On the Higher-Order q-Euler Numbers and Polynomials with Weight α

Abstract

The main purpose of this paper is to present a systemic study of some families of higher-order q-Euler numbers and polynomials with weight α. In particular, by using the fermionic p-adic q-integral on ℤ_𝑝, we give a new concept of q-Euler numbers and polynomials with weight α.

1. Introduction

Let 𝑝 be a fixed odd prime. Throughout this paper ℤ_𝑝, ℚ_𝑝, ℂ, and ℂ_𝑝, will, respectively, denote the ring of 𝑝-adic rational integers, the field, of 𝑝-adic rational numbers, the complex number field and the completion of algebraic closure of ℚ_𝑝. Let ℕ be the set of natural numbers and ℤ₊=ℕ∪{0}. Let 𝜈_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝜈𝑝(𝑝)}=𝑝⁻¹ (see [1–14]). When one speaks of 𝑞-extension, 𝑞 can be regarded as an indeterminate, complex number 𝑞∈ℂ, or 𝑝-adic number 𝑞∈ℂ_𝑝; it is always clear from context. If 𝑞∈ℂ, we assume |𝑞|<1. If 𝑞∈ℂ_𝑝, then we assume |1−𝑞|_𝑝<1 (see [1–14]).

In this paper, we use the notation of 𝑞-number as follows: [𝑥]_𝑞=1−𝑞^𝑥1−𝑞 (1.1)(see [1–14]). Note that lim_𝑞→1[𝑥]_𝑞=𝑥 for any 𝑥 with |𝑥|_𝑝≤1 in the 𝑝-adic case.

Let 𝐶(ℤ_𝑝) be the space of continuous functions on ℤ_𝑝. For 𝑓∈𝐶(ℤ_𝑝), the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 is defined by 𝐼_−𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥)−𝑞^𝑥,=lim_𝑁→∞[2]_𝑞2^𝑝𝑁−1_𝑥=0𝑓(𝑥)−𝑞^𝑥 (1.2) (see [4–7]).

From (1.2), we note that 𝑞𝐼_−𝑞𝑓₁+𝐼_−𝑞𝑓=[2]_𝑞𝑓(0), (1.3) where 𝑓₁(𝑥)=𝑓(𝑥+1).

It is well known that the ordinary Euler polynomials are defined by 2𝑒^𝑡𝑒+1^𝑥𝑡=𝑒^{𝐸(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛,𝑛! (1.4) with the usual convention of replacing 𝐸^𝑛(𝑥) by 𝐸_𝑛(𝑥).

In the special case, 𝑥=0 and 𝐸_𝑛(0)=𝐸_𝑛 are called the 𝑛th Euler numbers (see [1–14]).

By (1.5), we get the following recurrence relation as follows:𝐸₀=1,(𝐸+1)^𝑛+𝐸=2,if𝑛=0,0,if𝑛>0. (1.5)

Recently, (ℎ,𝑞)-Euler numbers are defined by 𝐸^(ℎ)_0,𝑞=21+𝑞^ℎ,𝑞^ℎ𝑞𝐸_𝑞^(ℎ)+1^𝑛+𝐸_𝑞^(ℎ)=2,if𝑛=0,0,if𝑛>0, (1.6) with the usual convention about replacing (𝐸_𝑞^(ℎ))^𝑛 by 𝐸^(ℎ)_𝑛,𝑞 (see [1–12]).

Note that lim_𝑞→1𝐸^(ℎ)_𝑛,𝑞=𝐸_𝑛.

For 𝛼∈ℕ, the weight 𝑞-Euler numbers are also defined by 𝐸^(𝛼)_0,𝑞=1,𝑞𝑞^𝛼𝐸_𝑞^(𝛼)+1^𝑛+𝐸^(𝛼)_𝑛,𝑞=[2]_𝑞,if𝑛=0,0,if𝑛>0, (1.7) with the usual convention about replacing (𝐸_𝑞^(𝛼))^𝑛 by 𝐸^(𝛼)_𝑛,𝑞 (see [4]).

The purpose of this paper is to present a systemic study of some families of higher-order 𝑞-Euler numbers and polynomials with weight 𝛼. In particular, by using the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝, we give a new concept of 𝑞-Euler numbers and polynomials with weight 𝛼.

2. Higher-Order 𝑞-Euler Numbers and Polynomials with Weight 𝛼

For ℎ∈ℤ,𝛼,𝑘∈ℕ, and 𝑛∈ℤ₊, let us consider the expansion of higher-order 𝑞-Euler polynomials with weight 𝛼 as follows:𝐸^(𝛼)_𝑛,𝑞()=ℎ,𝑘∣𝑥_{ℤ𝑝ℤ𝑝}⋯_𝑘-times𝑥₁+⋯+𝑥_𝑘+𝑥^𝑛_𝑞𝛼𝑞^{𝑥1(ℎ−1)+⋯+𝑥𝑘(ℎ−𝑘)}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘. (2.1) From (1.2) and (2.1), we note that:𝐸^(𝛼)_𝑛,𝑞()=[2]ℎ,𝑘∣𝑥^𝑘𝑞1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥1+𝑞^𝛼𝑙+ℎ⋯1+𝑞^{𝛼𝑙+ℎ−𝑘+1}. (2.2) In the special case, 𝑥=0, 𝐸^(𝛼)_𝑛,𝑞(ℎ,𝑘∣0)=𝐸^(𝛼)_𝑛,𝑞(ℎ,𝑘) are called the higher-order 𝑞-Euler numbers with weight 𝛼.

By (2.1), we get 𝐸^(𝛼)_𝑛,𝑞(ℎ,𝑘)=𝑞^𝛼−1𝐸^(𝛼)_𝑛+1,𝑞(ℎ−𝛼,𝑘)+𝐸^(𝛼)_𝑛,𝑞(ℎ−𝛼,𝑘). (2.3)

From (2.1) and (2.2), we have𝐸^(𝛼)_0,𝑞()=𝑚𝛼,𝑘+1_ℤ𝑝⋯_ℤ𝑝𝑞^{∑𝑘+1𝑗=1(𝑚𝛼−𝑗)𝑥𝑗}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘+1=^𝑚_𝑙=0𝑚𝑙𝑞^𝛼−1^𝑙_ℤ𝑝⋯_ℤ𝑝𝑥₁+⋯+𝑥_𝑘+1^𝑙_𝑞𝛼𝑞^{−∑𝑘+1𝑗=1𝑗𝑥𝑗}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘+1=^𝑚_𝑙=0𝑚𝑙𝑞^𝛼−1^𝑙𝐸^(𝛼)_𝑙,𝑞()=[2]0,𝑘+1_𝑞𝑘+11+𝑞^𝛼𝑚1+𝑞^𝛼𝑚−1⋯1+𝑞^{𝛼𝑚−𝑘}. (2.4)

From (2.1), we can derive the following equation:^𝑖_𝑗=0𝑖𝑗𝑞^𝛼−1^𝑗_ℤ𝑝⋯_ℤ𝑝𝑥₁+⋯+𝑥_𝑘𝑞^{𝑛−𝑖+𝑗}_𝛼𝑞^{(ℎ−𝛼−1)𝑥1+⋯+(ℎ−𝛼−𝑘)𝑥𝑘}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=_ℤ𝑝⋯_ℤ𝑝𝑥₁+⋯+𝑥_𝑘𝑞^𝑛−𝑖_𝛼𝑞^{(ℎ−1)𝑥1+⋯+(ℎ−𝑘)𝑥𝑘}𝑞^{𝛼(𝑥1+⋯+𝑥𝑘)(𝑖−1)}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=^𝑖−1_𝑗=0𝑞^𝛼−1^𝑗𝑗𝐸𝑖−1_{(𝛼)𝑛−𝑖+𝑗,𝑞}(ℎ,𝑘). (2.5)

By (2.1), (2.2), (2.3), and (2.4), we see that^𝑖_𝑗=0𝑞^𝛼−1^𝑗𝑖𝑗𝐸^(𝛼)_{𝑛−𝑖+𝑗,𝑞}(ℎ−𝛼,𝑘)=^𝑖−1_𝑗=0𝑞^𝛼−1^𝑗𝑗𝐸𝑖−1_{(𝛼)𝑛−𝑖+𝑗,𝑞}(ℎ,𝑘). (2.6)

Therefore, we obtain the following theorem.

Theorem 2.1. For 𝛼,𝑘∈ℕ and 𝑛,𝑖∈ℤ₊, one has ^𝑖_𝑗=0𝑖𝑗𝑞^𝛼−1^𝑗𝐸^(𝛼)_{𝑛−𝑖+𝑗,𝑞}(ℎ−𝛼,𝑘)=^𝑖−1_𝑗=0𝑞^𝛼−1^𝑗𝑗𝐸𝑖−1_{(𝛼)𝑛−𝑖+𝑗,𝑞}(ℎ,𝑘). (2.7)

By simple calculation, we easily see that ^𝑚_𝑗=0𝑚𝑗𝑞^𝛼−1^𝑗𝐸^(𝛼)_𝑗,𝑞()=[2]0,𝑘^𝑘𝑞1+𝑞^𝛼𝑚1+𝑞^𝛼𝑚−1⋯1+𝑞^{𝛼𝑚−𝑘+1}. (2.8)

3. Polynomials 𝐸^(𝛼)_𝑛,𝑞(0,𝑘∣𝑥)

We now consider the polynomials 𝐸^(𝛼)_𝑛,𝑞(0,𝑘∣𝑥) (in 𝑞^𝑥) by𝐸^(𝛼)_𝑛,𝑞(0,𝑘∣𝑥)=_ℤ𝑝⋯_{ℤ𝑝𝑘-times}𝑥+𝑥₁+⋯+𝑥_𝑘^𝑛_𝑞𝛼𝑞^{−∑𝑘𝑗=1𝑗𝑥𝑗}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘. (3.1)

By (3.1), we get𝑞^𝛼−1^𝑛𝐸^(𝛼)_𝑛,𝑞(0,𝑘∣𝑥)=[2]^𝑘_𝑞^𝑛_𝑙=0𝑛𝑙𝑞^𝛼𝑙𝑥(−1)^𝑛−𝑙11+𝑞^𝛼𝑙⋯1+𝑞^{𝛼𝑙−𝑘+1}. (3.2)

From (3.1) and (3.2), we can derive the following equation:_ℤ𝑝⋯_ℤ𝑝𝑞^{∑𝑘𝑗=1(𝛼𝑛−𝑗)𝑥𝑗+𝛼𝑛𝑥}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=^𝑛_𝑗=0𝑛𝑗[𝛼]^𝑗_𝑞𝑞−1^𝑗𝐸^(𝛼)_𝑗,𝑞(0,𝑘∣𝑥_ℤ𝑝),⋯_ℤ𝑝𝑞^{∑𝑘𝑗=1(𝛼𝑛−𝑗)𝑥𝑗+𝛼𝑛𝑥}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=[2]^𝑘_𝑞𝑞^𝛼𝑛𝑥1+𝑞^𝛼𝑛⋯1+𝑞^{𝛼𝑛−𝑘+1}. (3.3)

Therefore, by (3.2) and (3.3), we obtain the following theorem.

Theorem 3.1. For 𝛼∈ℕ and 𝑛,𝑘∈ℤ₊, one has 𝐸^(𝛼)_𝑛,𝑞()=[2]0,𝑘∣𝑥^𝑘𝑞[𝛼]^𝑛_𝑞1−𝑞^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥1−𝑞^{𝛼𝑙−𝑘+1}∶𝑞_{𝑘𝑛𝑙=0},𝑛𝑙[𝛼]^𝑙_𝑞𝑞−1^𝑙𝐸^(𝛼)_𝑙,𝑞()=𝑞0,𝑘∣𝑥^𝛼𝑛𝑥[2]^𝑘_𝑞−𝑞^{𝛼𝑛−𝑘+1}∶𝑞_𝑘, (3.4) where (𝑎∶𝑞)₀=1 and (𝑎∶𝑞)_𝑘=(1−𝑎)(1−𝑎𝑞)⋯(1−𝑎𝑞^𝑘−1).

Let 𝑑∈ℕ with 𝑑≡1(mod2). Then we have _ℤ𝑝⋯_ℤ𝑝𝑥+^𝑘_𝑗=1𝑥_𝑗^𝑛_𝑞𝛼𝑞^{−∑𝑘𝑗=1𝑗𝑥𝑗}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=[𝑑]^𝑛_𝑞𝛼[𝑑]^𝑘_−𝑞^𝑑−1_{𝑎1,…,𝑎𝑘=0}𝑞^{−∑𝑘𝑗=2(𝑗−1)𝑎𝑗}(−1)^{∑𝑘𝑗=1𝑎𝑗}×_ℤ𝑝⋯_ℤ𝑝𝑥+∑^𝑘_𝑗=1𝑎_𝑗𝑑+^𝑘_𝑗=1𝑥_𝑗^𝑛_𝑞𝛼𝑑𝑞^{−𝑑∑𝑘𝑗=1𝑗𝑥𝑗}𝑑𝜇_−𝑞𝑑𝑥₁⋯𝑑𝜇_−𝑞𝑑𝑥_𝑘. (3.5)

Thus, by (3.5), we obtain the following theorem.

Theorem 3.2. For 𝑑∈ℕ with 𝑑≡1(mod2), one has 𝐸^(𝛼)_𝑛,𝑞()=[𝑑]0,𝑘∣𝑥^𝑛𝑞𝛼[𝑑]^𝑘_−𝑞^𝑑−1_{𝑎1,…,𝑎𝑘=0}𝑞^{−∑𝑘𝑗=2(𝑗−1)𝑎𝑗}(−1)^{∑𝑘𝑗=1𝑎𝑗}𝐸^(𝛼)_{𝑛,𝑞𝑑}0,𝑘∣𝑥+𝑎₁+⋯+𝑎_𝑘𝑑. (3.6) Moreover, 𝐸^(𝛼)_𝑛,𝑞(0,𝑘∣d𝑥)=[𝑑]^𝑛_𝑞𝛼[𝑑]^𝑘_−𝑞^𝑑−1_{𝑎1,…,𝑎𝑘=0}𝑞^{−∑𝑘𝑗=2(𝑗−1)𝑎𝑗}(−1)^{∑𝑘𝑗=1𝑎𝑗}𝐸^(𝛼)_{𝑛,𝑞𝑑}𝑎0,𝑘∣𝑥+¹+⋯+𝑎_𝑘𝑑. (3.7)

By (3.1), we get 𝐸^(𝛼)_𝑛,𝑞(0,𝑘∣𝑥)=^𝑛_𝑙=0𝑛𝑙[𝑥]_𝑞^𝑛−𝑙_𝛼𝑞^𝛼𝑙𝑥𝐸^(𝛼)_𝑙,𝑞(0,𝑘), (3.8) where 𝐸^(𝛼)_𝑛,𝑞(0,𝑘∣0)=𝐸^(𝛼)_𝑛,𝑞(0,𝑘).

Thus, we note that 𝐸^(𝛼)_𝑛,𝑞0,𝑘∣𝑥+𝑦=^𝑛_𝑙=0𝑛𝑙𝑦_𝑞^𝑛−𝑙_𝛼𝑞^𝛼𝑙𝑦𝐸^(𝛼)_𝑙,𝑞(0,𝑘∣𝑥). (3.9)

4. Polynomials 𝐸^(𝛼)_𝑛,𝑞(ℎ,1∣𝑥)

Let us define polynomials 𝐸^(𝛼)_𝑛,𝑞(ℎ,1∣𝑥) as follows: 𝐸^(𝛼)_𝑛,𝑞(ℎ,1∣𝑥)=_ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝛼𝑞^{𝑥1(ℎ−1)}𝑑𝜇_−𝑞𝑥₁. (4.1)

From (4.1), we have 𝐸^(𝛼)_𝑛,𝑞()=[2]ℎ,1∣𝑥^𝑞1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥11+𝑞^𝛼𝑙+ℎ. (4.2)

By the calculation of the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝, we see that𝑞^𝛼𝑥_ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝛼𝑞^{𝑥1(ℎ−1)}𝑑𝜇_−𝑞𝑥₁=𝑞^𝛼−1_ℤ𝑝𝑥+𝑥_1𝑞^𝑛+1_𝛼𝑞^{𝑥1(ℎ−𝛼−1)}𝑑𝜇_−𝑞𝑥₁+_ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝛼𝑞^{𝑥1(ℎ−𝛼−1)}𝑑𝜇_−𝑞𝑥₁. (4.3)

Thus, by (4.3), we obtain the following theorem.

Theorem 4.1. For 𝛼∈ℕ and ℎ∈ℤ, one has 𝑞^𝛼𝑥𝐸^(𝛼)_𝑛,𝑞(ℎ,1∣𝑥)=𝑞^𝛼−1𝐸^(𝛼)_𝑛+1,𝑞(ℎ−𝛼−1,1∣𝑥)+𝐸^(𝛼)_𝑛,𝑞(ℎ−𝛼−1,1∣𝑥). (4.4)

It is easy to show that𝐸^(𝛼)_𝑛,𝑞(ℎ,1∣𝑥)=_ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝛼𝑞^{𝑥1(ℎ−1)}𝑑𝜇_−𝑞𝑥₁=^𝑛_𝑙=0𝑛𝑙[𝑥]_𝑞^𝑛−𝑙_𝛼𝑞^𝛼𝑙𝑥_ℤ𝑝𝑥₁^𝑙_𝑞𝛼𝑞^{𝑥1(ℎ−1)}𝑑𝜇_−𝑞𝑥₁=^𝑛_𝑙=0𝑛𝑙[𝑥]_𝑞^𝑛−𝑙_𝛼𝑞^𝛼𝑙𝑥𝐸^(𝛼)_𝑙,𝑞()=𝑞ℎ,1_𝛼𝑥𝐸_𝑞^(𝛼)(ℎ,1)+[𝑥]_𝑞𝛼^𝑛,for𝑛≥1, (4.5) with the usual convention about replacing (𝐸_𝑞^(𝛼)(ℎ,1))^𝑛 by 𝐸^(𝛼)_𝑛,𝑞(ℎ,1).

From 𝑞𝐼_−𝑞(𝑓₁)+𝐼_−𝑞(𝑓)=[2]_𝑞𝑓(0), we have𝑞^ℎ_ℤ𝑝𝑥+𝑥₁+1^𝑛_𝑞𝛼𝑞^{𝑥1(ℎ−1)}𝑑𝜇_−𝑞𝑥₁+_ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝛼𝑞^{𝑥1(ℎ−1)}𝑑𝜇_−𝑞𝑥₁=[2]_𝑞[𝑥]^𝑛_𝑞𝛼. (4.6)

By (4.3) and (4.6), we get 𝑞^ℎ𝐸^(𝛼)_𝑛,𝑞(ℎ,1∣𝑥+1)+𝐸^(𝛼)_𝑛,𝑞(ℎ,1∣𝑥)=[2]_𝑞[𝑥]^𝑛_𝑞𝛼. (4.7)

For 𝑥=0 in (4.7), we have𝑞^ℎ𝐸^(𝛼)_𝑛,𝑞(ℎ,1∣1)+𝐸^(𝛼)_𝑛,𝑞()=[2]ℎ,1^𝑞,if𝑛=0,0,if𝑛>0. (4.8)

Therefore, by (4.8), we obtain the following theorem.

Theorem 4.2. For ℎ∈ℤ and 𝑛∈ℤ₊, one has 𝑞^ℎ𝑞^𝛼𝐸_𝑞^(𝛼)(ℎ,1)+1^𝑛+𝐸^(𝛼)_𝑛,𝑞()=[2]ℎ,1^𝑞,if𝑛=0,0,if𝑛>0, (4.9) with the usual convention about replacing (𝐸_𝑞^(𝛼)(ℎ,1))^𝑛 by 𝐸^(𝛼)_𝑛,𝑞(ℎ,1).

From the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝, we easily get 𝐸^(𝛼)_0,𝑞(ℎ,1)=_ℤ𝑝𝑞^{𝑥1(ℎ−1)}𝑑𝜇_−𝑞𝑥₁=[2]_𝑞[2]_𝑞ℎ. (4.10)

By (4.1), we see that 𝐸^(𝛼)_{𝑛,𝑞−1}(ℎ,1∣1−𝑥)=_ℤ𝑝1−𝑥+𝑥₁^𝑛_𝑞−𝛼𝑞^{−𝑥1(ℎ−1)}𝑑𝜇_−𝑞−1𝑥₁=(−1)^𝑛𝑞^{𝛼𝑛+ℎ−1}[2]_𝑞1−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥11+𝑞^𝛼𝑙+ℎ=(−1)^𝑛𝑞^{𝛼𝑛+ℎ−1}𝐸^(𝛼)_{𝑛,𝑞−1}(ℎ,1∣𝑥). (4.11)

Therefore, by (4.11), we obtain the following theorem.

Theorem 4.3. For 𝛼∈ℕ, ℎ∈ℤ, and 𝑛∈ℤ₊, one has 𝐸^(𝛼)_{𝑛,𝑞−1}(ℎ,1∣1−𝑥)=(−1)^𝑛𝑞^{𝛼𝑛+ℎ−1}𝐸^(𝛼)_𝑛,𝑞(ℎ,1∣𝑥). (4.12) In particular, for 𝑥=1, one gets 𝐸^(𝛼)_𝑛,𝑞(ℎ,1)=(−1)^𝑛𝑞^{𝛼𝑛+ℎ−1}𝐸^(𝛼)_𝑛,𝑞()=()ℎ,1∣1−1^𝑛−1𝑞^𝛼𝑛−1𝐸^(𝛼)_𝑛,𝑞(ℎ,1)if𝑛≥1. (4.13)

Let 𝑑∈ℕ with 𝑑≡1(mod2). Then one has _ℤ𝑝𝑞^{𝑥1(ℎ−1)}𝑥+𝑥₁^𝑛_𝑞𝛼𝑑𝜇_−𝑞𝑥₁=[𝑑]^𝑛_𝑞𝛼[𝑑]_−𝑞^𝑑−1_𝑎=0𝑞^ℎ𝑎(−1)^𝑎_ℤ𝑝𝑥+𝑎𝑑+𝑥₁^𝑛_𝑞𝛼𝑑𝑞^{𝑥1(ℎ−1)𝑑}𝑑𝜇_−𝑞𝑑𝑥₁. (4.14)

Therefore, by (4.14), we obtain the following theorem.

Theorem 4.4 (Multiplication formula). For 𝑑∈ℕ with 𝑑≡1(mod2), we have 𝐸^(𝛼)_𝑛,𝑞()=[𝑑]ℎ,1∣𝑥^𝑛𝑞𝛼[𝑑]_−𝑞^𝑑−1_𝑎=0𝑞^ℎ𝑎(−1)^𝑎𝐸^(𝛼)_{𝑛,𝑞𝑑}ℎ,1∣𝑥+𝑎𝑑. (4.15)

5. Polynomials 𝐸^(𝛼)_𝑛,𝑞(ℎ,𝑘∣𝑥) and 𝑘=ℎ

In (2.1), we know that𝐸^(𝛼)_𝑛,𝑞()=ℎ,𝑘∣𝑥_ℤ𝑝⋯_ℤ𝑝𝑥₁+⋯+𝑥_𝑘+𝑥^𝑛_𝑞𝛼𝑞^{(ℎ−1)𝑥1+⋯+(ℎ−𝑘)𝑥𝑘}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘. (5.1) Thus, we get𝑞^𝛼−1^𝑛𝐸^(𝛼)_𝑛,𝑞(ℎ,𝑘∣𝑥)=[2]^𝑘_𝑞^𝑛_𝑙=0𝑛𝑙(−1)^𝑛−𝑙𝑞^𝛼𝑙𝑥1+𝑞^𝛼𝑙+ℎ⋯1+𝑞^{𝛼𝑙+ℎ−𝑘+1},𝑞^ℎ_ℤ𝑝⋯_ℤ𝑝𝑥+1+𝑥₁+⋯+𝑥_𝑘^𝑛_𝑞𝛼𝑞^{(ℎ−1)𝑥1+⋯+(ℎ−𝑘)𝑥𝑘}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=−_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑘^𝑛_𝑞𝛼𝑞^{(ℎ−1)𝑥1+⋯+(ℎ−𝑘)𝑥𝑘}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘+[2]_𝑞ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₂+⋯+𝑥_𝑘^𝑛_𝑞𝛼𝑞^{(ℎ−2)𝑥2+⋯+(ℎ−𝑘)𝑥𝑘}𝑑𝜇_−𝑞𝑥₂⋯𝑑𝜇_−𝑞𝑥_𝑘. (5.2)

Therefore, by (2.1) and (5.2), we obtain the following theorem.

Theorem 5.1. For ℎ∈ℤ,𝛼∈ℕ, and 𝑛∈ℤ₊, one has 𝑞^ℎ𝐸^(𝛼)_𝑛,𝑞(ℎ,𝑘∣𝑥+1)+𝐸^(𝛼)_𝑛,𝑞(ℎ,𝑘∣𝑥)=[2]_𝑞E^(𝛼)_𝑛,𝑞(ℎ−1,𝑘−1∣𝑥). (5.3)

Note that 𝑞^𝛼𝑥_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑘^𝑛_𝑞𝛼𝑞^{ℎ𝑥1+(ℎ−1)𝑥2+⋯+(ℎ+1−𝑘)𝑥𝑘}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=𝑞^𝛼−1_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑘𝑞^𝑛+1_𝛼𝑞^{(ℎ−𝛼)𝑥1+⋯+(ℎ+1−𝛼−𝑘)𝑥𝑘}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘+_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑘^𝑛_𝑞𝛼𝑞^{(ℎ−𝛼)𝑥1+⋯+(ℎ+1−𝛼−𝑘)𝑥𝑘}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=𝑞^𝛼−1𝐸^(𝛼)_𝑛+1,𝑞(ℎ+1−𝛼,𝑘∣𝑥)+𝐸^(𝛼)_𝑛,𝑞(ℎ+1−𝛼,𝑘∣𝑥). (5.4)

Therefore, by (5.4), we obtain the following theorem.

Theorem 5.2. For 𝑛∈ℤ₊, one has 𝑞^𝛼𝑥𝐸^(𝛼)_𝑛,𝑞(ℎ+1,𝑘∣𝑥)=𝑞^𝛼−1𝐸^(𝛼)_𝑛+1,𝑞(ℎ+1−𝛼,𝑘∣𝑥)+𝐸^(𝛼)_𝑛,𝑞(ℎ+1−𝛼,𝑘∣𝑥). (5.5)

Let 𝑑∈ℕ with 𝑑≡1(mod2). Then we get _ℤ𝑝⋯_ℤ𝑝𝑥+^𝑘_𝑗=1𝑥_𝑗^𝑛_𝑞𝛼𝑞^{∑𝑘𝑗=1(ℎ−𝑗)𝑥𝑗}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=[𝑑]^𝑛_𝑞𝛼[𝑑]^𝑘_−𝑞^𝑑−1_{𝑎1,…,𝑎𝑘=0}𝑞^{ℎ∑𝑘𝑗=1𝑎𝑗−∑𝑘𝑗=2(𝑗−1)𝑎𝑗}(−1)^{∑𝑘𝑗=1𝑎𝑗}×_ℤ𝑝⋯_ℤ𝑝𝑥+∑^𝑘_𝑗=1𝑎_𝑗𝑑+^𝑘_𝑗=1𝑥_𝑗^𝑛_𝑞𝛼𝑑𝑞^{𝑑∑𝑘𝑗=1(ℎ−𝑗)𝑥𝑗}𝑑𝜇_−𝑞𝑑𝑥₁⋯𝑑𝜇_−𝑞𝑑𝑥_𝑘. (5.6)

Therefore, by (5.6), we obtain the following theorem.

Theorem 5.3. For 𝑑∈ℕ with 𝑑≡1(mod2), one has 𝐸^(𝛼)_𝑛,𝑞()=[𝑑]ℎ,𝑘∣𝑑𝑥_𝑛𝑞𝛼[𝑑]^𝑘_−𝑞^𝑑−1_{𝑎1,…,𝑎𝑘=0}𝑞^{ℎ∑𝑘𝑗=1𝑎𝑗−∑𝑘𝑗=2(𝑗−1)𝑎𝑗}(−1)^{∑𝑘𝑗=1𝑎𝑗}𝐸^(𝛼)_{𝑛,𝑞𝑑}𝑎ℎ,𝑘∣𝑥+¹+⋯+𝑎_𝑘𝑑. (5.7)

Let 𝐸^(𝛼)_𝑛,𝑞(𝑘,𝑘∣𝑥)=𝐸^(𝛼)_𝑛,𝑞(𝑘∣𝑥). Then we get 𝑞^𝛼−1^𝑛𝐸^(𝛼)_𝑛,𝑞(),=𝑘∣𝑥_𝑛𝑙=0𝑛𝑙(−1)^𝑛−𝑙𝑞^𝛼𝑙𝑥[2]^𝑘_𝑞1+𝑞^{𝛼𝑙+𝑘}⋯1+𝑞^𝛼𝑙+1_ℤ𝑝⋯_ℤ𝑝𝑘−𝑥+𝑥₁+⋯+𝑥_𝑘^𝑛_𝑞−𝛼𝑞^{−(𝑘−1)𝑥1−⋯−(𝑘−𝑘)𝑥𝑘}𝑑𝜇_−𝑞−1𝑥₁⋯𝑑𝜇_−𝑞−1𝑥_𝑘=𝑞^{𝛼2𝑘+1−𝑘}1−𝑞^−𝛼𝑛[2]^𝑘_𝑞^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥11+𝑞^𝛼𝑙+1⋯1+𝑞^{𝛼𝑙+𝑘}=(−1)^𝑛𝑞^𝑛𝛼𝑞^{𝛼2𝑘+1−𝑘}[2]^𝑘_𝑞1−𝑞^𝛼𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝛼𝑙𝑥1+𝑞^𝛼𝑙+1⋯1+𝑞^{𝛼𝑙+𝑘}=(−1)^𝑛𝑞^{𝛼2𝑛+𝑘+1−𝑘}𝐸^(𝛼)_𝑛,𝑞(𝑘∣𝑥). (5.8)

Therefore, by (5.8), we obtain the following theorem.

Theorem 5.4. For 𝑛∈ℤ₊, one has 𝐸^(𝛼)_{𝑛,𝑞−1}(𝑘∣𝑘−𝑥)=(−1)^𝑛𝑞^{𝛼2𝑛+𝑘+1−𝑘}𝐸^(𝛼)_𝑛,𝑞(𝑘∣𝑥). (5.9)

Let 𝑥=𝑘 in Theorem 5.4. Then we see that𝐸^(𝛼)_{𝑛,𝑞−1}(𝑘∣0)=(−1)^𝑛𝑞^{𝛼2𝑛+𝑘+1−𝑘}𝐸^(𝛼)_𝑛,𝑞(𝑘∣𝑘). (5.10)

From (4.6) and Theorem 5.1, we note that𝑞^𝑘𝐸^(𝛼)_𝑛,𝑞(𝑘∣𝑥+1)+𝐸^(𝛼)_𝑛,𝑞(𝑘∣𝑥)=[2]_𝑞𝐸^(𝛼)_𝑛,𝑞(𝑘−1∣𝑥). (5.11)

It is easy to show that 𝑞^𝛼−1^𝑛𝐸^(𝛼)_𝑛,𝑞(𝑘∣0)=^𝑛_𝑙=0𝑛l(−1)^𝑙+𝑛[2]^𝑘_𝑞1+𝑞^𝛼𝑙+1⋯1+𝑞^{𝛼𝑙+𝑘}. (5.12)

By simple calculation, we get^𝑛_𝑙=0𝑛𝑙𝑞^𝛼−1^𝑙_ℤ𝑝⋯_ℤ𝑝𝑥₁+⋯+𝑥_𝑘^𝑙_𝑞𝛼𝑞^{∑𝑘𝑙=1(𝑘−𝑙)𝑥𝑙}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=[2]^𝑘_𝑞1+𝑞^{𝛼𝑛+𝑘}1+𝑞^{𝛼𝑛+𝑘−1}⋯1+𝑞^𝛼𝑛+1. (5.13)

From (5.13), we note that^𝑛_𝑙=0𝑛𝑙𝑞^𝛼−1^𝑙𝐸^(𝛼)_𝑙,𝑞()=[2]𝑘∣0^𝑘𝑞1+𝑞^{𝛼𝑛+𝑘}1+𝑞^{𝛼𝑛+𝑘−1}⋯1+𝑞^𝛼𝑛+1,𝐸^(𝛼)_𝑛,𝑞(𝑘∣𝑥)=_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑘^𝑛_𝑞𝛼𝑞^{(𝑘−1)𝑥1+⋯+(𝑘−𝑘)𝑥𝑘}𝑑𝜇_−𝑞𝑥₁⋯𝑑𝜇_−𝑞𝑥_𝑘=^𝑛_𝑙=0𝑛𝑙𝑞^𝛼𝑙𝑥𝐸^(𝛼)_𝑙,𝑞(𝑘∣0)[𝑥]_𝑞^𝑛−𝑙_𝛼=𝑞^𝑥𝛼𝐸_𝑞^(𝛼)(𝑘∣0)+[𝑥]_𝑞𝛼^𝑛for𝑛∈ℤ₊, (5.14) with the usual convention about replacing (𝐸_𝑞^(𝛼)(𝑘∣0))^𝑛 by 𝐸^(𝛼)_𝑛,𝑞(𝑘∣0).

Put 𝑥=0 in (5.11); we get 𝑞^𝑘𝐸^(𝛼)_𝑛,𝑞(𝑘∣1)+𝐸^(𝛼)_𝑛,𝑞(𝑘∣0)=[2]_𝑞𝐸^(𝛼)_𝑛,𝑞(𝑘−1∣0). (5.15)

Thus, we have𝑞^𝑘𝑞^𝛼𝐸_𝑞^(𝛼)(𝑘∣0)+1^𝑛+𝐸^(𝛼)_𝑛,𝑞(𝑘∣0)=[2]_𝑞𝐸^(𝛼)_𝑛,𝑞(𝑘−1∣0), (5.16) with the usual convention about replacing (𝐸_𝑞^(𝛼)(𝑘∣0))^𝑛 by 𝐸^(𝛼)_𝑛,𝑞(𝑘∣0).

Acknowledgment

This work was supported by the Dong-A University research fund.