# Some Identities between the Extendedq-Bernstein Polynomials with Weight α and q-Bernoulli Polynomials with Weight (α,β)

Authors: H. Y. Lee; C. S. Ryoo
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Using bosonicp-adic q-integral on ℤp, we give some interesting relationships between q-Bernoulli numbers with weight (α,β) and q-Bernstein polynomials with weight α. Also, using q-Bernstein polynomials with two variables, we derive some interesting properties associated with q-Bernoulli numbers with weight (α,β).


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## Body

## 1.   Introduction

In many areas the Bernstein polynomials have many applications: in statistics, in computer applications, in approximations of functions, in numerical analysis, inp-adic analysis, and  in the solution of differential equations. In particular, recognition of human speech has long been a hot issue among artificial intelligence and signal processing researchers, and q-Bernstein polynomials, which are mathematical operators, have been applied for pattern recognition, and a new method has thus been developed. For q-Bernoulli numbers and polynomials, several results have been studied by Carlitz (see [1, 2]), Kim (see [3–7]), Ozden et al. (see [8]), and M.-S. Kim et al. (see [9]). Bernoulli numbers and polynomials possess many interesting properties and arise in many areas of mathematics, mathematical physics, and statistical physics. Recently, many mathematicians have studied in the area of Bernoulli numbers and polynomials. First, we introduce the q-Bernoulli numbers and polynomials with weight (α,β) as extended measure. And we investigate some identities due to the q-Bernoulli numbers and polynomials with weight (α,β). Second, we consider the p-adic analogue of the extended q-Bernstein polynomials on ℤp and investigate some properties of several extended q-Bernstein polynomials by using the bosonic p-adic q-integral. Finally, we investigate the  relation of Bernstein polynomials and Bernoulli numbers and polynomials using p-adic q-integral on ℤp.Throughout this paper, we use the following notations. Byℤp we denote the ring of p-adic rational integers, ℚp denotes the field of p-adic rational numbers, ℂp denotes the completion of algebraic closure of ℚp, ℕ denotes the set of natural numbers, ℤ denotes the ring of rational integers, ℚ denotes the field of rational numbers, ℂ denotes the set of complex numbers, and ℤ+=ℕ∪{0}. Let νp be the normalized exponential valuation of ℂp with |p|p=p-νp(p)=p-1. When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q∈ℂ, or p-adic number q∈ℂp. If q∈ℂ one normally assumes that |q|<1. If q∈ℂp, we normally assume that |q-1|p<p-1/(p-1) so that qx=exp(xlogq) for |x|p≤1. Throughout this paper we use the following notation:
(1)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q
(see [1–22]). The q-numbers [x]q satisfy many simple relations, all easily verified, as below:
(2)[x+y]q=[x]+qx[y]q,[x]q-1=q1-x[x]q,[-x]q=-q-x[x]q,[-x]q-1=-q[x]q,[1-x]q=1-[x]q-1.For a fixed positive integerf with (f,p)=1, let
(3)X=Xf=limN←(ZfpNℤ),X1=ℤp,X*=⋃0<a<fpN(a,p)=1‍(a+fpNℤp),a+fpNℤp={x∈X∣x≡a(modfpN)},
where a∈ℤ and 0≤a<fpN (see [3]). We say that f is a uniformly differential function at a point a∈ℤp and denotes this property by f∈UD(ℤp) if the difference quotients
(4)Ff(x,y)=f(x)-f(y)x-y
have a lim(x,y)→(a,a)Ff(x,y)=f′(a). For f∈UD(ℤp), let us begin with the expression
(5)1[pN]q∑x=0pN-1‍f(x)qx=∑0≤x<pN‍f(x)μq(x+pNℤp)
representing a q-analogue of the Riemann sum for f. The integral of f on ℤp is defined as the limit (N→∞) of the sums (if exists). The p-adic q-integral (or q-Volkenborn integrals of f∈UD(ℤp)) is defined by T. Kim (see [4]) as follows
(6)Iq(f)=∫X‍f(x)dμq(x)=∫Zp‍f(x)dμq(x)=limN→∞1[pN]q∑0≤x<pN‍f(x)qx
(see [4]). If we take f1(x)=f(x+1) in (6), then we easily see that
(7)qβIqβ(f1)=Iqβ(f)+(qβ-1)f(0)+qβ-1logqβf′(0).
From (6), we obtain
(8)qβnIqβ(fn)=Iqβ(f)+(qβ-1)∑l=0n-1‍qβlf(l)+qβ-1logqβ∑l=0n-1‍f′(l),
where fn(x)=f(x+n) (cf. [1–19]).Carlitz’sq-Bernoulli numbers βk,q can be defined recursively by β0,q=1 and by the value that
(9)q(qβq+1)k-βk,q={1,ifk=10,ifn>1,
with the usual convention of replacing βqi=βi,q.Carliz’sq-Bernoulli polynomials are also defined by
(10)βn,q(x)=(qxβ+[x]q)k=∑i=0k‍(ki)qixβi,q[x]qk-i.It is well known that(11)βn,q=∫ℤp‍[x]qndμq(x),βn,q(x)=∫ℤp‍[x+y]qndμq(y),n∈Z+,
where βn,q(x) are called the nth Carlitz’s q-Bernoulli polynomials.In this paper, we define theq-Bernoulli numbers and polynomials with weight (α,β) as below.Forα,β∈ℤ and q∈ℂp with |1-q|p<1, the q-Bernoulli numbers βn,q(α,β) with weight (α,β) are defined by
(12)βn,q(α,β)=∫ℤp‍[x]qαndμqβ(x).Forα,β∈ℤ and q∈ℂp with |1-q|p<1, the q-Bernoulli polynomials with weight (α,β) are also defined by
(13)βn,q(α,β)(x)=∫ℤp‍[x+y]qαndμqβ(y).


## 2. The Properties of theq-Bernoulli Numbers and Polynomials with Weight (α,β)

Our primary goal of this section is to find some properties of theq-Bernoulli numbers with weight (α,β), respectively, βn,q(α,β), βn,q(α,β)(x). This is a very important process to find the relationship between Bernoulli numbers and polynomials and Bernstein polynomials. In this section, by using the bosonic p-adic q-integral on ℤp, we obtain some properties.Forα∈ℤ and q∈ℂp with |1-q|p<1, q-Bernoulli numbers with weight (α,β) are represented as follows by simple calculus:
(14)∫ℤp‍[x]qαndμqβ(x)=limN→∞1[pN]qβ∑x=0pN-1‍[x]qαn(qβ)x=(11-qα)n∑l=0n‍(nl)(-1)l1+(α/β)l[1+(α/β)l]qβ.We set(15)Fq(α,β)(t)=∑n=0∞‍βn,q(α,β)tnn!.
By (14) and (15), we have
(16)Fq(α,β)(t)=∑n=0∞‍βn,q(α,β)tnn!=∑n=0∞‍(11-qα)n∑l=0n‍(nl)(-1)l1+(α/β)l[1+(α/β)l]qβtnn!=(1-qβ){∑m=0∞‍qβme[m]qαt-αβt∑m=0∞‍q(α+β)m1-qαe[m]qαt}.Note that forα=β=1,
(17)βn,q(1,1)(x)=Bn,q(x),
where Bn,q(x) are the nth Carliz’s q-Bernoulli polynomials. In particular, x=0, βn,q(1,1)(0)=βn,q are the nth Carliz’s q-Bernoulli numbers.Forα∈ℤ and q∈ℂp with |1-q|p<1, since [x+y]qα=[x]qα+qαx[y]qα, we get the formula as below:
(18)βn,q(α,β)(x)=∫ℤp‍[y+x]qαndμqβ(y)=([x]qα+qαxβq(α,β))n,
with usual convention about replacing (βq(α,β))n by (βn,q(α,β)). Also, by (6) and the bosonic p-adic q-integral on ℤp, we get the finite summation formula as below:
(19)βn,q(α,β)(x)=∫ℤp‍[y+x]qαndμqβ(y)=(11-qα)n∑l=0n‍(nl)(-1)lqαxl1+(α/β)l[1+(α/β)l]qβ.Therefore, by (18) and (19), we have the following theorem.Theorem 1.
Forα,β∈ℤ and q∈ℂp with |1-q|p<1, one has
(20)βn,q(α,β)(x)=([x]qα+qαxβq(α,β))n=(11-qα)n∑l=0n‍(nl)(-1)lqαxl1+(α/β)l[1+(α/β)l]qβ,
and if x=0, βn,q(α,β)(0)=βn,q(α,β).We also get from Theorem1 that for n∈ℤ+,
(21)βn,q-1(α,β)(1-x)=(-1)nqαnβn,q(α,β)(x).From (7) we get the following formula:
(22)∑n=0∞‍qββn,q(α,β)(1)-βn,q(α,β)tnn!=qβ-1+t
and by comparison of coefficients we obtain as below:
(23)qββn,q(α,β)(1)-βn,q(α,β)={qβ-1ifn=01ifn=10ifn≥2.From (8), we get the following formula:
(24)qβnβk,q(α,β)(n)=βk,q(α,β)+(qβ-1)∑l=0n-1‍qβl[l]qαk+αβ[βα]qα∑l=0n-1‍kq(α+β)l[l]qαk-1.
Specially, we consider when n=1 and n=2, then we obtain the following Theorem.Theorem 2.
Forα,β∈ℤ and q∈ℂp with |1-q|p<1, one has
(25)qββk,q(α,β)(1)=βk,q(α,β),ifn=1,q2ββk,q(α,β)(2)=βk,q(α,β)+(qβ-1)(qβ)+kαβ[αβ]qαq(α+β),ifn=2.By the bosonicp-adic q-integral on ℤp and (21), for n∈ℕ(26)∫ℤp‍[1-x]q-αndμqβ(x)=(-1)nqαn∫ℤp‍[x-1]qαndμqβ(x)=(-1)nqαnβn,q(α,β)(-1)=βn,q-1(α,β)(2).Therefore, from Theorem2 and (26), we obtain the following theorem.Theorem 3.
Forα,β∈ℤ and q∈ℂp with |1-q|p<1, one has
(27)∫ℤp‍[1-x]q-αndμqβ(x)=q2ββn,q-1(α,β)+1-qβ+nq(β-α)αβ[βα]q-α.


## 3. Properties ofq-Bernstein Polynomials

In this section, we introduce theq-Bernstein polynomials and we get some properties to use. Let C[0,1] be the set of continuous functions on [0,1]. Then, the classical Bernstein polynomials of degree n for f∈C[0,1] are defined by
(28)𝔹n(f)=∑k=0n‍f(kn)Bk,n(x),0≤x≤1,
where 𝔹n(f) is called the Bernstein operator and
(29)Bk,n(x)=(nk)xk(1-x)n-k
are called the Bernstein basis polynomials (or the Bernstein polynomials of degree n). In  recent years, Gupta et al. have studied the generation function for q-Bernstein polynomials (see [19]). Their generating function for B~k,n(x∣q)=(nk)[x]qk[1-x]qn-k is given by
(30)Fk,q(t,x)=tke[1-x]qt[x]qkk!=∑n=k∞‍B~k,n(x∣q)tnn!,
where [x]q=(1-qx)/(1-q). Observe that
(31)limq→1B~k,n(x∣q)=Bk,n(x).Also,q-Bernstein polynomials with weight α, B~k,n(α)(x∣q) are studied by T. Kim (see [7]), and the formula is as below:
(32)B~k,n(α)(x∣q)=(nk)[x]qαk[1-x]qαn-k.Letf be continuous functions on [0,1]. q-Bernstein operator is defined as below:
(33)𝔹~n,q(f∣x)=∑k=0n‍f(kn)B~k,n(x,q)=∑k=0n‍f(kn)(nk)∑s=kn‍(-1)s-k[x]qs.From now on, we introduce some properties ofq-Bernstein polynomials with weight α and q-Bernstein operator. It is easy to show that these properties are true by some calculus are(1)
B~n-k,n(α)(1-xq-1)=B~k,n(α)(x∣q),(2)
[1-x]q-αB~k,n-1(α)(x∣q)+[x]qαB~k-1,n-1(α)(x∣q)=B~k,n(α)(x∣q),(3)
(d/dx)B~k,n(α)(x∣q)=(-nαqαxlogq/(1-qα))(B~k,n(α)(x∣q)-B~k-1,n-1(α)(x∣q)),(4)
𝔹~n,q(f∣x)=∑k=0n‍f(k/n)B~k,n(α)(x∣q).In the above property 4, we consider special cases f=1, f=t, f=t2 as below.Iff=1,
(34)𝔹~n,q(1∣x)=1.Iff=t,
(35)𝔹~n,q(t∣x)=[x]qα∑s=0n-1‍(n-1s)[x]qαs(1-[x]qα)n-1-s=[x]qα.Iff=t2,
(36)𝔹~n,q(t2∣x)=∑k=0n‍k2n2(nk)[x]qαk[1-x]q-αn-k=∑s=0n-1‍s+1n(n-1s)[x]qαs+1[1-x]q-αn-s-1=n-1n[x]qα2+[x]qαn,
then one has that limn→∞𝔹~n,q(t2∣x)=[x]qα2.From definition ofB~k,n(α)(x∣q) with n∈ℕ, k∈ℤ+ and x∈[0,1], for 0≤k≤n, we have that
(37)n-knB~k,n(α)(x∣q)+k+1nB~k+1,n(α)(x∣q)=B~k,n-1(α)(x∣q).It is possible to write[x]qαk as a linear combination of B~k,n(x∣q) by using the degree evaluation formula and mathematical induction:
(38)∑k=1n‍(k1)(n1)B~k,n(α)(x∣q)=[x]qα.
By the same method, we get
(39)∑k=1n‍(k2)(n2)B~k,n(α)(x∣q)=[x]qα2.
Continuing this process, we have the following:
(40)∑k=jn‍(kj)(nj)B~n,k(α)(x∣q)=[x]q(α)j.Forx∈[0,1], n∈ℕ, k∈ℤ+,
(41)B~n,k(α)=[x]qαk1+[x]qα(nk)(1-(-[x]qα)n-k+1).
By some calculus, we obtain the following:
(42)B~n,k(α)=∑l=0k‍(nk)(kl)(-1)l1-(-[x]qα)n+l-k+11+[x]qα.
Hence, from (41) and (42) we have the following theorem.Theorem 4.
Forα∈ℤ, q∈ℂp with |1-q|p<1 and n,k∈ℕ, one has
(43)[x]qαk(1-(-[x]qα)n-k+1)=∑l=0k‍(kl)(-1)l(1-(-[x]qα)n+l-k+1).


## 4.   Relations ofq-Bernoulli Numbers and Polynomials with Weight (α,β) and q-Bernstein Polynomials

In this section, assume thatq∈ℂp with |1-q|p<1. With bosonic p-adic q-integral on ℤp, we get from Theorem 2 and (26) that
(44)∫ℤp‍B~k,n(α)(x∣q)dμqβ(x)=∫ℤp‍(nk)[x]qαk[1-x]q-αn-kdμqβ(x)=(nk)∑l=0k‍(kl)(-1)l∫ℤp‍[1-x]q-αn-ldμqβ(x)=(nk)∑l=0k‍(kl)(-1)lβn-l,q-1(α,β)(2)=(nk)∑l=0k‍(kl)(-1)l×{q2ββn-l,q-1(α,β)+1-qβ+(n-l)qβ-ααβ[βα]q-α}.Therefore, we have the following theorem.Theorem 5.
Forα,β∈ℤ, q∈ℂp with |1-q|p<1 and n,k∈ℕ, one has
(45)∫ℤp‍B~k,n(α)(x∣q)dμqβ(x)=(nk)∑l=0k‍(kl)(-1)lβn-l,q-1(α,β)(2)=(nk)∑l=0k‍(kl)(-1)l×{q2ββn-l,q-1(α,β)+1-qβ+(n-l)qβ-ααβ[βα]q-α}.Using[1-x]q-α=(1-[x]qα), we get the other identities as below:
(46)∫ℤp‍B~k,n(α)(x∣q)dμqβ(x)=∑l=0n-k‍(nk)(n-kl)(-1)lβk+l,q(α,β).From Theorem5 and (46), we obtain the following:
(47)∑l=0k‍(kl)(-1)lβn-l,q-1(α,β)(2)=∑l=0n-k‍(nk)(n-kl)(-1)lβk+l,q(α,β).Specially, ifk=n, then
(48)∑l=0n‍(nl)(-1)lβn-l,q-1(α,β)(2)=βn,q(α,β).
If n=k+1, then
(49)∑l=0k‍(kl)(-1)lβk+1-l,q-1(α,β)(2)=βk,q(α,β)-βk+1,q(α,β).Note that form,n,k∈ℤ+,
(50)∫ℤp‍B~k,n(α)(x∣q)B~k,m(α)(x∣q)dμqβ(x)=(nk)(mk)∫ℤp‍[x]qα2k[1-x]q-αm+n-2kdμqβ(x)=(nk)(mk)∑l=0m+n-2k‍(m+n-2kl)(-1)l∫ℤp‍[x]qα2k+ldμqβ(x)=(nk)(mk)∑l=0m+n-2k‍(m+n-2kl)(-1)lβ2k+l,q(α,β).
Also,
(51)∫ℤp‍B~k,n(α)(x∣q)B~k,m(α)(x∣q)dμqβ(x)=(nk)(mk)∑l=02k‍(-1)l(2kl)∫ℤp‍[1-x]q-αm+n-2kdμqβ(x)=(nk)(mk)∑l=02k‍(-1)l(2kl)βm+n-l(α,β)(2)=(nk)(mk)∑l=02k‍(-1)l(2kl)×{q2ββm+n-l(α,β)+1-qβ+(m+n-l)qβ-ααβ[βα]q-α}.Hence, one has the theorem as below.Theorem 6.
Forα,β∈ℤ, q∈ℂp with |1-q|p<1 and n,k∈ℕ, one has
(52)∫ℤp‍B~k,n(α)(x∣q)B~k,m(α)(x∣q)dμqβ(x)=(nk)(mk)∑l=0m+n-2k‍(m+n-2kl)(-1)lβ2k+l,q(α,β)=(nk)(mk)∑l=02k‍(-1)l(2kl)βm+n-l(α,β)(2).
Also, from Theorem 6, we get the identity as below:
(53)∑l=0m+n-2k‍(m+n-2kl)(-1)lβ2k+l,q(α,β)=∑l=02k‍(-1)l(2kl)βm+n-l(α,β)(2).
Continuing this process, we obtain the following theorem.Theorem 7.
Forα,β∈ℤ, q∈ℂp with |1-q|p<1 and n,k∈ℕ, one has
(54)∫ℤp‍B~k,n1(α)(x∣q)B~k,n2(α)(x∣q)⋯B~k,ns(α)(x∣q)dμqβ(x)=∏i=1s‍(nik)∑l=0N-ks‍(N-ksl)(-1)lβks+l,q(α,β),∫ℤp‍B~k,n1(α)(x∣q)B~k,n2(α)(x∣q)⋯B~k,ns(α)(x∣q)dμqβ(x)=∏i=1s‍(nik)∑l=0ks‍(ksl)(-1)ks-lβN-l,q-1(α,β)(2),
where N=n1+n2+⋯+ns.Hence, from Theorem7, we get the corollary as below.Corollary 8.
Forα,β∈ℤ, q∈ℂp with |1-q|p<1 and n,k∈ℕ, one has
(55)∑l=0N-ks‍(N-ksl)(-1)lβks+l,q(α,β)=∑l=0ks‍(ksl)(-1)ks-lβN-l,q-1(α,β)(2),
where N=n1+n2+⋯+ns.


## 5. Extension ofq-Bernstein Polynomials with Weight α

In this section, we introduce the extended Kim’sq-Bernstein polynomials of order n and the extended Kim’s q-Bernstein operator of order n. Also, we define the extended Kim’s q-Bernstein polynomials of order n with weight α and the extended Kim’s q-Bernstein operator of order n with weight α and we investigate the properties of these. Finally, we investigate the relation of q-Bernoulli numbers with weight (α,β) and extension of q-Bernstein polynomials with weight α. We assume that q∈ℝ with 0<q<1. Let C[0,1] be the set of continuous function on [0,1] and f∈C[0,1]. For n,k∈ℤ+ and x1,x2∈[0,1], the extended Kim’s q-Bernstein polynomials of order n are as below:
(56)B~k,n(x1,x2∣q)=(nk)[x1]qk[1-x2]q-1n-k.Forn,k∈ℤ+, x1,x2∈[0,1] and f∈C[0,1], the extended Kim’s q-Bernstein operator of order n is as below:
(57)𝔹~n,q(f∣x1,x2)=∑k=0n‍f(kn)(nk)[x1]qk[1-x2]q-1n-k=∑k=0n‍f(kn)B~k,n(x1,x2∣q).Forn,k∈ℤ+, x1,x2∈[0,1], f∈C[0,1] and α∈ℕ, we define the extended Kim’s q-Bernstein polynomials of order n with weight α and the extended Kim’s q-Bernstein operator of order n with weight α as below:
(58)B~k,n(α)(x1,x2∣q)=(nk)[x1]qαk[1-x2]q-αn-k,𝔹~n,q(α)(f∣x1,x2)=∑k=0n‍f(kn)(nk)[x1]qαk[1-x2]q-αn-k=∑k=0n‍f(kn)B~k,n(α)(x1,x2∣q).Properties ofB~k,n(α)(x1,x2∣q) and 𝔹~n,q(α)(f∣x1,x2) are same to B~k,n(x1,x2∣q) and 𝔹~n,q(fx1,x2), respectively. So we will enumerate these properties without proof as below.Forα,q,n∈ℕ, x1,x2∈[0,1], (1)
B~n-k,n(α)(1-x2,1-x1q-1)=B~k,n(α)(x1,x2q),(2)
[1-x2]q-αB~k,n-1(α)(x1,x2∣q)+[x1]qαB~k-1,n-1(α)(x1,x2q)=B~k,n(α)(x1,x2q),(3)
(∂/∂x1)B~k,n(α)(x1,x2q)=(αlogq/(qα-1))qαx1nB~k-1,n-1(α)(x1,x2q),(4)
(∂/∂x2)B~k,n(α)(x1,x2q)=αlogq/(1-qα)qαx2nB~k,n-1(α)(x1,x2q),(5)
(1/(1+[x1]qα-[x2]qα)n-j)∑k=jn‍((kj)/(nj))B~k,n(α)(x1,x2∣q)=[x1]qαj.Taking double bosonic p-adic q-integral on ℤp, we get from (26) and (27) that
(59)∬ℤpB~k,n(α)(x1,x2∣q)dμqβ(x1)dμqβ(x2)=(nk)βn,q(α,β)×(q2ββn-k,q-1(α,β)+1-qβ+(n-k)qβ-ααβ[βα]q-α).
Thus, from (59), we have the following theorem.Theorem 9.
Forα,β∈ℤ, q∈ℂp with |1-q|p<1 and n,k∈ℕ, one has(60)
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.We get from theq-symmetric properties of the q-Bernstein polynomials that for n,k∈ℤ+(61)∬ℤpB~k,n(α)(x1,x2∣q)dμqβ(x1)dμqβ(x2)=∬ℤp(nn-k)[1-x2]q-αn-k[x1]qαkdμqβ(x1)dμqβ(x2)=(nk)∫ℤp‍[1-x2]q-αn-kdμqβ(x2)×(∑l=0k‍(kl)(-1)k+l∫ℤp‍[1-x1]q-αk-ldμqβ(x1))=(nk)βn-k,q-1(α)(2)×(1-kβ1,q-1(α,β)(2)+∑l=0k-2‍(kl)(-1)k+lβk-l,q-1(α,β)(2)).Note that form,n,k∈ℤ+(62)∬ℤpB~k,n(α)(x1,x2∣q)B~k,m(α)(x1,x2∣q)dμqβ(x1)dμqβ(x2)=(nk)(mk)β2k,q(α,β)∫ℤp‍[1-x2]q-αm+n-2kdμqβ(x2).Note also that form,n,k∈ℤ+(63)∬ℤpB~k,n(α)(x1,x2∣q)B~k,m(α)(x1,x2∣q)dμqβ(x1)dμqβ(x2)=(nk)(mk)∫ℤp‍[1-x2]q-αm+n-2kdμqβ(x2)×∑l=02k‍(2kl)(-1)2k-l×∫ℤp‍[1-x1]q-α2k-ldμqβ(x1)=(nk)(mk)∫ℤp‍[1-x2]q-αm+n-2kdμqβ(x2)×(1-2kβ1,q(α,β)(2)+∑l=02k-2‍(2kl)(-1)2k-lβ2k-l,q(α,β)(2)).Therefore, from (63) and (65), we have the following theorem.Theorem 10.
Forα,β∈ℤ, q∈ℂp with |1-q|p<1 and n,k∈ℕ, one has
(64)β2k,q(α,β)=1-2kβ1,q(α,β)(2)+∑l=02k-2‍(2kl)(-1)2k-lβ2k-l,q(α,β)(2).Note that forn1,n2,…,ns,k∈ℤ+ and s∈ℕ(65)∬ℤp∏i=1sB~k,ni(α)(x1,x2∣q)dμqβ(x1)dμqβ(x2)=∬ℤp∏i=1s(nik)[x1]qαsk[1-x2]q-αM-skdμqβ(x2)=∏i=1s(nik)βsk,q(α,β)∫ℤp‍[1-x2]q-αM-skdμqβ(x2),
where n1+n2+⋯+ns=M.Note also from the binomial theorem thatn1,n2,…,ns,k∈ℤ+ and s∈ℕ,
(66)∬ℤp∏i=1sB~k,ni(α)(x1,x2∣q)dμqβ(x1)dμqβ(x2)=∏i=1s(nik)∫ℤp‍∑l=0sk‍(skl)(-1)sk-l[1-x1]q-αsk-ldμqβ(x1)×∫ℤp‍[1-x2]q-αM-skdμqβ(x2),
where n1+n2+⋯+ns=M.Hence, from (65) and (66), we see that the following theorem holds.Theorem 11.
Forα,β∈ℤ, q∈ℂp with |1-q|p<1 and n,k∈ℕ, one has
(67)βsk,q(α,β)=1-skβ1,q(α,β)(2)+∑l=0sk-2‍(skl)(-1)sk-lβsk-l,q(α,β)(2).



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*Source: 604364-2013-11-24.xml*