0,my)tnmn[n]q!=∑n=1∞‍(∑k=0n‍[nk]q1mn-1-k𝔊k,q(α)(x,0)-m𝔊n,q(α)(x,0)∑k=0n‍[nk]q1mn-1-k)tn-1[n]q!∑n=0∞‍𝔅n,q(0,my)tnmn[n]q!=∑n=0∞‍(∑k=0n+1‍[n+1k]qmk𝔊k,q(α)(x,0)-mn+1𝔊n+1,q(α)(x,0)∑k=0n+1‍[n+1k]q)tnmn[n+1]q!∑n=0∞‍𝔅n,q(0,my)tnmn[n]q!=∑n=0∞‍∑k=0n‍1mn[k+1]q(∑j=0k+1‍[k+1j]qmj𝔊j,q(α)(x,0)-mk+1𝔊k+1,q(α)(x,0)∑j=0k+1‍[k+1j]qmj𝔊j,q(α))𝔅n-k,q(0,my)tn[n]q!.
It remains to use Property 1.8.Since𝔊n,q(α)(x,y) is not symmetric with respect to x and y, we can prove a different form of the previously mentioned theorem. It should be stressed out that Theorems 2.1 and 2.2 coincide in the limiting case when q→1-.Theorem 2.2.
Forn∈ℕ0, the following relationship
(2.4)𝔊n,q(α)(x,y)=∑k=0n‍[nk]q1mn-k-1[k+1]q[2[k+1]q∑j=0k‍[kj]q(1m-1)qk-j𝔊j,q(α-1)(0,y)-∑j=0k+1‍[k+1j]q(1m-1)qk+1-j𝔊j,q(α)(0,y)-𝔊k+1,q(0,y)]×𝔅n-k,q(mx,0)
holds true between the q-Genocchi and the q-Bernoulli polynomials.Proof.
The proof is based on the following identity:(2.5)(2teq(t)+1)αeq(tx)Eq(ty)=(2teq(t)+1)αEq(ty)·eq(t/m)-1t·teq(t/m)-1·eq(tmmx).Next we discuss some special cases of Theorems2.1 and 2.2. By noting that
(2.6)𝔊j,q(0)(0,y)=q(1/2)j(j-1)yj,𝔊j,q(0)(x,-1)=(x-1)qj,
we deduce from Theorems 2.1 and 2.2 Corollary 2.3 below.Corollary 2.3.
Forn∈ℕ0, m∈ℕ the following relationship
(2.7)𝔊n,q(x,y)=∑k=0n‍[nk]q1mn-k-1[k+1]q[2[k+1]q∑j=0k‍[kj]q(1m-1)qk-jq(1/2)j(j-1)yj-∑j=0k+1‍[k+1j]q(1m-1)qk+1-j𝔊j,q(0,y)-𝔊k+1,q(0,y)]×𝔅n-k,q(mx,0),𝔊n,q(x,y)=∑k=0n‍[nk]q1mn-k-1[k+1]q[2[k+1]q∑j=0k‍[kj]q1mk-j(x-1)qj-∑j=0k+1‍[k+1j]q1mk+1-j𝔊j,q(x,-1)-𝔊k+1,q(x,0)]×𝔅n-k,q(0,my)
holds true between the q-Bernoulli polynomials and q-Euler polynomials.Corollary 2.4.
Forn∈ℕ0, m∈ℕ the following relationship holds true:
(2.8)Gn(x+y)=∑k=0n‍(nk)2k+1((k+1)yk-Gk+1,q(y))Bn-k(x),(2.9)Gn(x+y)=∑k=0n‍(nk)1mn-k-1(k+1)[2(k+1)Gk(y+1m-1)-Gk+1(y+1m-1)-Gk+1(y)]Bn-k,q(mx)
between the classical Genocchi polynomials and the classical Bernoulli polynomials.Note that the formula (2.9) is new for the classical polynomials.In terms of theq-Genocchi numbers 𝔊k,q(α), by setting y=0 in Theorem 2.1, we obtain the following explicit relationship between the q-Genocchi polynomials 𝔊k,q(α) of order α and the q-Bernoulli polynomials.Corollary 2.5.
The following relationship holds true:(2.10)𝔊n,q(α)(x,0)=∑k=0n‍[nk]q1mn-k-1[k+1]q[2[k+1]q∑j=0k‍[kj]q(1m-1)qk-j𝔊j,q(α-1)-∑j=0k+1‍[k+1j]q(1m-1)qk+1-j𝔊j,q(α)-𝔊k+1,q(α)]𝔅n-k,q(mx,0).Corollary 2.6.
Forn∈ℕ0 the following relationship holds true:
(2.11)𝔊n,q(x,y)=∑k=0n‍[nk]q2[k+1]q[[k+1]qq(1/2)k(k-1)yk-𝔊k+1,q(0,y)]𝔅n-k,q(x,0).Corollary 2.7.
Forn∈ℕ0 the following relationship holds true:
(2.12)𝔊n,q(x,0)=-∑k=0n‍[nk]q2[k+1]q𝔊k+1,q𝔅n-k,q(x,0),𝔊n,q=-∑k=0n‍[nk]q2[k+1]q𝔊k+1,q𝔅n-k,q.



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*Source: 169348-2012-09-06.xml*