Abstract

Recently mathematicians have studied some interesting relations between 𝑞-Genocchi numbers, 𝑞-Euler numbers, polynomials, Bernstein polynomials, and 𝑞-Bernstein polynomials. In this paper, we give some interesting identities of the twisted 𝑞-Genocchi numbers, polynomials, and 𝑞-Bernstein polynomials with weighted 𝛼.

1. Introduction

Throughout this paper, let 𝑝 be a fixed odd prime number. The symbols 𝑝, 𝑝, and 𝑝 denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of 𝑝. Let be the set of natural numbers and let +={0}. As a well-known definition, the 𝑝-adic absolute value is given by |𝑥|𝑝=𝑝𝑟, where 𝑥=𝑝𝑟𝑡/𝑠 with (𝑡,𝑝)=(𝑠,𝑝)=(𝑡,𝑠)=1. When one talks of 𝑞-extension, 𝑞 is variously considered as an indeterminate, a complex number 𝑞, or a 𝑝-adic number 𝑞𝑝. In this paper we assume that 𝑞𝑝 with |1𝑞|𝑝<1.

We assume that UD(𝑝) is the space of the uniformly differentiable function on 𝑝. For 𝑓UD(𝑝), Kim defined the fermionic 𝑝-adic 𝑞-integral on 𝑝 as follows:𝐼𝑞(𝑓)=𝑝𝑓(𝑥)𝑑𝜇𝑞(𝑥)=lim𝑁1𝑝𝑁𝑝𝑞𝑁1𝑥=0𝑓(𝑥)(𝑞)𝑥.(1.1) For 𝑛, let 𝑓𝑛(𝑥)=𝑓(𝑥+𝑛) be translation. As a well known equation, by (1.1), we have 𝑞𝑛𝑝𝑓(𝑥+𝑛)𝑑𝜇𝑞(𝑥)=(1)𝑛𝐼𝑞[2](𝑓)+𝑞𝑛1𝑙=0(1)𝑛1𝑙𝑞𝑙𝑓(𝑙),(1.2) compared [14]. Throughout this paper we use the notation: [𝑥]𝑞=1𝑞𝑥,[𝑥]1𝑞𝑞=1(𝑞)𝑥,1+𝑞(1.3)(cf. [116]). lim𝑞1[𝑥]𝑞=𝑥 for any 𝑥 with |𝑥|𝑝1 in the present 𝑝-adic case. To investigate relation of the twisted 𝑞-Genocchi numbers and polynomials with weight 𝛼 and the Bernstein polynomials with weight 𝛼, we will use useful property for [𝑥]𝑞𝛼 as follows;[𝑥]𝑞𝛼[]=11𝑥𝑞𝛼,[]1𝑥𝑞𝛼[𝑥]=1𝑞𝛼.(1.4)

The twisted 𝑞-Genocchi numbers and polynomials with weight 𝛼 are defined by the generating function as follows, respectively: 𝐺(𝛼)𝑛,𝑞,𝑤=𝑛𝑝𝜙𝑤([𝑥]𝑥)𝑞𝑛1𝛼𝑑𝜇𝑞(𝑥),(1.5)𝐺(𝛼)𝑛,𝑞,𝑤(𝑥)=𝑛𝑝𝜙𝑤[](𝑦)𝑦+𝑥𝑞𝑛1𝛼𝑑𝜇𝑞(𝑦).(1.6) In the special case, 𝑥=0, 𝐺(𝛼)𝑛,𝑞,𝑤(0)=𝐺(𝛼)𝑛,𝑞,𝑤 are called the 𝑛th twisted 𝑞-Genocchi numbers with weight 𝛼 (see [9]).

Let 𝐶𝑝𝑛={𝑤𝑤𝑝𝑛=1} be the cyclic group of order 𝑝𝑛 and let 𝑇𝑝=lim𝑛𝐶𝑝𝑛=𝑛1𝐶𝑝𝑛,(1.7) see [9, 1215].

Kim defined the 𝑞-Bernstein polynomials with weight 𝛼 of degree 𝑛 as follows: 𝐵(𝛼)𝑛,𝑘𝑛𝑘[𝑥](𝑥)=𝑘𝑞𝛼[]1𝑥𝑞𝑛𝑘𝛼,where[]𝑥0,1,𝑛,𝑘+,(1.8) compare [4, 7].

In this paper, we investigate some properties for the twisted 𝑞-Genocchi numbers and polynomials with weight 𝛼. By using these properties, we give some interesting identities on the twisted 𝑞-Genocchi polynomials with weight 𝛼 and 𝑞-Bernstein polynomials with weight 𝛼.

2. Some Identities on the Twisted 𝑞-Genocchi Polynomials with Weight 𝛼 and 𝑞-Bernstein Polynomials with Weight 𝛼

From (1.8), we can derive the following recurrence formula for the twisted 𝑞-Genocchi numbers with weight 𝛼: 𝐺(𝛼)0,𝑞,𝑤=0,𝑞𝑤𝐺(𝛼)𝑛,𝑞,𝑤(1)+𝐺(𝛼)𝑛,𝑞,𝑤=[2]𝑞,if𝑛=1,0,if𝑛>1,(2.1)𝐺(𝛼)0,𝑞,𝑤=0,𝑞𝑤1+𝑞𝛼𝐺(𝛼)𝑞,𝑤𝑛+𝑞𝛼𝐺(𝛼)𝑛,𝑞,𝑤=𝑞𝛼[2]𝑞,if𝑛=1,0,if𝑛>1,(2.2)𝑞𝛼𝑥𝐺(𝛼)𝑛+1,𝑞,𝑤[𝑥](𝑥)=𝛼𝑞+𝑞𝛼𝑥𝐺(𝛼)𝑞,𝑤𝑛+1(2.3) with usual convention about replacing (𝐺(𝛼)𝑞,𝑤)𝑛 by 𝐺(𝛼)𝑛,𝑞,𝑤.

By (1.5), we easily get 𝐺(𝛼)𝑛,𝑞,𝑤[2](𝑥)=𝑛𝑞11𝑞𝛼𝑛1𝑛1𝑙=0𝑙𝑛1(1)𝑙𝑞𝛼𝑥𝑙11+𝑤𝑞𝛼𝑙+1.(2.4) By (2.4), we obtain the theorem below.

Theorem 2.1. Let 𝑛+. For 𝑤𝑇𝑝, one has 𝐺(𝛼)𝑛,𝑞,𝑤(𝑥)=(1)𝑛1𝑤1𝑞𝛼(1𝑛)𝐺(𝛼)𝑛,𝑞1,𝑤1(1𝑥).(2.5)

By (2.1), (2.2), and (2.3) we note that 𝐺(𝛼)𝑛,𝑞,𝑤=𝑞𝑤𝐺(𝛼)𝑛,𝑞,𝑤(1)=𝑛𝑤𝑞𝐺(𝛼)1,𝑞,𝑤+𝑤2𝑞𝑛2𝛼𝑙=2𝑛𝑙𝑞𝛼𝑙𝐺(𝛼)𝑙,𝑞,𝑤(1)=𝑛𝑤𝑞𝐺(𝛼)1,𝑞,𝑤+𝑤2𝑞𝑛22𝛼𝑙=2𝑛𝑙𝑞𝛼𝑙1+𝑞𝛼𝐺(𝛼)𝑞,𝑤𝑙=𝑛𝑤𝑞𝐺(𝛼)1,𝑞,𝑤+𝑤2𝑞22𝛼[2]𝑞𝛼+𝑞2𝛼𝐺(𝛼)𝑞,𝑤𝑛𝑛𝑤2𝑞2𝐺(𝛼)1,𝑞,𝑤=𝑛𝑤𝑞𝐺(𝛼)1,𝑞,𝑤+𝑤2𝑞2𝐺(𝛼)𝑛,𝑞,𝑤(2)𝑛𝑤2𝑞2𝐺(𝛼)1,𝑞,𝑤.(2.6)

Therefore, by (2.6), we obtain the theorem below.

Theorem 2.2. For 𝑛 with 𝑛>1, one has 𝐺(𝛼)𝑛,𝑞,𝑤(2)=𝑤2𝑞2𝐺(𝛼)𝑛,𝑞,𝑤+𝑤1𝑞1𝑛[2]𝑞+𝑛[2]1+𝑞𝑤𝑞.1+𝑞𝑤(2.7)

By (1.6) and Theorem 2.2, 𝐺(𝛼)𝑛+1,𝑞,𝑤(2)=𝑛+1𝑝𝜙𝑤[](𝑦)𝑦+2𝑛𝑞𝛼𝑑𝜇𝑞=1(𝑦)([2]𝑛+1𝑛+1)𝑞+(1+𝑞𝑤𝑛+1)𝑤1𝑞1[2]𝑞1+𝑞𝑤+𝑤2𝑞2𝐺(𝛼)𝑛+1,𝑞,𝑤=[2]𝑛+1𝑞1+𝑞𝑤+𝑤1𝑞1[2]𝑞1+𝑞𝑤+𝑤2𝑞2𝐺(𝛼)𝑛+1,𝑞,𝑤.𝑛+1(2.8) Hence, we obtain the corollary below.

Corollary 2.3. For 𝑛, one has 𝑝𝜙𝑤[](𝑦)𝑦+2𝑛𝑞𝛼𝑑𝜇𝑞[2](𝑦)=𝑞1+𝑞𝑤+𝑤1𝑞1[2]𝑞1+𝑞𝑤+𝑤2𝑞2𝐺(𝛼)𝑛+1,𝑞,𝑤.𝑛+1(2.9)

By fermionic integral on 𝑝,Theorems 2.1 and 2.2, we note that𝑝𝜙𝑤([]𝑥)1𝑥𝑛𝑞𝛼𝑑𝜇𝑞(𝑥)=(1)𝑛𝑞𝛼𝑛𝑝𝜙𝑤([]𝑥)𝑥1𝑛𝑞𝛼𝑑𝜇𝑞(𝑥)=(1)𝑛𝑞𝛼𝑛𝐺(𝛼)𝑛+1,𝑞,𝑤(1)𝑛+1=𝑤1𝐺(𝛼)𝑛+1,𝑞1,𝑤1(2)𝑛+1=𝑤1[2]𝑞11+𝑞1𝑤1[2]+𝑤𝑞𝑞11+𝑞1𝑤1+𝑤2𝑞2𝐺(𝛼)𝑛+1,𝑞1,𝑤1=[2]𝑛+1𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺(𝛼)𝑛+1,𝑞1,𝑤1.𝑛+1(2.10)

Therefore, we have the theorem below.

Theorem 2.4. For 𝑛 with 𝑛>1, one has 𝑝𝜙𝑤[](𝑥)1𝑥𝑛𝑞𝛼𝑑𝜇𝑞[2](𝑥)=𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺(𝛼)𝑛+1,𝑞1,𝑤1.𝑛+1(2.11)

By (1.4), Theorem 2.4, we take the fermionic 𝑝-adic invariant integral on 𝑝 for one q-Bernstein polynomials as follows: 𝑝𝜙𝑤(𝑥)𝐵𝑛,𝑘(𝑥,𝑞)𝑑𝜇𝑞(𝑥)=𝑝𝜙𝑤𝑛𝑘[𝑥](𝑥)𝑘𝑞𝛼[]1𝑥𝑞𝑛𝑘𝛼𝑑𝜇𝑞=𝑛𝑘(𝑥)𝑝𝜙𝑤([𝑥]𝑥)𝑘𝑞𝛼[𝑥]1𝑞𝛼𝑛𝑘𝑑𝜇𝑞(=𝑛𝑘𝑥)𝑛𝑘𝑙=0𝑙𝑛𝑘(1)𝑙𝐺(𝛼)𝑘+𝑙+1,𝑞,𝑤.𝑘+𝑙+1(2.12)

By symmetry of 𝑞-Bernstein polynomials with weight 𝛼 of degree 𝑛, we get the following formula; 𝑝𝜙𝑤(𝑥)𝐵𝑛,𝑘(𝑥,𝑞)𝑑𝜇𝑞(=𝑥)𝑝𝜙𝑤𝑛𝑘[𝑥](𝑥)𝑞𝑛𝑘𝛼[]1𝑥𝑘𝑞𝛼𝑑𝜇𝑞=(𝑥)𝑝𝜙𝑤𝑛𝑘[](𝑥)1𝑥𝑘𝑞𝛼[]11𝑥𝑞𝛼𝑛𝑘𝑑𝜇𝑞=𝑛𝑘(𝑥)𝑛𝑘𝑙=0𝑙𝑛𝑘(1)𝑛𝑘𝑙𝑝𝜙𝑤[](𝑥)1𝑥𝑞𝑛𝑙𝛼𝑑𝜇𝑞=𝑛𝑘(𝑥)𝑛𝑘𝑙=0𝑙𝑛𝑘(1)𝑛𝑘𝑙[2]𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺(𝛼)𝑛𝑙+1,𝑞1,𝑤1.𝑛𝑙+1(2.13)

Therefore, by (2.12) and (2.13), we have the theorem below.

Theorem 2.5. For 𝑛 with 𝑛>1, one has 𝑛𝑘𝑙=0𝑙𝑛𝑘(1)𝑙𝐺(𝛼)𝑘+𝑙+1,𝑞,𝑤=𝑘+𝑙+1𝑛𝑘𝑙=0𝑙(𝑛𝑘1)𝑛𝑘𝑙[2]𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺(𝛼)𝑛𝑙+1,𝑞1,𝑤1.𝑛𝑙+1(2.14) Also, we note that 𝑝𝜙𝑤(𝑥)𝐵𝑛,𝑘(𝑥,𝑞)𝑑𝜇𝑞(=𝑛𝑘𝑥)𝑛𝑘𝑙=0𝑙𝑛𝑘(1)𝑙𝐺(𝛼)𝑘+𝑙+1,𝑞,𝑤=𝑛𝑘𝑘+𝑙+1𝑝𝜙𝑤[](𝑥)1𝑥𝑞𝑛𝑘𝛼[𝑥]𝑘𝑞𝛼𝑑𝜇𝑞=𝑛𝑘(𝑥)𝑝𝜙𝑤[](𝑥)1𝑥𝑞𝑛𝑘𝛼[]11𝑥𝑞𝛼𝑘𝑑𝜇𝑞=𝑛𝑘(𝑥)𝑘𝑙=0𝑘𝑙(1)𝑘𝑙𝑝𝜙𝑤[](𝑥)1𝑥𝑞𝑛𝑙𝛼𝑑𝜇𝑞=𝑛𝑘(𝑥)𝑘𝑙=0𝑘𝑙(1)𝑘𝑙[2]𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺(𝛼)𝑛𝑙+1,𝑞1,𝑤1.𝑛𝑙+1(2.15)

Therefore, we have the theorem below.

Theorem 2.6. For 𝑛,𝑘+ with 𝑛>𝑘+1, one has 𝑝𝜙𝑤(𝑥)𝐵𝑘,𝑛(𝑥,𝑞)𝑑𝜇𝑞(=𝑛𝑘𝑥)𝑘𝑙=0𝑘𝑙(1)𝑘𝑙[2]𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺(𝛼)𝑛𝑙+1,𝑞1,𝑤1.𝑛𝑙+1(2.16)

By (2.11) and Theorem 2.6, we have the theorem below.

Theorem 2.7. Let 𝑛,𝑘+ with 𝑛>𝑘+1. Then one has 𝑛𝑘𝑙=0𝑙𝑛𝑘(1)𝑙𝐺(𝛼)𝑘+𝑙+1,𝑞,𝑤=𝑘+𝑙+1𝑘𝑙=0𝑘𝑙(1)𝑘𝑙[2]𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺(𝛼)𝑛𝑙+1,𝑞1,𝑤1.𝑛𝑙+1(2.17)

Let 𝑛1,𝑛2,𝑘+ with 𝑛1+𝑛2>2𝑘+1. Then we get 𝑝𝜙𝑤(𝑥)𝐵𝑛(𝛼)1,𝑘(𝑥,𝑞)𝐵𝑛(𝛼)2,𝑘(𝑥,𝑞)𝑑𝜇𝑞(=𝑛𝑥)1𝑘𝑛2𝑘2𝑘𝑙=0𝑙2𝑘(1)2𝑘𝑙𝑝𝜙𝑤[](𝑥)1𝑥𝑛1+𝑛2𝑞𝑙𝛼𝑑𝜇𝑞=𝑛(𝑥)1𝑘𝑛2𝑘2𝑘𝑙=0𝑙2𝑘(1)2𝑘𝑙[2]𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺𝑛(𝛼)1+𝑛2𝑙+1,𝑞1,𝑤1𝑛1+𝑛2.𝑙+1(2.18)

Therefore, we obtain the theorem below.

Theorem 2.8. For 𝑛1,𝑛2,𝑘+, one has 𝑝𝜙𝑤(𝑥)𝐵𝑛(𝛼)1,𝑘(𝑥,𝑞)𝐵𝑛(𝛼)2,𝑘(𝑥,𝑞)𝑑𝜇𝑞(=𝑛𝑥)1𝑘𝑛2𝑘2𝑘𝑙=0𝑙2𝑘(1)2𝑘𝑙[2]𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺𝑛(𝛼)1+𝑛2𝑙+1,𝑞1,𝑤1𝑛1+𝑛2=[2]𝑙+1𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺𝑛(𝛼)1+𝑛2𝑙+1,𝑞1,𝑤1𝑛1+𝑛2,𝑙+1if𝑘=0,𝑤𝑞2𝑛1𝑘𝑛2𝑘2𝑘𝑙=0𝑙2𝑘(1)2𝑘𝑙𝐺𝑛1+𝑛2𝑙+1,𝑞1,𝑤1𝑛1+𝑛2,𝑙+1if𝑘>0,(2.19)

By simple calculation, we easily see that𝑝𝜙𝑤(𝑥)𝐵𝑛(𝛼)1,𝑘(𝑥,𝑞)𝐵𝑛(𝛼)2,𝑘(𝑥,𝑞)𝑑𝜇𝑞(=𝑛𝑥)1𝑘𝑛2𝑘𝑛1+𝑛22𝑘𝑙=0(1)𝑙𝑛1+𝑛2𝑙2𝑘𝑝𝜙𝑤[𝑥](𝑥)𝑞2𝑘+𝑙𝛼𝑑𝜇𝑞=𝑛(𝑥)1𝑘𝑛2𝑘𝑛1+𝑛22𝑘𝑙=0(1)𝑙𝑛1+𝑛2𝑙𝐺2𝑘(𝛼)2𝑘+𝑙+1,𝑞,𝑤,2𝑘+𝑙+1where𝑛1,𝑛2,𝑘+.(2.20) Therefore, by (2.20) and Theorem 2.8, we obtain the theorem below.

Theorem 2.9. Let 𝑛1,𝑛2,𝑘+ with 𝑛1+𝑛2>2𝑘+1. Then one has 2𝑘𝑙=0𝑙2𝑘(1)2𝑘𝑙[2]𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺𝑛(𝛼)1+𝑛2𝑙+1,𝑞1,𝑤1𝑛1+𝑛2=𝑙+1𝑛1+𝑛22𝑘𝑙=0(1)𝑙𝑛1+𝑛2𝑙𝐺2𝑘(𝛼)2𝑘+𝑙+1,𝑞,𝑤.2𝑘+𝑙+1(2.21)

For 𝑛1,𝑛2,,𝑛𝑠,𝑘+, 𝑛1+𝑛2++𝑛𝑠>𝑠𝑘+1, and let 𝑠𝑖=1𝑛𝑖=𝑚, then by the symmetry of 𝑞-Bernstein polynomials with weight 𝛼, we see that 𝑝𝜙𝑤(𝑥)𝑠𝑖=1𝐵(𝛼)𝑘,𝑛𝑖(𝑥,𝑞)𝑑𝜇𝑞=(𝑥)𝑠𝑖=1𝑛𝑖𝑘𝑠𝑘𝑙=0𝑙(𝑠𝑘1)𝑠𝑘𝑙𝑝𝜙𝑤([]𝑥)1𝑥𝑞𝑚𝑙𝛼𝑑𝜇𝑞(=𝑥)𝑠𝑖=1𝑛𝑖𝑘𝑠𝑘𝑙=0𝑙𝑠𝑘(1)𝑠𝑘𝑙[2]𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺(𝛼)𝑚𝑙+1,𝑞1,𝑤1.𝑚𝑙+1(2.22)

Therefore, we have the theorem below.

Theorem 2.10. For 𝑛1,𝑛2,,𝑛𝑠,𝑘+ with 𝑛1+𝑛2++𝑛𝑠>𝑠𝑘+1, one has 𝑝𝜙𝑤(𝑥)𝑠𝑖=1𝐵(𝛼)𝑘,𝑛𝑖(𝑥,𝑞)𝑑𝜇𝑞=(𝑥)𝑠𝑖=1𝑛𝑖𝑘𝑠𝑘𝑙=0𝑙(𝑠𝑘1)𝑠𝑘𝑙[2]𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞+1+𝑞𝑤w𝑞2𝐺(𝛼)𝑚𝑙+1,𝑞1,𝑤1,𝑚𝑙+1(2.23) where 𝑛1++𝑛𝑠=𝑚.

In the same manner as in (2.15), we can get the following relation:𝑝𝜙𝑤(𝑥)𝑠𝑖=1𝐵(𝛼)𝑘,𝑛𝑖(𝑥,𝑞)𝑑𝜇𝑞=(𝑥)𝑠𝑖=1𝑛𝑖𝑘𝑝𝜙𝑤([𝑥]𝑥)𝑞𝑠𝑘𝛼𝑚𝑠𝑘𝑙=0(1)𝑙𝑙(𝑚𝑠𝑘1)𝑙[𝑥]𝑙𝑞𝛼𝑑𝜇𝑞(=𝑥)𝑠𝑖=1𝑛𝑖𝑘𝑚𝑠𝑘𝑙=0(1)𝑙𝑙𝐺𝑚𝑠𝑘(𝛼)𝑠𝑘+𝑙+1,𝑞,𝑤,𝑠𝑘+𝑙+1(2.24) where 𝑛1,𝑛2,,𝑛𝑠,𝑘+ with 𝑚=𝑛1+𝑛2++𝑛𝑠>𝑠𝑘+1.

By Theorem 2.10 and (2.13), we have the following corollary.

Corollary 2.11. Let 𝑚. For 𝑛1,𝑛2,,𝑛𝑠,𝑘+ with 𝑛1++𝑛𝑠>𝑚𝑘+1, one has 𝑠𝑘𝑙=0𝑙𝑠𝑘(1)𝑠𝑘𝑙[2]𝑞[2]1+𝑞𝑤+𝑤𝑞𝑞1+𝑞𝑤+𝑤𝑞2𝐺(𝛼)𝑚𝑙+1,𝑞1,𝑤1=𝑚𝑙+1𝑚𝑠𝑘𝑙=0(1)𝑙𝑙𝐺𝑚𝑠𝑘(𝛼)𝑠𝑘+𝑙+1,𝑞,𝑤,𝑠𝑘+𝑙+1(2.25) where 𝑛1++𝑛𝑠=𝑚.