Some Identities on Bernstein Polynomials Associated with 𝑞-Euler Polynomials

Abstract

We investigate some interesting properties of the 𝑞-Euler polynomials. The purpose of this paper is to give some relationships between Bernstein and 𝑞-Euler polynomials, which are derived by the 𝑝-adic integral representation of the Bernstein polynomials associated with 𝑞-Euler polynomials.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 denote the ring of 𝑝-adic integers, the field of 𝑝-adic numbers, and the field of 𝑝-adic completion of the algebraic closure of ℚ_𝑝, respectively (see [1–15]). Let ℕ be the set of natural numbers and ℤ₊=ℕ∪{0}. The normalized 𝑝-adic absolute value is defined by |𝑝|_𝑝=1/𝑝. As an indeterminate, we assume that 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1. Let UD(ℤ_𝑝) be the space of uniformly differentiable function on ℤ_𝑝. For 𝑓∈UD(ℤ_𝑝), the 𝑝-adic invariant integral on ℤ_𝑝 is defined by 𝐼₋₁(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=lim_{𝑝𝑁→∞}^𝑁−1_𝑥=0=lim_{𝑝𝑁→∞𝑁−1}𝑓(𝑥)𝜇₋₁𝑥+𝑝^𝑁ℤ_𝑝_𝑥=0𝑓(𝑥)(−1)^𝑥,(1.1) (see [7–10]). For 𝑛∈ℕ, we can derive the following integral equation from (1.1):𝐼₋₁𝑓_𝑛=(−1)^𝑛_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)+2^𝑛−1_𝑙=0(−1)^{𝑛−1−𝑙}𝑓(𝑙),(1.2) where 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛) (see [7–11]). As well-known definition, the Euler polynomials are given by the generating function as follows:2𝑒^𝑡+1𝑒^𝑥𝑡=𝑒^{𝐸(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡𝑛!,(1.3) (see [3, 5, 7–15]), with usual convention about replacing 𝐸^𝑛(𝑥) by 𝐸_𝑛(𝑥). In the special case 𝑥=0, 𝐸_𝑛(0)=𝐸_𝑛 are called the 𝑛th Euler numbers. From (1.3), we can derive the following recurrence formula for Euler numbers: 𝐸₀=1,(𝐸+1)^𝑛+𝐸_𝑛=20if𝑛=0,if𝑛>0,(1.4) (see [12]), with usual convention about replacing 𝐸^𝑛 by 𝐸_𝑛. By the definitions of Euler numbers and polynomials, we get 𝐸_𝑛(𝑥)=(𝐸+𝑥)^𝑛=^𝑛_𝑙=0𝑛𝑙𝑥^𝑛−𝑙𝐸_𝑙,(1.5) (see [3, 5, 7–15]). Let 𝐶[0,1] denote the set of continuous functions on [0,1]. For 𝑓∈𝐶[0,1], Bernstein introduced the following well-known linear positive operator in the field of real numbers ℝ: 𝔹_𝑛(𝑓∣𝑥)=^𝑛_𝑘=0𝑓𝑘𝑛𝑘𝑛𝑥^𝑘(1−𝑥)^𝑛−𝑘=^𝑛_𝑘=0𝑓𝑘𝑛𝐵^𝑘,𝑛(𝑥),(1.6) where (^𝑛𝑘)=𝑛(𝑛−1)⋯(𝑛−𝑘+1)/𝑘!=𝑛!/𝑘!(𝑛−𝑘)! (see [1, 2, 7, 11, 12, 14]). Here, 𝔹_𝑛(𝑓∣𝑥) is called the Bernstein operator of order 𝑛 for 𝑓. For 𝑘,𝑛∈ℤ₊, the Bernstein polynomials of degree 𝑛 are defined by𝐵_𝑘,𝑛(𝑥)=𝑘𝑛𝑥^𝑘(1−𝑥)^𝑛−𝑘,for𝑥∈[0,1].(1.7) In this paper, we study the properties of 𝑞-Euler numbers and polynomials. From these properties, we investigate some identities on the 𝑞-Euler numbers and polynomials. Finally, we give some relationships between Bernstein and 𝑞-Euler polynomials, which are derived by the 𝑝-adic integral representation of the Bernstein polynomials associated with 𝑞-Euler polynomials.

2. 𝑞-Euler Numbers and Polynomials

In this section, we assume that 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1. Let 𝑓(𝑥)=𝑞^𝑥𝑒^𝑥𝑡. From (1.1) and (1.2), we have_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=2𝑞𝑒^𝑡+1.(2.1) Now, we define the 𝑞-Euler numbers as follows:2𝑞𝑒^𝑡+1=𝑒^𝐸𝑞𝑡=^∞_𝑛=0𝐸_𝑛,𝑞𝑡^𝑛𝑛!,(2.2) with the usual convention about replacing 𝐸^𝑛𝑞 by 𝐸_𝑛,𝑞.

By (2.2), we easily get𝐸_0,𝑞=2𝑞+1,𝑞𝐸^𝑞+1^𝑛+𝐸_𝑛,𝑞=20if𝑛=0,if𝑛>0,(2.3) with usual convention about replacing 𝐸^𝑛𝑞 by 𝐸_𝑛,𝑞.

We note that2𝑞𝑒^𝑡+1=𝑒^𝑡+2𝑞⁻¹⋅21+𝑞=1+2𝑞^∞_𝑛=0𝐻_𝑛−𝑞⁻¹𝑡^𝑛𝑛!,(2.4) where 𝐻_𝑛(−𝑞⁻¹) is the 𝑛th Frobenius-Euler numbers.

From (2.1), (2.2), and (2.4), we have_ℤ𝑝𝑞^𝑥𝑒^𝑥𝑡𝑑𝜇₋₁(𝑥)=𝐸_𝑛,𝑞=21+𝑞𝐻^𝑛−𝑞⁻¹,for𝑛∈ℤ₊.(2.5) Now, we consider the 𝑞-Euler polynomials as follows:2𝑞𝑒^𝑡+1𝑒^𝑥𝑡=𝑒^{𝐸𝑞(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛,𝑞(𝑥)𝑡^𝑛𝑛!,(2.6) with the usual convention 𝐸^𝑛𝑞(𝑥) by 𝐸_𝑛,𝑞(𝑥).

From (1.2), (2.1), and (2.6), we get_ℤ𝑝𝑞^𝑥𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇₋₁(𝑦)=2𝑞𝑒^𝑡+1𝑒^𝑥𝑡=^∞_𝑛=0𝐸_𝑛,𝑞(𝑥)𝑡^𝑛𝑛!.(2.7) By comparing the coefficients on the both sides of (2.6) and (2.7), we get the following Witt's formula for the 𝑞-Euler polynomials as follows:_ℤ𝑝𝑞^𝑦(𝑥+𝑦)^𝑛𝑑𝜇₋₁(𝑦)=𝐸_𝑛,𝑞(𝑥)=^𝑛_𝑙=0𝑛𝑙𝑥^𝑛−𝑙𝐸_𝑙,𝑞.(2.8) From (2.6) and (2.8), we can derive the following equation:2𝑞𝑞𝑒^𝑡+1𝑒^{(1−𝑥)𝑡}=21+𝑞⁻¹𝑒^−𝑡𝑒^−𝑥𝑡=^∞_𝑛=0𝐸_{𝑛,𝑞−1}(𝑥)(−1)^𝑛𝑡𝑛!.(2.9) By (2.6) and (2.9), we obtain the following reflection symmetric property for the 𝑞-Euler polynomials.

Theorem 2.1. For 𝑛∈ℤ₊, one has (−1)^𝑛𝐸_{𝑛,𝑞−1}(𝑥)=𝑞𝐸_𝑛,𝑞(1−𝑥).(2.10)

From (2.5), (2.6), (2.7), and (2.8), we can derive the following equation: for 𝑛∈ℕ, 𝐸_𝑛,𝑞(2)=𝐸_𝑞+1+1^𝑛=^𝑛_𝑙=0𝑛𝑙𝐸^𝑙,𝑞=𝐸_0,𝑞(1)+1𝑞^𝑛_𝑙=1𝑛𝑙𝑞𝐸^𝑙,𝑞(1)=21+𝑞−1𝑞^𝑛_𝑙=1𝑛𝑙𝐸^𝑙,𝑞=21+𝑞+𝑞(12+𝑞)−1𝑞^𝑛_𝑙=0𝑛𝑙𝐸^𝑙,𝑞=2𝑞−𝑞1²𝑞𝐸_𝑛,𝑞(1)=2𝑞+𝑞1²𝐸_𝑛,𝑞,(2.11) by using recurrence formula (2.3). Therefore, we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℕ, one has 𝑞𝐸_𝑛,𝑞(2)=2+1𝑞𝐸^𝑛,𝑞.(2.12)

By using (2.5) and (2.8), we get _ℤ𝑝𝑞^−𝑥(1−𝑥)^𝑛𝑑𝜇₋₁(𝑥)=(−1)^𝑛_ℤ𝑝=(−1)_𝑛𝑞^−𝑥(𝑥−1)^𝑛𝑑𝜇₋₁(𝑥)𝐸_{𝑛,𝑞−1}(−1)=𝑞_ℤ𝑝(𝑥+2)^𝑛𝑑𝜇₋₁(𝑥)=𝑞2𝑞+𝑞1²𝐸_𝑛,𝑞=2+1𝑞𝐸^𝑛,𝑞=2+1𝑞_ℤ𝑝𝑥^𝑛𝑞^𝑥𝑑𝜇₋₁(𝑥),for𝑛>0.(2.13) Therefore, we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℕ, one has _ℤ𝑝𝑞^−𝑥(1−𝑥)^𝑛𝑑𝜇₋₁(𝑥)=2+1𝑞_ℤ𝑝𝑥^𝑛𝑞^𝑥𝑑𝜇₋₁(𝑥).(2.14)

By using Theorem 2.3, we will study for the 𝑝-adic integral representation on ℤ_𝑝 of the Bernstein polynomials associated with 𝑞-Euler polynomials in Section 3.

3. Bernstein Polynomials Associated with 𝑞-Euler Numbers and Polynomials

Now, we take the 𝑝-adic integral on ℤ_𝑝 for the Bernstein polynomials in (1.7) as follows:_ℤ𝑝𝐵_𝑘,𝑛(𝑥)𝑞^𝑥𝑑𝜇₋₁(𝑥)=_ℤ𝑝𝑘𝑛𝑥^𝑘=𝑘𝑛(1−𝑥)^𝑛−𝑘𝑞^𝑥𝑑𝜇₋₁(𝑥)_𝑛−𝑘_𝑗=0𝑛−𝑗𝑘(−1)^{𝑛−𝑘−𝑗}_ℤ𝑝=𝑘𝑛𝑥^𝑛−𝑗𝑞^𝑥𝑑𝜇₋₁(𝑥)_𝑛−𝑘_𝑗=0𝑛−𝑗𝑘(−1)^{𝑛−𝑘−𝑗}𝐸_{𝑛−𝑗,𝑞}=𝑘𝑛^𝑛−𝑘_𝑗=0𝑛−𝑗𝑘(−1)^𝑗𝐸_{𝑘+𝑗,𝑞},where𝑛,𝑘∈ℤ₊.(3.1) By the definition of Bernstein polynomials, we see that𝐵_𝑘,𝑛(𝑥)=𝐵_{𝑛−𝑘,𝑛}(1−𝑥),where𝑛,𝑘∈ℤ₊.(3.2) Let 𝑛,𝑘∈ℤ₊ with 𝑛>𝑘. Then, by (3.2), we get _ℤ𝑝𝑞^𝑥𝐵_𝑘,𝑛(𝑥)𝑑𝜇₋₁(𝑥)==𝑛_ℤ𝑝𝑞−𝑛^𝑥𝑘𝐵_{𝑛−𝑘,𝑛𝑘}(1−𝑥)𝑑𝜇₋₁(𝑥)_𝑗=0𝑘𝑗(−1)^𝑘−𝑗_ℤ𝑝=𝑘𝑛(1−𝑥)^𝑛−𝑗𝑞^𝑥𝑑𝜇₋₁(𝑥)_𝑘_𝑗=0𝑘𝑗(−1)^𝑘−𝑗2+𝑞_ℤ𝑝=𝑘𝑛𝑥^𝑛−𝑗𝑞^𝑥𝑑𝜇₋₁(𝑥)_𝑘=⎧⎪⎨⎪⎩2+𝑞𝐸_𝑗=0𝑘𝑗_{𝑛,𝑞−1}(−1)^𝑘−𝑗2+𝑞𝐸_{𝑛−𝑗,𝑞}⁻¹𝑘𝑛𝑞if𝑘=0,_𝑘_𝑗=0𝑘𝑗(−1)^𝑘−𝑗𝐸_{𝑛−𝑗,𝑞−1}if𝑘>0.(3.3) Thus, we obtain the following theorem.

Theorem 3.1. For 𝑛,𝑘∈ℤ₊ with 𝑛>𝑘, one has _ℤ𝑝𝑞^1−𝑥𝐵_𝑘,𝑛(𝑥)𝑑𝜇₋₁(𝑥)=⎧⎪⎨⎪⎩2𝑞+𝐸^𝑛,𝑞𝑘𝑛if𝑘=0,_𝑘_𝑗=0𝑘𝑗(−1)^𝑘−𝑗𝐸_{𝑛−𝑗,𝑞}if𝑘>0.(3.4)

By (3.1) and Theorem 3.1, we get the following corollary.

Corollary 3.2. For 𝑛,𝑘∈ℤ₊ with 𝑛>𝑘, one has ^𝑛−𝑘_𝑗=0𝑛−𝑗𝑘(−1)^𝑗𝐸_{𝑘+𝑗,𝑞−1}=⎧⎪⎪⎨⎪⎪⎩2+1𝑞𝐸^𝑛,𝑞if𝑘=0,_𝑘_𝑗=0𝑘𝑗(−1)^𝑘−𝑗1𝑞𝐸^{𝑛−𝑗,𝑞}if𝑘>0.(3.5)

For 𝑚,𝑛,𝑘∈ℤ₊ with 𝑚+𝑛>2𝑘. Then, we get _ℤ𝑝𝐵=_𝑘,𝑛(𝑘𝑛𝑥)𝐵𝑚𝑘_𝑘,𝑚(𝑥)_2𝑘𝑞^−𝑥𝑑𝜇₋₁(𝑥)_𝑗=02𝑗𝑘(−1)^𝑗+2𝑘_ℤ𝑝=𝑘𝑛𝑚𝑘_2𝑘𝑞^−𝑥(1−𝑥)^{𝑛+𝑚−𝑗}𝑑𝜇₋₁(𝑥)_𝑗=02𝑗𝑘(−1)^𝑗+2𝑘=𝑘𝑛𝑚𝑘_2𝑘𝑞_ℤ𝑝(𝑥+2)^{𝑛+𝑚−𝑗}𝑞^𝑥𝑑𝜇₋₁(𝑥)_𝑗=02𝑗𝑘(−1)^𝑗+2𝑘𝑞2𝑞+𝑞1²𝐸_{𝑛+𝑚−𝑗,𝑞}=⎧⎪⎪⎨⎪⎪⎩2+1𝑞𝐸^{𝑛+𝑚,𝑞}𝑘𝑛𝑚𝑘_2𝑘if𝑘=0,_𝑗=02𝑗𝑘(−1)^𝑗+2𝑘1𝑞𝐸^{𝑛+𝑚−𝑗,𝑞}if𝑘>0.(3.6) Therefore, we obtain the following theorem.

Theorem 3.3. For 𝑚,𝑛,𝑘∈ℤ₊ with 𝑚+𝑛>2𝑘, one has _ℤ𝑝𝐵_𝑘,𝑛(𝑥)𝐵_𝑘,𝑚(𝑥)𝑞^1−𝑥𝑑𝜇₋₁(𝑥)=⎧⎪⎨⎪⎩2𝑞+𝐸^{𝑛+𝑚,𝑞}𝑘𝑛𝑚𝑘_2𝑘if𝑘=0,_𝑗=02𝑗𝑘(−1)^𝑗+2𝑘𝐸_{𝑛+𝑚−𝑗,𝑞}if𝑘>0.(3.7)

By using binomial theorem, for 𝑚,𝑛,𝑘∈ℤ₊, we get_ℤ𝑝𝐵_𝑘,𝑛(𝑥)𝐵_𝑘,𝑚(𝑥)𝑞^1−𝑥𝑑𝜇₋₁(𝑥)(3.8)=𝑘𝑛𝑚𝑘^{𝑛+𝑚−2𝑘}_𝑗=0𝑛+𝑚𝑗−2𝑘(−1)^𝑗_ℤ𝑝=𝑞𝑘𝑛𝑚𝑘_{𝑛+𝑚−2𝑘}𝑥^𝑗+2𝑘𝑞^1−𝑥𝑑𝜇₋₁(𝑥)_𝑗=0𝑛+𝑚𝑗−2𝑘(−1)^𝑗𝐸_{𝑗+2𝑘,𝑞−1}.(3.9) By comparing the coefficients on the both sides of (3.8) and Theorem 3.3, we obtain the following corollary.

Corollary 3.4. Let 𝑚,𝑛,𝑘∈ℤ₊ with 𝑚+𝑛>2𝑘. Then, we get ^𝑛+^𝑚−2𝑘_𝑗=0𝑛+𝑚𝑗−2𝑘(−1)^𝑗𝐸_{𝑗+2𝑘,𝑞−1}=⎧⎪⎪⎨⎪⎪⎩2+1𝑞𝐸^{𝑛+𝑚,𝑞}1if𝑘=0,𝑞^2𝑘_𝑗=02𝑗𝑘(−1)^𝑗+2𝑘𝐸_{𝑛+𝑚−𝑗,𝑞}if𝑘>0.(3.10)

For 𝑠∈ℕ, let 𝑛₁,𝑛₂,…,𝑛_𝑠, 𝑘∈ℤ₊ with 𝑛₁+𝑛₂+⋯+𝑛_𝑠>𝑠𝑘. By induction, we get _ℤ𝑝𝐵=_𝑘,𝑛1(𝑥)⋯𝐵_{𝑘,𝑛𝑠}(𝑥)𝑞^−𝑥𝑑𝜇₋₁(𝑥)_𝑠_𝑖=1𝑛𝑘^𝑖_ℤ𝑝=𝑥^𝑠𝑘(1−𝑥)^{𝑛1+⋯+𝑛𝑠−𝑠𝑘}𝑞^−𝑥𝑑𝜇₋₁(𝑥)_𝑠_𝑖=1𝑛𝑘^𝑖^𝑠𝑘_𝑗=0𝑠𝑗𝑘(−1)^{𝑠𝑘+𝑗}_ℤ𝑝=(1−𝑥)^{𝑛1+⋯+𝑛𝑠−𝑗}𝑞^−𝑥𝑑𝜇₋₁(𝑥)_𝑠_𝑖=1𝑛𝑘^𝑖^𝑠𝑘_𝑗=0𝑠𝑗𝑘(−1)^{𝑠𝑘+𝑗}=𝑞_ℤ𝑝(𝑥+2)^{𝑛1+⋯+𝑛𝑠−𝑗}𝑞^𝑥𝑑𝜇₋₁(𝑥)_𝑠_𝑖=1𝑛𝑘^𝑖^𝑠𝑘_𝑗=0𝑠𝑗𝑘(−1)^{𝑠𝑘+𝑗}𝑞2𝑞+𝑞1²𝐸_{𝑛1+⋯+𝑛𝑠−𝑗,𝑞}=⎧⎪⎪⎨⎪⎪⎩2+1𝑞𝐸^{𝑛1+⋯+𝑛𝑠,𝑞}if𝑘=0,_𝑠_𝑖=1𝑛𝑘^𝑖1𝑞^𝑠𝑘_𝑗=0𝑠𝑗𝑘(−1)^{𝑠𝑘+𝑗}𝐸_{𝑛1+⋯+𝑛𝑠−𝑗,𝑞}if𝑘>0.(3.11) Therefore, we obtain the following theorem.

Theorem 3.5. Let 𝑠∈ℕ. For 𝑛₁,𝑛₂,…,𝑛_𝑠,𝑘∈ℤ₊ with 𝑛₁+𝑛₂+⋯+𝑛_𝑠>𝑠𝑘, one has _ℤ𝑝^𝑠_𝑖=1𝐵_{𝑘,𝑛𝑖}(𝑥)𝑞^1−𝑥𝑑𝜇₋₁(𝑥)=⎧⎪⎨⎪⎩2𝑞+𝐸^{𝑛1+𝑛2+⋯+𝑛𝑠,𝑞}if𝑘=0,_𝑠_𝑖=1𝑛𝑘^𝑖^𝑠𝑘_𝑗=0𝑠𝑗𝑘(−1)^{𝑠𝑘+𝑗}𝐸_{𝑛1+𝑛2+⋯+𝑛𝑠−𝑗,𝑞}if𝑘>0.(3.12)

For 𝑛₁,𝑛₂,…,𝑛_𝑠,𝑘∈ℤ₊ by binomial theorem, we get_ℤ𝑝^𝑠_𝑖=1=𝑛𝐵_{𝑘,𝑛𝑖}(𝑥)𝑞^−𝑥𝑑𝜇₋₁(𝑥)₁𝑘⋯𝑛𝑘^𝑠^{𝑛1+⋯+𝑛𝑠−𝑠𝑘}_𝑗=0𝑛¹+⋯+𝑗𝑛_𝑠−𝑠𝑘(−1)_𝑗_ℤ𝑝=𝑛𝑥^{𝑗+𝑠𝑘}𝑞^−𝑥𝑑𝜇₋₁(𝑥)₁𝑘⋯𝑛𝑘^𝑠^{𝑛1+⋯+𝑛𝑠−𝑠𝑘}_𝑗=0𝑛¹+⋯+𝑗𝑛_𝑠−𝑠𝑘(−1)_𝑗𝐸_{𝑗+𝑠𝑘,𝑞−1}.(3.13)

By using (3.13) and Theorem 3.5, we obtain the following corollary.

Corollary 3.6. Let 𝑠∈ℕ. For 𝑛₁,𝑛₂,…,𝑛_𝑠,𝑘∈ℤ₊ with 𝑛₁+𝑛₂+⋯+𝑛_𝑠>𝑠𝑘, one has ^𝑛1+⋯^{+𝑛𝑠−𝑠𝑘}_𝑗=0𝑛¹+⋯+𝑗𝑛_𝑠−𝑠𝑘(−1)_𝑗𝐸_{𝑗+𝑠𝑘,𝑞−1}=⎧⎪⎪⎨⎪⎪⎩2+1𝑞𝐸^{𝑛1+𝑛2+⋯+𝑛𝑠,𝑞}1if𝑘=0,𝑞^𝑠𝑘_𝑗=0𝑠𝑗𝑘(−1)^{𝑠𝑘+𝑗}𝐸_{𝑛1+𝑛2+⋯+𝑛𝑠−𝑗,𝑞}if𝑘>0.(3.14)