# On the Symmetries of theq-Bernoulli Polynomials

Authors: Taekyun Kim
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

Kupershmidt and Tuenter have introduced reflection symmetries
for theq-Bernoulli numbers and the Bernoulli polynomials in (2005), (2001),
respectively. However, they have not dealt with congruence properties for these
numbers entirely. Kupershmidt gave a quantization of the reflection symmetry
for the classical Bernoulli polynomials. Tuenter derived a symmetry of power
sum polynomials and the classical Bernoulli numbers. In this paper, we study
the new symmetries of the q-Bernoulli numbers and polynomials, which are
different from Kupershmidt's and Tuenter's results. By using our symmetries
for the q-Bernoulli polynomials, we can obtain some interesting relationships
between q-Bernoulli numbers and polynomials.


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## Body

## 1. Introduction

Letp be a fixed prime. Throughout this paper, ℤp, ℚp, ℂ,
and ℂp will, respectively, denote the ring of p-adic rational integer, the field of p-adic rational numbers, the complex number
field, and the completion of algebraic closure of ℚp.
The p-adic absolute value in ℂp is normalized so that |p|p<p−1. Let q be variously considered as an indeterminate, a
complex number q∈ℂ,
or a p-adic number q∈ℂp.
If q∈ℂp,
we assume that |1−q|p<1. We say that f is uniformly differentiable function at a
point a∈ℤp,
and we denote this property by f∈UD(ℤp) if the difference quotients, (1.1)Ff:ℤp×ℤp→ℤpbyFf(x,y)=f(x)−f(y)x−y, have a limit l=f′(a) as (x,y)→(a,a).
The p-adic invariant integral on ℤp is defined as (1.2)I(f)=∫ℤpf(x)dx=limN→∞1pN∑x=0pN−1f(x) [1–22]. From this integral, we derive
several further interesting properties of symmetry for the q-Bernoulli numbers and polynomials in this
paper. Kupershmidt [14] and Tuenter [20] have introduced reflection symmetries
for the q-Bernoulli numbers and the Bernoulli
polynomials. However, they have not dealt with congruence properties for these
numbers entirely. Kupershmidt gave a quantization of the reflection symmetry
for the classical Bernoulli polynomials. Tuenter derived a symmetry of power
sum polynomials and the classical Bernoulli numbers. In this paper, we study
the new symmetries of the q-Bernoulli numbers and polynomials, which are
different from Kupershmidt's and Tuenter's results. By using our symmetries for
the q-Bernoulli polynomials, we can obtain some
interesting relationships between q-Bernoulli numbers and polynomials.


## 2. On the Symmetries of theq-Bernoulli Polynomials

Forf∈UD(ℤp),
the p-adic invariant integral on ℤp is defined as (2.1)I(f)=∫ℤpf(x)dx=limN→∞1pN∑x=0pN−1f(x). Let f1(x) be a translation with f1(x)=f(x+1).
Then, we have (2.2)I(f1)=I(f)+f′(0). From (2.2), we can also
derive (2.3)I(fn)=I(f)+∑i=0n−1f′(i),f′(i)=df(i)dx. Let f(x)=qxetx,
then we have (2.4)∫ℤpqxetxdx=t+logqqet−1. It is known that the q-Bernoulli polynomials are defined
as (2.5)t+logqqet−1ext=∑n=0∞Bn,q(x)tnn! [17, 19]. Now we define an integral
representation for the q-extension of Bernoulli numbers as
follows: (2.6)∫ℤpqxetxdx=logq+tqet−1=∑n=0∞Bn,qtnn!.From (2.3), (2.4), and (2.6), we can derive  (2.7)∫ℤpqy(x+y)ndy=Bn,q(x),∫ℤpqxxndx=Bn,q. By (2.3), we easily see
that (2.8)1logq+t(∫ℤpqn+xe(n+x)tdx−∫ℤpqxextdx)=qnent−1t+logq∫ℤpqxextdx=qnent−1qet−1=∑i=0n−1qieit=∑k=0∞(∑i=0n−1ikqi)tkk!.In (2.2), it is not difficult to show
that (2.9)1logq+t(∫ℤpqn+xe(n+x)tdx−∫ℤpqxextdx)=n∫ℤpextqxdx∫ℤpenxtqnxdx. For each integer k≥0,
let (2.10)Sk,q(n)=0k+1kq+2kq2+⋯+qnnk.From (2.8) and (2.9), we derive (2.11)1logq+t(∫ℤpqn+xe(n+x)tdx−∫ℤpqxextdx)=n∫ℤpextqxdx∫ℤpenxtqnxdx=∑k=0∞Sk,q(n−1)tkk!.From (2.11), we note that (2.12)Bk,q(n)−Bk,q=kSk−1,q(n−1)+logqSk,q(n−1),wherek,n∈ℕ. Let w1,w2∈ℕ,
then we have (2.13)∬ℤpe(w1x1+w2x2)tqw1x1+w2x2dx1dx2∫ℤpew1w2xtqw1w2xdx=(t+logq)qw1w2ew1w2t−1(qw1ew1t−1)(qw2ew2t−1). By (2.11), we see
that (2.14)w1∫ℤpextqxdx∫ℤpqw1xew1xtdx=∑l=0∞(∑k=0w1−1klqk)tll!=∑l=0∞Sl,q(w1−1)tll!.Let(2.15)T(w1,w2;x,t)=∬ℤpqw1x1+w2x2e(w1x1+w2x2+w1w2x)tdx1dx2∫ℤpew1w2x3tqw1w2x3dx3, then we have (2.16)T(w1,w2;x,t)=(t+logq)ew1w2xt(qw1w2ew1w2t−1)(qw1ew1t−1)(qw2ew2t−1).From (2.15) we derive (2.17)T(w1,w2;x,t)=(1w1∫ℤpew1(x1+w2x)tqw1x1dx1)(w1∫ℤpew2x2tqw2x2dx2∫ℤpew1w2xtqw1w2xdx).By (2.5), (2.14), and (2.17), we see that (2.18)T(w1,w2;x,t)=1w1(∑i=0∞Bi,qw1(w2x)w1itii!)(∑l=0∞Sl,qw2(w1−1)w2ltll!)=∑n=0∞(∑i=0n(ni)Bi,qw1(w2x)Sn−i,qw2(w1−1)w1i−1w2n−i)tnn!. By the symmetry of p-adic invariant integral on ℤp,
we also see that (2.19)T(w1,w2;x,t)=(1w2∫ℤpew2(x2+w1x)tqw2x2dx2)(w2∫ℤpew1x1tqw1x1dx1∫ℤpew1w2xtqw1w2xdx)=∑n=0∞(∑i=0n(ni)Bi,qw2(w1x)Sn−i,qw1(w2−1)w2i−1w1n−i)tnn!. By comparing the coefficients tn/n! on the both sides of (2.18) and (2.19), we obtain
the following theorem.Theorem 2.1.
For allw1,w2(∈ℕ),
we have (2.20)∑i=0n(ni)Bi,qw1(w2x)Sn−i,qw2(w1−1)w1i−1w2n−i=∑i=0n(ni)Bi,qw2(w1x)Sn−i,qw1(w2−1)w2i−1w1n−i, where (ni) is the binomial coefficient.If we takew2=1 in Theorem 2.1, then we have (2.21)Bn,q(w1x)=∑i=0n(ni)Bi,qw1(x)Sn−i,q(w1−1)w1i−1.Therefore, we obtain the following corollary.Corollary 2.2.
Forn≥0,
we have (2.22)Bn,q(w1x)=∑i=0n(ni)Bi,qw1(x)Sn−i,q(w1−1)w1i−1.By (2.17), (2.18), and (2.19), we also see that (2.23)T(w1,w2;x,t)=(ew1w2xtw1∫ℤpew1x1tqw1x1dx1)(w1∫ℤpew2x2tqw2x2dx2∫ℤpew1w2xtqw1w2xdx)=(ew1w2xtw1∫ℤpew1x1tqw1x1dx1)(∑i=0w1−1qw2iew2it)=1w1∑i=0w1−1qw2i∫ℤpe(x1+w2x+(w2/w1)i)tw1qxw1dx1=∑n=0∞(∑i=0w1−1Bn,qw1(w2x+w2w1i)w1n−1qw2i)tnn!.From the symmetry ofT(w1,w2;x,t),
we can also derive (2.24)T(w1,w2;x,t)=∑n=0∞(∑i=0w2−1Bn,qw2(w1x+w1w2i)w2n−1qw1i)tnn!.By comparing the coefficientstn/n! on the both sides of (2.23) and (2.24), we obtain
the following theorem.Theorem 2.3.
Forn∈ℤ+, w1,w2∈ℕ,
we have (2.25)∑i=0w1−1Bn,qw1(w2x+w2w1i)w1n−1qw2i=∑i=0w2−1Bn,qw2(w1x+w1w2i)w2n−1qw1i.Remark 2.4.
Settingw2=1 in Theorem 2.3, we get the multiplication
theorem for the q-Bernoulli polynomials as follows: (2.26)Bn,q(w1x)=w1n−1∑i=0w1−1Bn,qw1(x+iw1)qi.
I cannot obtain the extended formulae of Theorems2.1 and 2.3 related to the Carlitz's q-Bernoulli numbers and polynomials. So, we
suggest the following two questions.Question.
Find the extended formulae of Theorems2.1
and 2.3, which are related to the Carlitz's q-Bernoulli numbers and polynomials.Question.
Find the twisted formulae of Theorems2.1 and 2.3, which are related to the twisted Carlitz's q-Bernoulli polynomials.Remark 2.5.
In
[12], q-Volkenborn integral is defined by (2.27)Iq(f)=∫ℤpf(x)dμq(x)=limN→∞1[pN]q∑x=0pN−1f(x)qx.
Thus, we note that Carlitz'sq-Bernoulli numbers can be written by (2.28)βn,q=∫ℤp[x]qndμq(x),Witt's  type  formula.



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*Source: 914367-2008-10-12.xml*