On the 𝑞-Euler Numbers and Polynomials with Weight 0

Abstract

The purpose of this paper is to investigate some properties of 𝑞-Euler numbers and polynomials with weight 0. From those 𝑞-Euler numbers with weight 0, we derive some identities on the 𝑞-Euler numbers and polynomials with weight 0.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of ℚ_𝑝. The 𝑝-adic absolute value is defined by |𝑥|_𝑝=1/𝑝^𝑟 where 𝑥=𝑝^𝑟𝑠/𝑡 for 𝑠,𝑡∈ℤ with (𝑝,𝑡)=(𝑝,𝑠)=1 and 𝑟∈ℚ. In this paper, we assume that 𝛼∈ℚ and 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1. As well-known definition, the Euler polynomials are defined by2𝑒^𝑡𝑒+1^𝑥𝑡=𝑒^{𝐸(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛,𝑛! (1.1) with the usual convention about replacing 𝐸^𝑛(𝑥) by 𝐸_𝑛(𝑥) (see [1–15]).

In this special case, 𝑥=0, 𝐸_𝑛(0)=𝐸_𝑛 are called the 𝑛th Euler numbers (see [1]). Recently, the 𝑞-Euler numbers with weight 𝛼 are defined by𝐸^(𝛼)_0,𝑞=1,𝑞𝑞^𝛼𝐸_𝑞^(𝛼)+1^𝑛+𝐸^(𝛼)_𝑛,𝑞=0if𝑛>0, (1.2) with the usual convention about replacing (𝐸_𝑞^(𝛼))^𝑛 by 𝐸^(𝛼)_𝑛,𝑞 (see [3, 12]). The 𝑞-number of 𝑥 is defined by [𝑥]_𝑞=(1−𝑞^𝑥)/(1−𝑞) (see [1–15]). Note that lim_𝑞→1[𝑥]_𝑞=𝑥. Let us define the notation of 𝑞-Euler numbers with weight 0 as 𝐸⁽⁰⁾_𝑛,𝑞=𝐸_𝑛,𝑞. The purpose of this paper is to investigate some interesting identities on the 𝑞-Euler numbers with weight 0.

2. On the Extended 𝑞-Euler Numbers of Higher-Order with Weight 0

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 Kim as follows:𝐼_𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞[2]_𝑞1+𝑞^{𝑝𝑁𝑝𝑁−1}_𝑥=0𝑓(𝑥)−𝑞^𝑥, (2.1) (see [1–12]). By (2.1), we get𝑞^𝑛𝐼_𝑞𝑓_𝑛+(−1)^𝑛−1𝐼_𝑞𝑓=[2]_𝑞^𝑛−1_𝑙=0(−1)^{𝑛−1−𝑙}𝑓(𝑙)𝑞^𝑙, (2.2) where 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛) and 𝑛∈ℕ (see [4, 5]).

By (1.2), (2.1), and (2.2), we see that_ℤ𝑝[𝑥]^𝑛_𝑞𝛼𝑑𝜇_−𝑞(𝑥)=𝐸^(𝛼)_𝑛,𝑞=[2]_𝑞1−𝑞^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑙=0𝑛𝑙(−1)^𝑙11+𝑞^𝛼𝑙+1. (2.3)

In the special case, 𝑛=1, we get_ℤ𝑝𝑒^𝑥𝑡𝑑𝜇_−𝑞(𝑥)=[2]_𝑞𝑞𝑒^𝑡=+11+𝑞⁻¹𝑒^𝑡+𝑞⁻¹=^∞_𝑛=0𝐻_𝑛−𝑞⁻¹𝑡^𝑛,𝑛! (2.4) where 𝐻_𝑛(−𝑞⁻¹) are the 𝑛th Frobenius-Euler numbers. From (2.4), we note that the 𝑞-Euler numbers with weight 0 are given by𝐸_𝑛,𝑞=_ℤ𝑝𝑥^𝑛𝑑𝜇_−𝑞(𝑥)=𝐻_𝑛−𝑞⁻¹,for𝑛∈ℤ₊. (2.5)

Therefore, by (2.5), we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊, one has 𝐸_𝑛,𝑞=𝐻_𝑛−𝑞⁻¹, (2.6) where 𝐻_𝑛(−𝑞⁻¹) are called the 𝑛th Frobenius-Euler numbers.

Let us define the generating function of the 𝑞-Euler numbers with weight 0 as follows:𝐹_𝑞(𝑡)=^∞_𝑛=0𝐸_𝑛,𝑞𝑡^𝑛.𝑛! (2.7)

Then, by (2.3) and (2.7), we get𝐹_𝑞(𝑡)=[2]_𝑞^∞_𝑚=0(−1)^𝑚𝑞^𝑚𝑒^𝑚𝑡=1+𝑞𝑞𝑒^𝑡.+1 (2.8)

Now we define the 𝑞-Euler polynomials with weight 0 as follows:^∞_𝑛=0𝐸_𝑛,𝑞(𝑥)𝑡^𝑛=𝑛!1+𝑞𝑞𝑒^𝑡𝑒+1^𝑥𝑡. (2.9)

Thus, (2.4) and (2.9), we get_ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇_−𝑞𝑦=1+𝑞𝑞𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛! (2.10)

From (2.10), we have^∞_𝑛=0𝐸_𝑛,𝑞(𝑥)𝑡^𝑛=𝑛!1+𝑞⁻¹𝑒^𝑡+𝑞⁻¹𝑒^𝑥𝑡=^∞_𝑛=0𝐻_𝑛−𝑞⁻¹𝑡,𝑥^𝑛,𝑛! (2.11) where 𝐻_𝑛(−𝑞⁻¹,𝑥) are called the 𝑛th Frobenius-Euler polynomials (see [9]).

Therefore, by (2.11), we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℤ₊, one has 𝐸_𝑛,𝑞(𝑥)=_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇_−𝑞(𝑥)=𝐻_𝑛−𝑞⁻¹,𝑥, (2.12) where 𝐻_𝑛(−𝑞⁻¹,𝑥) are called the 𝑛th Frobenius-Euler polynomials.

From (2.2) and Theorem 2.2, we note that𝑞^𝑛𝐻_𝑚−𝑞⁻¹,𝑛+𝐻_𝑚−𝑞⁻¹=[2]_𝑞^𝑛−1_𝑙=0(−1)^𝑙𝑙^𝑚𝑞^𝑙, (2.13) where 𝑛∈ℕ with 𝑛≡1 (mod 2).

Therefore, by (2.13), we obtain the following corollary.

Corollary 2.3. For 𝑛∈ℕ, with 𝑛≡1 (mod 2) and 𝑚∈ℤ₊, one has 𝑞^𝑛𝐻_𝑚−𝑞⁻¹,𝑛+𝐻_𝑚−𝑞⁻¹=[2]_𝑞^𝑛−1_𝑙=0(−1)^𝑙𝑙^𝑚𝑞^𝑙. (2.14)

In particular, 𝑞=1, we get 𝐸_𝑚(𝑛)+𝐸_𝑚=2∑^𝑛−1_𝑙=0(−1)^𝑙𝑙^𝑚, where 𝐸_𝑚 and 𝐸_𝑚(𝑛) are called the 𝑚th Euler numbers and polynomials which are defined by2𝑒^𝑡=+1^∞_𝑚=0𝐸_𝑚𝑡^𝑚,2𝑚!𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑚=0𝐸_𝑚(𝑥)𝑡^𝑚.𝑚! (2.15)

By (2.2), we easily see that𝑞_ℤ𝑝𝑓(𝑥+1)𝑑𝜇_−𝑞(𝑥)+_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=[2]_𝑞𝑓(0). (2.16)

Thus, by (2.16), we get[2]_𝑞=𝑞_ℤ𝑝𝑒^(𝑥+1)𝑡𝑑𝜇_−𝑞(𝑥)+_ℤ𝑝𝑒^𝑥𝑡𝑑𝜇_−𝑞(𝑥)=^∞_𝑛=0𝑞_ℤ𝑝(𝑥+1)^𝑛𝑑𝜇_−𝑞(𝑥)+_ℤ𝑝𝑥^𝑛𝑑𝜇_−𝑞(𝑥𝑡)^𝑛=𝑛!_∞𝑛=0𝑞𝐻_𝑛−𝑞⁻¹,1+𝐻_𝑛−𝑞⁻¹𝑡^𝑛.𝑛! (2.17)

Therefore, by (2.16), we obtain the following theorem.

Theorem 2.4. For 𝑛∈ℤ₊, one has 𝑞𝐻_𝑛−𝑞⁻¹,1+𝐻_𝑛−𝑞⁻¹=1+𝑞,if𝑛=0,0,if𝑛>0, (2.18)

where 𝐻_𝑛(−𝑞⁻¹,𝑥) are called the 𝑛th Frobenius-Euler polynomials and 𝐻_𝑛(−𝑞⁻¹) are called the 𝑛th Frobenius-Euler numbers. In particular, 𝑞=1, we have𝐸_𝑛(1)+𝐸_𝑛=2,if𝑛=0,0,if𝑛>0, (2.19) where 𝐸_𝑛 are called the 𝑛th Euler numbers.

From (2.5) and Theorem 2.2, we note that𝐸_𝑛,𝑞(𝑥)=_ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇_−𝑞𝑦=^𝑛_𝑙=0𝑛𝑙^ℤ𝑝𝑦^𝑙𝑑𝜇_−𝑞𝑦𝑥^𝑛−𝑙=^𝑛_𝑙=0𝑛𝑙𝐸_𝑛,𝑞𝑥^𝑛−𝑙=𝑥+𝐸_𝑞^𝑛, (2.20) where the usual convention about replacing (𝐸_𝑞)^𝑙 by 𝐸_𝑙,𝑞. By Theorems 2.2 and 2.4, we get𝑞𝐸_𝑛,𝑞(1)+𝐸_𝑛,𝑞=[2]_𝑞,if𝑛=0,0,if𝑛>0. (2.21)

From (2.20) and (2.21), we have𝑞𝐸_𝑞+1^𝑛+𝐸_𝑛,𝑞=[2]_𝑞,if𝑛=0,0,if𝑛>0. (2.22)

For 𝑛∈ℕ, by (2.20) and (2.22), we have𝑞²𝐸_𝑛,𝑞(2)=𝑞²𝐸_𝑞+1+1^𝑛=𝑞^2𝑛_𝑙=1𝑛𝑙𝐸_𝑞+1^𝑙+𝑞1+𝑞−𝐸_0,𝑞=𝑞+𝑞²−𝑞^𝑛_𝑙=0𝑛𝑙𝐸_𝑙,𝑞=𝑞+𝑞²−𝑞𝐸_𝑞+1^𝑛=𝑞+𝑞²+𝐸_𝑛,𝑞−𝑞[2]_𝑞𝛿_0,𝑛. (2.23)

Therefore, by (2.23), we obtain the following theorem.

Theorem 2.5. For 𝑛∈ℕ, one has 𝑞²𝐸_𝑛,𝑞(2)=𝑞+𝑞²+𝐸_𝑛,𝑞. (2.24)

For 𝑛∈ℤ₊, we have𝐸_{𝑛,𝑞−1}(1−𝑥)=_ℤ𝑝1−𝑥+𝑥₁^𝑛𝑑𝜇_−𝑞−1𝑥₁=(−1)^𝑛_ℤ𝑝𝑥₁+𝑥^𝑛𝑑𝜇_−𝑞𝑥₁=(−1)^𝑛𝐸_𝑛,𝑞(𝑥). (2.25)

Therefore, by (2.25), we obtain the following theorem.

Theorem 2.6. For 𝑛∈ℤ₊, one has 𝐸_{𝑛,𝑞−1}(1−𝑥)=(−1)^𝑛𝐸_𝑛,𝑞(𝑥). (2.26)

From (2.20), we have_ℤ𝑝(1−𝑥)^𝑛𝑑𝜇_−𝑞(𝑥)=(−1)^𝑛_ℤ𝑝(𝑥−1)^𝑛𝑑𝜇_−𝑞(𝑥)=(−1)^𝑛𝐸_𝑛,𝑞(−1). (2.27)

By Theorem 2.6 and (2.27), we get_ℤ𝑝(1−𝑥)^𝑛𝑑𝜇_−𝑞(𝑥)=𝐸_{𝑛,𝑞−1}(2)=1+𝑞+𝑞²𝐸_{𝑛,𝑞−1}if𝑛>0. (2.28)

Therefore, by (2.28), we obtain the following theorem.

Theorem 2.7. For 𝑛∈ℕ, one has _ℤ𝑝(1−𝑥)^𝑛𝑑𝜇_−𝑞(𝑥)=1+𝑞+𝑞²𝐸_{𝑛,𝑞−1}. (2.29)

Let 𝐶(ℤ_𝑝) be the space of continuous functions on ℤ_𝑝. For 𝑓∈𝐶(ℤ_𝑝), 𝑝-adic analogue of Bernstein operator of order 𝑛 for 𝑓 is given by𝔹_𝑛𝑓∣𝑥=^𝑛_𝑘=0𝐵_𝑘,𝑛(𝑥)𝑓𝑘𝑛=^𝑛_𝑘=0𝑓𝑘𝑛𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘, (2.30) where 𝑛,𝑘∈ℤ₊ (see [1, 6, 7]).

For 𝑛,𝑘∈ℤ₊, 𝑝-adic Bernstein polynomial of degree 𝑛 is defined by𝐵_𝑘,𝑛(𝑥)=𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘,𝑥∈ℤ_𝑝 (2.31) (see [1, 6, 7]).

Let us take the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 for one Bernstein polynomials in (2.31) as follows:_ℤ𝑝𝐵_𝑘,𝑛(𝑥)𝑑𝜇_−𝑞(𝑥)=𝑛𝑘^ℤ𝑝𝑥^𝑘(1−𝑥)^𝑛−𝑘𝑑𝜇_−𝑞(𝑥)=𝑛𝑘^{𝑛−𝑘𝑙=0}𝑙()𝑛−𝑘−1^𝑙_ℤ𝑝𝑥^𝑘+𝑙𝑑𝜇_−𝑞(𝑥)=𝑛𝑘^{𝑛−𝑘𝑙=0}𝑙()𝑛−𝑘−1^𝑙𝐸_{𝑘+𝑙,𝑞}. (2.32)

By simple calculation, we easily get_ℤ𝑝𝐵_𝑘,𝑛(𝑥)𝑑𝜇_−𝑞(𝑥)=_ℤ𝑝𝐵_{𝑛−𝑘,𝑛}(1−𝑥)𝑑𝜇_−𝑞(𝑥)=𝑛𝑘^𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙_ℤ𝑝(1−𝑥)^𝑛−𝑙𝑑𝜇_−𝑞(𝑥)=𝑛𝑘^𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙1+𝑞+𝑞²𝐸_{𝑛−𝑙,𝑞−1}=𝑛𝑘^𝑘𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝑞²𝐸_{𝑛−𝑙,𝑞−1}+[2]_𝑞𝑛𝑘(−1)^𝑘𝛿_0,𝑘if𝑛>𝑘. (2.33)

Therefore, by (2.32) and (2.33), we obtain the following theorem.

Theorem 2.8. For 𝑛∈ℤ₊ with 𝑛>𝑘>0, one has ^{𝑛−𝑘𝑙=0}𝑙()𝑛−𝑘−1^𝑙𝐸_{𝑘+𝑙,𝑞}=^𝑘_𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝑞²𝐸_{𝑛−𝑙,𝑞−1}. (2.34)

In particular, 𝑘=0, we get^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝐸_𝑙,𝑞=𝑞²𝐸_{𝑛,𝑞−1}+[2]_𝑞. (2.35)

By Theorems 2.1 and 2.2, we get^{𝑛−𝑘𝑙=0}𝑙𝑛−𝑘(−1)^𝑙𝐻_𝑘+𝑙−𝑞⁻¹=^𝑘_𝑙=0𝑘𝑙(−1)^𝑘+𝑙𝑞²𝐻_𝑛−𝑙−𝑞, (2.36) where 𝑛,𝑘∈ℤ₊ with 𝑛>𝑘>0.