# Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

Authors: Taekyun Kim; Seog-Hoon Rim; Byungje Lee
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

By the properties ofp-adic invariant integral on ℤp,
we establish various identities concerning the generalized Bernoulli numbers and
polynomials. From the symmetric properties of p-adic invariant integral on ℤp, we give some interesting relationship between the power sums and the generalized
Bernoulli polynomials.


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## Body

## 1. Introduction

Letp be a  fixed prime number. Throughout this paper, the symbols ℤ, ℤp, ℚp, and ℂp will denote the ring of rational integers, 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}. Let vp be the normalized exponential valuation of ℂp with |p|p=p-vp(p)=1/p. Let UD(ℤp) be the space of uniformly differentiable function on ℤp. For f∈UD(ℤp), the p-adic invariant integral on ℤp is defined as
(1.1)I(f)=∫ℤpf(x)dx=limN→∞1pN∑x=0pN-1f(x),
(see [1]). From  the definition (1.1), we have
(1.2)I(f1)=I(f)+f′(0),wheref′(0)=df(x)dx|x=0,f1(x)=f(x+1).
Let fn(x)=f(x+n),  (n∈ℕ). Then we can derive the following equation  from (1.2):
(1.3)I(fn)=I(f)+∑i=0n−1f′(i),
(see [1]). It is well known that the ordinary Bernoulli polynomials Bn(x) are defined as
(1.4)tet-1ext=∑n=0∞Bn(x)tnn!,
(see [1–25]),  and the Bernoulli number Bn are defined as Bn=Bn(0).Letd be a fixed positive integer. For n∈ℕ, we set
(1.5)X=Xd=lim←N(ℤ/dpNℤ),X1=ℤp;X*=⋃0<a<dp,(a,p)=1(a+dpℤp);a+dpNℤp={x∈X∣x≡a(moddpN)},
where a∈ℤ lies in 0≤a<dpN. It is easy to see that
(1.6)∫Xf(x)dx=∫ℤpf(x)dx,forf∈UD(ℤp).In [14], the Witt's formula for the Bernoulli numbers are given by
(1.7)∫ℤpxndx=Bn,n∈ℤ+.
Let χ be the Dirichlet's character with conductor d∈ℕ. Then the generalized Bernoulli polynomials attached to χ are defined as
(1.8)∑a=1dχ(a)teatedt-1ext=∑n=0∞Bn,χ(x)tnn!,
(see [22]), and the generalized Bernoulli numbers attached to χ, Bn,χ are defined as Bn,χ=Bn,χ(0).In this paper, we investigate the interesting identities of symmetry for the generalized Bernoulli numbers and polynomials attached toχ by using the properties of p-adic invariant integral on ℤp. Finally, we will give relationship between the power sum polynomials and the generalized Bernoulli numbers attached to χ.


## 2. Symmetry of Power Sum and the Generalized Bernoulli Polynomials

Letχ be the Dirichlet character with conductor d∈ℕ. From (1.3), we note that
(2.1)∫Xχ(x)extdx=t∑i=0d-1χ(i)eitedt-1=∑n=0∞Bn,χtnn!,
where Bn,χ(x) are the nth generalized Bernoulli numbers attached to χ. Now, we also see that the generalized Bernoulli polynomials attached to χ are given by
(2.2)∫Xχ(y)e(x+y)tdy=t∑i=0d-1χ(i)eitedt-1ext=∑n=0∞Bn,χ(x)tnn!.
By (2.1) and (2.2), we easily see that
(2.3)∫Xχ(x)xndx=Bn,χ,∫Xχ(y)(x+y)ndy=Bn,χ(x).
From (2.2),  we have
(2.4)Bn,χ(x)=∑ℓ=0n(nℓ)Bℓ,χxn-ℓ.
From (2.2), we can also derive
(2.5)∫Xχ(x)extdx=∑i=0d-1χ(i)tedt-1e(i/d)dt=∑n=0∞(dn−1∑i=0d-1χ(i)Bn(id))tnn!.
Therefore, we obtain the following lemma.Lemma 2.1.
Forn∈ℤ+,  one has
(2.6)∫Xχ(x)xndx=Bn,χ=dn−1∑i=0d-1χ(i)Bi(id).We observe that(2.7)1t(∫Xχ(x)e(nd+x)tdx-∫Xextχ(x)dx)=nd∫Xχ(x)extdx∫Xendxtdx=endt-1edt-1(∑i=0d-1χ(i)eit).
Thus, we have
(2.8)1t(∫Xχ(x)e(nd+x)tdx-∫Xχ(x)extdx)=∑k=0∞(∑ℓ=0nd-1χ(ℓ)ℓk)tkk!.
Let us define the p-adic functional Tk(χ,n) as follows:
(2.9)Tk(χ,n)=∑ℓ=0nχ(ℓ)ℓk,fork∈ℤ+.
By (2.8) and (2.9), we see that
(2.10)1t(∫Xχ(x)e(nd+x)tdx-∫Xχ(x)extdx)=∑n=0∞(Tk(χ,nd-1))tkk!.
By using Taylor expansion in (2.10), we have
(2.11)∫Xχ(x)(dn+x)kdx-∫Xχ(x)xkdx=kTk-1(χ,nd-1),fork,n,d∈ℕ.
That is,
(2.12)Bk,χ(nd)-Bk,χ=kTk-1(χ,nd-1).
Let w1,w2,d∈ℕ. Then we consider the following integral equation:
(2.13)d∬Xχ(x1)χ(x2)e(w1x1+w2x2)tdx1dx2∫Xedw1w2xtdx=t(edw1w2t-1)(ew1dt-1)(ew2dt-1)(∑a=0d-1χ(a)ew1at)(∑b=0d-1χ(b)ew2bt).
From (2.7) and (2.10), we note that
(2.14)dw1∫Xχ(x)extdx∫Xedw1xtdx=∑k=0∞(Tk(χ,dw1-1))tkk!.
Let us consider the p-adic functional Tχ(w1,w2) as follows:
(2.15)Tχ(w1,w2)=d∬Xχ(x1)χ(x2)e(w1x1+w2x2+w1w2x)tdx1dx2∫Xedw1w2x3tdx3.
Then we see that Tχ(w1,w2) is symmetric in w1 and w2, and
(2.16)Tχ(w1,w2)=t(edw1w2t-1)ew1w2xt(ew1dt-1)(ew2dt-1)(∑a=0d-1χ(a)ew1at)(∑b=0d-1χ(b)ew2bt).
By (2.15) and (2.16), we have(2.17)Tχ(w1,w2)=(1w1∫Xχ(x1)ew1(x1+w2x)tdx1)(dw1∫Xχ(x2)ew2x2tdx2∫Xedw1w2xtdx)=(1w1∑i=0∞Bi,χ(w2x)w1itii!)(∑k=0∞Tk(χ,dw1-1)w2ktkk!)=1w1(∑ℓ=0∞(∑i=0ℓBi,χ(w2x)Tℓ-i(χ,dw1-1)w1iw2ℓ-iℓ!i!(ℓ-i)!)tℓℓ!)=∑ℓ=0∞(∑i=0ℓ(ℓi)Bi,χ(w2x)Tℓ-i(χ,dw1-1)w1i-1w2ℓ-i)tℓℓ!.
From the symmetric property of Tχ(w1,w2) in w1 and w2, we note that
(2.18)Tχ(w1,w2)=(1w2∫Xχ(x2)ew2(x2+w1x)tdx2)(dw2∫Xχ(x1)ew1x1tdx1∫Xedw1w2xtdx)=(1w2∑i=0∞Bi,χ(w1x)w2itii!)(∑k=0∞Tk(χ,dw2-1)w1ktkk!)=1w2(∑ℓ=0∞(∑i=0ℓBi,χ(w1x)w2iTℓ-i(χ,dw2-1)w1ℓ-iℓ!i!(ℓ-i)!)tℓℓ!)=∑ℓ=0∞(∑i=0ℓ(ℓi)w2i-1w1ℓ-iBi,χ(w1x)Tℓ-i(χ,dw2-1))tℓℓ!.
By comparing the coefficients on the both sides of (2.17) and (2.18), we obtain the following theorem.Theorem 2.2.
Forw1,w2,d∈ℕ,   one has
(2.19)∑i=0ℓ(ℓi)Bi,χ(w2x)Tℓ-i(χ,dw1-1)w1i-1w2ℓ-i=∑i=0ℓ(ℓi)Bi,χ(w1x)Tℓ-i(χ,dw2-1)w2i-1w1ℓ-i.Letx=0 in Theorem 2.2. Then we have
(2.20)∑i=0ℓ(ℓi)Bi,χTℓ-i(χ,dw1-1)w1i-1w2ℓ-i=∑i=0ℓ(ℓi)Bi,χTℓ-i(χ,dw2-1)w2i-1w1ℓ-i.
By (2.14) and (2.16), we also see that
(2.21)Tχ(w1,w2)=(ew1w2xtw1∫Xχ(x1)ew1x1tdx1)(dw1∫Xχ(x2)ew2x2tdx2∫Xedw1w2xtdx)=(ew1w2xtw1∫Xχ(x1)ew1x1tdx1)(edw1w2t-1ew2dt-1)(∑i=0d-1χ(i)ew2it)=(ew1w2xtw1∫Xχ(x1)ew1x1tdx1)(∑ℓ=0w1-1∑i=0d-1ew2(i+ℓd)tχ(i+ℓd))=(ew1w2xtw1∫Xχ(x1)ew1x1tdx1)(∑i=0dw1-1ew2itχ(i))=1w1∑i=0dw1-1χ(i)∫Xχ(x1)ew1(x1+w2x+(w2/w1)i)tdx1=1w1∑i=0dw1-1χ(i)∑k=0∞Bk,χ(w2x+w2w1i)w1ktkk!=∑k=0∞(∑i=0dw1-1χ(i)Bk,χ(w2x+w2w1i)w1k-1)tkk!.
From the symmetric property of Tχ(w1,w2) in w1 and w2, we can also derive the following equation:
(2.22)Tχ(w1,w2)=(ew1w2xtw2∫Xχ(x2)ew2x2tdx2)(dw2∫Xχ(x1)ew1x1tdx1∫Xedw1w2xtdx)=(ew1w2xtw2∫Xχ(x2)ew2x2tdx2)(edw1w2t-1ew1dt-1)(∑i=0d-1χ(i)ew1it)=(ew1w2xtw2∫Xχ(x2)ew2x2tdx2)(∑ℓ=0w2-1ew1dℓt)(∑i=0d-1χ(i)ew1it)=1w2∑i=0dw2-1χ(i)∫Xχ(x2)ew2(x2+w1x+(w1/w2)i)tdx2=1w2∑i=0dw2-1χ(i)∑k=0∞Bk,χ(w1x+w1w2i)w2ktkk!=∑k=0∞{∑i=0dw2-1χ(i)Bk,χ(w1x+w1w2i)w2k-1}tkk!.
By comparing the coefficients on the both sides of (2.21) and (2.22), we obtain the following theorem.Theorem 2.3.
Forw1,w2,d∈ℕ,  one has
(2.23)∑i=0dw1-1χ(i)Bk,χ(w2x+w2w1i)w1k-1=∑i=0dw2-1χ(i)Bk,χ(w1x+w1w2i)w2k-1.Remark 2.4.
Letx=0 in Theorem 2.3. Then we see that
(2.24)∑i=0dw1-1χ(i)Bk,χ(w2w1i)w1k-1=∑i=0dw2-1χ(i)Bk,χ(w1w2i)w2k-1.
If we takew2=1, then we have
(2.25)∑i=0dw1-1χ(i)Bk,χ(iw1)w1k-1=∑i=0d-1χ(i)Bk,χ(w1i).Remark 2.5.
Letχ be trivial character. Then we can easily derive the “multiplication theorem for Bernoulli polynomials’’  from Theorems 2.2 and 2.3 (see [14]).



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*Source: 848943-2009-07-06.xml*