# On theq-Bernoulli Numbers and Polynomials with Weight α

Authors: T. Kim; J. Choi
Journal: Abstract and Applied Analysis
 (2011)

---

## Abstract

We present a systemic study of some families of higher-orderq-Bernoulli numbers and polynomials with weight α. From these studies, we derive some interesting identities on the q-Bernoulli numbers and polynomials with weight α.


---

## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this paper, ℤp, ℚp, and ℂp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of ℚp, respectively. The p-adic norm of ℂp is defined as |x|p=p-r, where x=prm/n with (p,m)=(p,n)=1, r∈ℚ and m,n∈ℤ. Let ℕ and ℤ be the set of natural numbers and integers, respectively, ℤ+=ℕ∪{0}. Let q∈ℂp with |1-q|p<p-1/(p-1). The notation of q-number is defined by [x]w=(1-wx)/(1-w) and [x]q=(1-qx)/(1-q), (see [1–13]).As the well known definition, the Bernoulli polynomials are defined by(1.1)tet-1ex=∑n=0∞Bn(x)tnn!.
In the special case, x=0, Bn(0)=Bn are called the nth Bernoulli numbers. That is, the recurrence formula for the Bernoulli numbers is given by (1.2)B0=1,(B+1)n-Bn={1ifn=1,0ifn>1,
with the usual convention about replacing Bi with Bi.In [1, 2], q-extension of Bernoulli numbers are defined by Carlitz as follows:(1.3)β0,q=1,q(qβ+1)n-βn,q={1ifn=1,0ifn>1,
with the usual convention about replacing βi with βi,q.By (1.2) and (1.3), we get limq→1βi,q=Bi. In this paper, we assume that α∈ℕ.In [7], the q-Bernoulli numbers with weight α are defined by Kim as follows:(1.4)β̃0,q(α)=1,q(qαβ̃α+1)n-β̃n,q(α)={α[α]qifn=1,0ifn>1,
with the usual convention about replacing (β̃(α))i with β̃i,q(α).LetUD(ℤp) be the space of uniformly differentiable functions on ℤp. For f∈UD(ℤp), the p-adic q-integral on ℤp is defined as (1.5)Iq(f)=∫Zpf(x)dμq(x)=lim⁡N→∞1[pN]q∑x=0pN-1f(x)qx,
(see[4, 5]). From (1.5), we note that(1.6)qnIq(fn)=Iq(f)+(q-1)∑l=0n-1qlf(l)+q-1log⁡q∑l=0n-1qlf′(l),
where fn(x)=f(x+n) and f′(l)=(df(x)/dx)|x=l.By (1.4), (1.5), and (1.6), we set (1.7)β̃n,q(α)=∫Zp[x]qαndμq(x),wheren∈Z+,
(see[7]). The q-Bernoulli polynomials are also given by (1.8)β̃n,q(α)(x)=∫Zp[x+y]qαndμq(x)=∑l=0n(nl)[x]qαn-lqαlxβ̃l,q(α).The purpose of this paper is to derive a new concept of higher-orderq-Bernoulli numbers and polynomials with weight α from the fermionic p-adic q-integral on ℤp. Finally, we present a systemic study of some families of higher-order q-Bernoulli numbers and polynomials with weight α.


## 2. Higher Orderq-Bernoulli Numbers with Weight α

Letβ∈ℤ and α∈ℕ in this paper. For k∈ℕ and n∈ℤ+, we consider the expansion of higher-order q-Bernoulli polynomials with weight α as follows: (2.1)β̃n,q(β,k∣α)(x)=∫Zp⋯∫Zp[x1+⋯+xk+x]qαnqx1(β-1)+⋯+xk(β-k)dμq(x1)⋯dμq(xk).From (2.1), we note that (2.2)β̃n,q(β,k∣α)(x)=(1-q)k-n[α]qn∑l=0n(nl)(-1)lqαlx∏i=0k-1(αl+β-i)∏i=0k-1(1-qαl+β-i)=1(1-q)n[α]qn∑l=0n(nl)(-1)lqαlx(αl+βk)k!(αl+βk)q[k]q!,
where (αl)q=((1-qα)(1-qα-1)⋯(1-qα-l+1))/((1-q)(1-q2)⋯(1-ql)) and [k]q!=[k]q⋯[2]q[1]q.Therefore, we obtain the following theorem.Theorem 2.1.
Forn∈ℤ+ and k∈ℕ, we have
(2.3)β̃n,q(β,k∣α)(x)=1(1-q)n[α]qn∑l=0n(nl)(-1)lqαlx(αl+βk)k!(αl+βk)q[k]q!.In the special case,x=0, β̃n,q(β,k∣α)(0)=β̃n,q(β,k∣α) are called the nth higher order q-Bernoulli numbers with weight α.From (2.1) and (2.2), we can derive (2.4)β̃n,q(β,k∣α)=(qα-1)β̃n+1,q(β-α,k∣α)+β̃n,q(β-α,k∣α).
By Theorem 2.1 and (2.4), we get (2.5)β̃0,q(mα,k∣α)=∫Zp⋯∫Zpq∑j=1k(αm-j)xjdμq(x1)⋯dμq(xk)=∑l=0m(ml)(qα-1)l∫Zp⋯∫Zp[x1+⋯+xk]qαlq-∑j=1kjxjdμq(x1)⋯dμq(xk)=∑l=0m(ml)(qα-1)lβ̃l,q(0,k∣α)=(1-q)k∏i=0k-1(1-qαm-k+1+i)=(1-q)k∑l=0k(k+l-1l)qq(αm-k+1)l.
From (2.1), we have (2.6)∑j=0i(ij)(qα-1)j∫Zp⋯∫Zp[x1+⋯+xk]qαn-i+jq(β-α-1)x1+⋯+(β-α-k)xkdμq(x1)⋯dμq(xk)=∫Zp⋯∫Zp[x1+⋯+xk]qαn-iq(β-1)x1+⋯+(β-k)xkqα(x1+⋯+xk)(i-1)dμq(x1)⋯dμq(xk)=∑j=0i-1(i-1j)(qα-1)jβ̃n-i+j,q(β,k∣α).Thus, we obtain the following theorem.Theorem 2.2.
Fori∈ℕ, we have
(2.7)∑j=0i(ij)(qα-1)jβ̃n-i+j,q(β-α,k∣α)=∑j=0i-1(i-1j)(qα-1)jβ̃n-i+j,q(β,k∣α).It is easy to show that∑j=0m(mj)(qα-1)jβ̃j,q(0,k∣α)=(1-q)k∑l=0k(k+l-1l)qq(αm-k+1)l=(1-q)k∏i=0k-1(1-qαm-k+1+i).


## 3. Polynomialsβ̃n,q(0,k∣α)(x)

In this section, we consider the polynomialsβ̃n,q(0,k|α)(x) as follows: (3.1)β̃n,q(0,k∣α)(x)=∫Zp⋯∫Zp[x1+⋯+xk+x]qαnq-∑j=1kjxjdμq(x1)⋯dμq(xk).From (3.1), we can easily derive the following equation: (3.2)β̃n,q(0,k|α)(x)=(1-q)k(1-qα)n∑l=0n(nl)(-1)lqαlx∏i=0k(αl-i)∏i=0k-1(1-qαl-i)=1(1-q)n[α]qn∑l=0n(nl)(-1)lqαlx(αlk)k!(αlk)q[k]q!.
By (3.1) and (3.2), we get (3.3)∫Zp⋯∫Zpq∑j=1k(αn-j)xj+αnxdμq(x1)⋯dμq(xk)=∑l=0n(nl)[α]ql(q-1)lβ̃l,q(0,k∣α)(x),(3.4)∫Zp⋯∫Zpq∑l=1k(αn-l)xl+αnxdμq(x1)⋯dμq(xk)=qαnx(1-q)k(∏j=0k-1(αn-j))∏j=0k-1(1-qαn-j)=qαnx(αnk)k!(αnk)q[k]q!.
Therefore, by (3.3) and (3.4), we obtain the following theorem.Theorem 3.1.
Forn∈ℤ+, we have
(3.5)(1-q)nβ̃n,q(0,k∣α)(x)=1[α]qn∑l=0n(nl)(-1)lqαlx(αlk)k!(αlk)q[k]q!.
Moreover,
(3.6)∑l=0n(nl)[α]ql(q-1)lβ̃l,q(0,k∣α)(x)=qαnx(αnk)k!(αnk)q[k]q!.Letd∈ℕ. Then, we have (3.7)∫Zp⋯∫Zp[x+x1+⋯+xk]qαnq-∑j=1kjxjdμq(x1)⋯dμq(xk)=[d]qαn[d]qk∑a1,⋯,ak=0d-1q-∑j=2k(j-1)aj×∫Zp⋯∫Zp[x+∑j=1kajd+∑i=1kxi]qαdnq-d∑j=1kjxjdμqd(x1)⋯dμqd(xk).
Thus, by (3.1) and (3.7), we obtain the following theorem.Theorem 3.2.
Ford,k∈ℕ, and n∈ℤ+, we have
(3.8)β̃n,q(0,k∣α)(x)=[d]qαn[d]qk∑a1,…,ak=0d-1q-∑j=2k(j-1)ajβ̃n,qα(0,k∣α)(x+a1+⋯+akd).From (3.1), we note that (3.9)β̃n,q(0,k|α)(x)=∑l=0n(nl)[x]qαn-lqαlxβ̃l,q(0,k∣α),β̃n,q(0,k∣α)(x+y)=∑l=0n(nl)[y]qαn-lqαlxβ̃l,q(0,k∣α)(x).


## 4. Polynomialsβ̃n,q(h,1|α)(x)

Forh∈ℤ, let us define weighted (h,q)-Bernoulli polynomials β̃n,q(h,1∣α)(x) as follows: (4.1)β̃n,q(h,1∣α)(x)=∫Zp[x+x1]qαnqx1(h-1)dμq(x1).
By (4.1), we easily see that (4.2)β̃n,q(h,1∣α)(x)=1[α]qn(1-q)n∑l=0n(nl)(-1)lqαlxαl+h[αl+h]q.
Therefore, by (4.2), we obtain the following theorem.Theorem 4.1.
Forh∈ℤ and n∈ℤ+, we have
(4.3)β̃n,q(h,1|α)(x)=1[α]qn(1-q)n∑l=0n(nl)(-1)lqαlxαl+h[αl+h]q.From (4.1), we can derive the following equation: (4.4)qαx∫Zp[x+x1]qαnqx1(h-1)dμq(x1)=(qα-1)∫Zp[x+x1]qαn+1qx1(h-α-1)dμq(x1)+∫Zp[x+x1]qαnqx1(h-α-1)dμq(x1).
By (4.4), we easily get (4.5)qαxβ̃n,q(h,1∣α)(x)=(qα-1)β̃n+1,q(h-α-1,1∣α)(x)+β̃n,q(h-α-1,1∣α)(x).
From (4.1), we have (4.6)β̃n,q(h,1|α)(x)=∫Zp[x+x1]qαnqx1(h-1)dμq(x1)=∑l=0n(nl)[x]qαn-lqαlxβ̃l,q(h,1∣α),
where β̃l,q(h,1∣α)(0)=β̃l,q(h,1∣α).By (4.6), we get the following recurrence formula: (4.7)β̃n,q(h,1∣α)(x)=(qαxβ̃q(h,1∣α)+[x]qα)n,forn≥1,
with the usual convention about replacing (β̃q(h,1∣α))n with β̃n,q(h,1∣α).From (1.6), we note that (4.8)qIq(f1)=Iq(f)+(q-1)f(0)+q-1log⁡⁡qf′(0).
For h∈ℤ+, by (4.8), we have (4.9)qh∫Zpf(x+1)q(h-1)xdμq(x)=∫Zpf(x)dμq(x)+(q-1)hf(0)+q-1log⁡⁡qf′(0).
If h∈{-1,-2,-3,…}, then we get (4.10)qh∫Zpf(x+1)q(h-1)xdμq(x)=∫Zpf(x)dμq(x)+(1-q)hf(0)+q-1log⁡⁡qf′(0).
Let h∈ℤ+. By (4.9), we get (4.11)qh∫Zp[x+x1+1]qαnq(h-1)x1dμq(x1)-∫Zp[x+x1]qαnq(h-1)x1dμq(x1)=(q-1)h[x]qαn+α[α]q[x]qαn-1qαx.
From (4.6) and (4.11), we note that (4.12)qhβ̃n,q(h,1∣α)(x+1)-β̃n,q(h,1|α)(x)=(q-1)h[x]qαn+nα[α]q[x]qαn-1qαx.
If we take x=0 in (4.12), then we have (4.13)β̃0,q(h,1|α)=h[h]q,qhβ̃n,q(h,1∣α)(1)-β̃n,q(h,1∣α)={α[α]qifn=1,0ifn>1.
Therefore, by (4.12) and (4.13), we obtain the following theorem.Theorem 4.2.
Forh∈ℤ+, we have
(4.14)β̃0,q(h,1∣α)=h[h]q,qhβ̃n,q(h,1∣α)(1)-β̃n,q(h,1∣α)={α[α]qifn=1,0ifn>1.By (4.7) and Theorem 4.2, we obtain the following corollary.Corollary 4.3.
Forh∈ℤ+, we have
(4.15)β̃0,q(h,1∣α)=h[h]q,qh(qαβ̃q(h,1∣α)+1)n-β̃n,q(h,1∣α)={α[α]qifn=1,0ifn>1,
with the usual convention about replacing (β̃q(h,1∣α))n with β̃n,q(h,1∣α).From (4.1), we have (4.16)β̃0,q(h,1∣α)=∫Zpqx1(h-1)dμq(x1)=h[h]q,ifh∈Z+.
It is not difficult to show that (4.17)β̃n,q-1(h,1∣α)(1-x)=∫Zp[1-x+x1]q-αnq-x1(h-1)dμq-1(x1)=(-1)nqαn+h-1∫Zp[x+x1]qαqx1(h-1)dμq(x1)=(-1)nqαn+h-1β̃n,q(h,1∣α)(x).
Therefore, by (4.17), we obtain the following theorem.Theorem 4.4.
Forh,n∈ℤ+, we have
(4.18)β̃n,q-1(h,1∣α)(1-x)=(-1)nqαn+h-1β̃n,q(h,1∣α)(x).Forx=1 in Theorem 4.4, we get (4.19)β̃n,q-1(h,1∣α)=(-1)nqαn+h-1β̃n,q(h,1∣α)(1)=(-1)nqαn-1β̃n,q(h,1∣α)ifn>1.
Therefore, by (4.19), we obtain the following corollary.Corollary 4.5.
Forh∈ℤ+ and n∈ℕ with n>1, we have
(4.20)β̃n,q-1(h,1∣α)=(-1)nqαn-1β̃n,q(h,1∣α).Letd∈ℕ. By (4.1), we see that (4.21)∫Zpq(h-1)x1[x+x1]qαndμq(x1)=[d]qαn[d]q∑a=0d-1qha∫Zp[x+ad+x1]qαdnqx1(h-1)ddμqα(x1).
By (4.1) and (4.21), we obtain the following equation: (4.22)β̃n,q(h,1∣α)(x)=[d]qαn[d]q∑a=0d-1qhaβ̃n,q(h,1∣α)(x+ad),
where d∈ℕ and h∈ℤ+.


## 5. Polynomialsβ̃n,q(h,k|α)(x) and h=k

From (2.1), we note that (5.1)β̃n,q(h,k|α)(x)=∫Zp⋯∫Zp[x+x1+⋯+xk]qαnq(h-1)x1+⋯+(h-k)xkdμq(x1)⋯dμq(xk)=1(1-q)n[α]qn∑l=0n(nl)(-1)lqαlx(αl+h)⋯(αl+h-k+1)[αl+h]q⋯[αl+h-k+1]q,(5.2)qh∫Zp⋯∫Zp[x+1+x1+⋯+xk]qαnq(h-1)x1+⋯+(h-k)xkdμq(x1)⋯dμq(xk)=∫Zp⋯∫Zp[x+x1+⋯+xk]qαnq(h-1)x1+⋯+(h-k)xkdμq(x1)⋯dμq(xk)+(q-1)h∫Zp⋯∫Zp[x+x2+⋯+xk]qαnq(h-1)x2+⋯+(h-k)xkdμq(x2)⋯dμq(xk)+nα[α]qqαx∫Zp⋯∫Zp[x+x2+⋯+xk]qαn-1×qα(h-1)x2+⋯+(h-k+1)xkdμq(x2)⋯dμq(xk).
From (5.1), we have (5.3)qhβ̃n,q(h,k∣α)(x+1)=β̃n,q(h,k∣α)(x)+(q-1)hβ̃n,q(h-1,k-1∣α)(x)+qαxnα[α]qβ̃n,q(h,k-1∣α)(x).
Therefore, by (5.3), we obtain the following theorem.Theorem 5.1.
Forh,n∈ℤ+ and k∈ℕ, we have
(5.4)qhβ̃n,q(h,k∣α)(x+1)-β̃n,q(h,k∣α)(x)=(q-1)hβ̃n,q(h-1,k-1∣α)(x)+nqαxα[α]qβ̃n,q(h,k-1∣α)(x).It is easy to show that(5.5)qαx∫Zp⋯∫Zp[x+x1+⋯+xk]qαnqhx1+(h-1)x2+⋯+(h+1-k)xkdμq(x1)⋯dμq(xk)=(qα-1)∫Zp⋯∫Zp[x+x1+⋯+xk]qαn+1q(h-α)x1+⋯+(h-α+1-k)xkdμq(x1)⋯dμq(xk)+∫Zp⋯∫Zp[x+x1+⋯+xk]qαnq(h-α)x1+⋯+(h-α+1-k)xkdμq(x1)⋯dμq(xk)=(qα-1)β̃n+1,q(h+1-α,k∣α)(x)+β̃n,q(h+1-α,k∣α)(x).
Thus, by (5.5), we obtain the following proposition.Proposition 5.2.
Forh,n∈ℤ+, we have
(5.6)qαxβ̃n,q(h+1,k∣α)(x)=(qα-1)β̃n+1,q(h+1-α,k∣α)(x)+β̃n,q(h+1-α,k∣α)(x).Ford∈ℕ, we get (5.7)∫Zp⋯∫Zp[x+∑j=1kxj]qαnq∑j=1k(h-j)xjdμq(x1)⋯dμq(xk)=[d]qαn[d]qk∑a1,…,ak=0d-1qh∑j=1kaj-∑j=2k(j-1)aj×∫Zp⋯∫Zp[x+∑j=1kajd+∑j=1kxj]qαdnqd∑j=1k(h-j)xjdμqd(x1)⋯dμqd(xk).
Thus, we obtain (5.8)β̃n,q(h,k∣α)(x)=[d]qαn[d]qk∑a1,…,ak=0d-1qh∑j=1kaj-∑j=2k(j-1)ajβ̃n,qd(h,k∣α)(x+a1+⋯+akd).
Equation (5.8) is multiplication formula for the q-Bernoulli polynomials of order (h,k) with weight α.Let us defineβ̃n,q(k,k∣α)(x)=β̃n,q(k∣α)(x). Then we see that (5.9)β̃n,q(k∣α)(x)=1(1-q)n[α]qn∑l=0n(nl)(-1)lqαlx(αl+k)⋯(αl+1)[αl+k]q⋯[αl+1]q,(5.10)∫Zp⋯∫Zp[k-x+x1+⋯xk]q-αnq-(k-1)x1-⋯-(k-k)xkdμq-1(x1)⋯dμq-1(xk)=(-1)nqαn-k+(k+12)(1-q)n[α]qn∑l=0n(nl)(-1)lqαlx(αl+k)⋯(αl+1)[αl+k]q⋯[αl+1]q=(-1)nqαn-k+(k+12)β̃n,q(k∣α)(x).
Therefore, by (5.9) and (5.10), we obtain the following theorem.Theorem 5.3.
Forn∈ℤ+ and k∈ℕ, we have
(5.11)β̃n,q-1(k∣α)(k-x)=(-1)nqαn-k+(k+12)β̃n,q(k∣α)(x).Letx=k in Theorem 5.3. Then we see that (5.12)β̃n,q-1(k∣α)=(-1)nqαn-k+(k+12)β̃n,q(k∣α)(k).
From Theorem 5.1, we can derive the following equation: (5.13)qkβ̃n,q(k∣α)(x+1)-β̃n,q(k∣α)(x)=k(q-1)β̃n,q(k-1∣α)(x)+nqαxα[α]qβ̃n,q(k,k-1∣α)(x).
By (5.9), we easily get (5.14)β̃n,q(k∣α)=1(1-q)n[α]qn∑l=0n(nl)(-1)l(αl+k)⋯(αl+1)[αl+k]q⋯[αl+1]q.
From the definition of p-adic q-integral on ℤp, we note that (5.15)∑l=0n(nl)(qα-1)l∫Zp⋯∫Zp[x1+⋯+xk]qαlq∑l=0k(k-l)xldμq(x1)⋯dμq(xk)=(αn+k)⋯(αn+1)[αn+k]q⋯[αn+1]q=(αnk)k!(αnk)q[k]q!.
Thus, by (5.15), we get (5.16)∑l=0n(nl)(qα-1)lβ̃n,q(k∣α)=(αnk)k!(αnk)q[k]q!.
By the definition of polynomial β̃n,q(k∣α)(x), we see that (5.17)β̃n,q(k∣α)(x)=∫Zp⋯∫Zp[x+x1+⋯xk]qαnq(k-1)x1+⋯+(k-k)xkdμq(x1)⋯dμq(xk)=∑l=0n(nl)qαlxβ̃l,q(k∣α)[x]qn-l=(qαxβ̃q(k∣α)+[x]qα)n,wheren∈Z+,
with the usual convention about replacing (β̃q(k∣α))n with β̃n,q(k∣α).Letx=0 in (5.13). Then we have (5.18)qkβ̃n,q(k∣α)(1)-β̃n,q(k∣α)=k(q-1)β̃n,q(k-1∣α)+nα[α]qβ̃n,q(k,k-1∣α).



---

*Source: 392025-2011-10-01.xml*