On 𝑞-Euler Numbers Related to the Modified 𝑞-Bernstein Polynomials

Abstract

We consider q-Euler numbers, polynomials, and q-Stirling numbers of first and second kinds. Finally, we investigate some interesting properties of the modified q-Bernstein polynomials related to q-Euler numbers and q-Stirling numbers by using fermionic p-adic integrals on ℤ_𝑝.

1. Introduction

Let 𝐶[0,1] be the set of continuous functions on [0,1]. The classical Bernstein polynomials of degree 𝑛 for 𝑓∈𝐶[0,1] are defined by𝔹_𝑛(𝑓)=^𝑛_𝑘=0𝑘𝑓(𝑛)𝐵_𝑘,𝑛(𝑥),0≤𝑥≤1, (1.1) where 𝔹_𝑛(𝑓) is called the Bernstein operator and𝐵_𝑘,𝑛𝑛𝑘(𝑥)=𝑥^𝑘(𝑥−1)^𝑛−𝑘 (1.2) are called the Bernstein basis polynomials (or the Bernstein polynomials of degree 𝑛) (see [1]). Recently, Acikgoz and Araci have studied the generating function for Bernstein polynomials (see [2, 3]). Their generating function for 𝐵_𝑘,𝑛(𝑥) is given by𝐹^(𝑘)𝑡(𝑡,𝑥)=^𝑘𝑒^{(1−𝑥)𝑡}𝑥^𝑘𝑘!=^∞_𝑛=0𝐵_𝑘,𝑛𝑡(𝑥)^𝑛,𝑛! (1.3) where 𝑘=0,1,… and 𝑥∈[0,1]. Note that 𝐵_𝑘,𝑛𝑛𝑘𝑥(𝑥)=^𝑘(1−𝑥)^𝑛−𝑘,if𝑛≥𝑘,0,if𝑛<𝑘, (1.4) for 𝑛=0,1,… (see [2, 3]).

Let 𝑝 be an 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 the algebraic closure of ℚ_𝑝, respectively. Let 𝑣_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝⁻¹.

Throughout this paper, we use the following notation: [𝑥]_𝑞=1−𝑞^𝑥1−𝑞,[𝑥]_−𝑞=1−(−𝑞)^𝑥1+𝑞 (1.5) (cf. [4–7]). Let ℕ be the natural numbers and ℤ₊=ℕ∪{0}. Let UD(ℤ_𝑝) be the space of uniformly differentiable function on ℤ_𝑝.

Let 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<𝑝^{−1/(𝑝−1)} and 𝑥∈ℤ_𝑝. Then 𝑞-Bernstein type operator for 𝑓∈UD(ℤ_𝑝) is defined by (see [8, 9])𝔹_𝑛,𝑞(𝑓)=^𝑛_𝑘=0𝑘𝑓(𝑛)𝑛𝑘[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘=^𝑛_𝑘=0𝑘𝑓(𝑛)𝐵_𝑘,𝑛(𝑥,𝑞), (1.6) for 𝑘,𝑛∈ℤ₊, where𝐵_𝑘,𝑛𝑛𝑘(𝑥,𝑞)=[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘 (1.7) is called the modified 𝑞-Bernstein polynomials of degree 𝑛. When we put 𝑞→1 in (1.7), [𝑥]^𝑘_𝑞→𝑥^𝑘,[1−𝑥]_𝑞^𝑛−𝑘→(1−𝑥)^𝑛−𝑘, and we obtain the classical Bernstein polynomial, defined by (1.2). We can deduce very easily from (1.7) that𝐵_𝑘,𝑛(𝑥,𝑞)=[1−𝑥]_𝑞𝐵_{𝑘,𝑛−1}(𝑥,𝑞)+[𝑥]_𝑞𝐵_{𝑘−1,𝑛−1}(𝑥,𝑞) (1.8) (see [8]). For 0≤𝑘≤𝑛, derivatives of the 𝑛th degree modified 𝑞-Bernstein polynomials are polynomials of degree 𝑛−1:dd𝑥𝐵_𝑘,𝑛(𝑥,𝑞)=𝑛(𝑞^𝑥𝐵_{𝑘−1,𝑛−1}(𝑥,𝑞)−𝑞^1−𝑥𝐵_{𝑘,𝑛−1}(𝑥,𝑞))ln𝑞𝑞−1 (1.9) (see [8]).

The Bernstein polynomials can also be defined in many different ways. Thus, recently, many applications of these polynomials have been looked for by many authors. In the recent years, the 𝑞-Bernstein polynomials have been investigated and studied by many authors in many different ways (see [1, 8, 9] and references therein [10, 11]). In [11], Phillips gave many results concerning the 𝑞-integers and an account of the properties of 𝑞-Bernstein polynomials. He gave many applications of these polynomials on approximation theory. In [2, 3], Acikgoz and Araci have introduced several type Bernstein polynomials. The Acikgoz and Araci paper to announced in the conference is actually motivated to write this paper. In [1], Simsek and Acikgoz constructed a new generating function of the 𝑞-Bernstein type polynomials and established elementary properties of this function. In [8], Kim et al. proposed the modified 𝑞-Bernstein polynomials of degree 𝑛, which are different 𝑞-Bernstein polynomials of Phillips. In [9], Kim et al. investigated some interesting properties of the modified 𝑞-Bernstein polynomials of degree 𝑛 related to 𝑞-Stirling numbers and Carlitz's 𝑞-Bernoulli numbers.

In the present paper, we consider 𝑞-Euler numbers, polynomials, and 𝑞-Stirling numbers of first and second kinds. We also investigate some interesting properties of the modified 𝑞-Bernstein polynomials of degree 𝑛 related to 𝑞-Euler numbers and 𝑞-Stirling numbers by using fermionic 𝑝-adic integrals on ℤ_𝑝.

2. 𝑞-Euler Numbers and Polynomials Related to the Fermionic 𝑝-Adic Integrals on ℤ_𝑝

For 𝑁≥1, the fermionic 𝑞-extension 𝜇_𝑞 of the 𝑝-adic Haar distribution 𝜇_Haar,𝜇_−𝑞(𝑎+𝑝^𝑁ℤ_𝑝)=(−𝑞)^𝑎[𝑝^𝑁]_−𝑞, (2.1) is known as a measure on ℤ_𝑝, where 𝑎+𝑝^𝑁ℤ_𝑝={𝑥∈ℚ_𝑝∣|𝑥−𝑎|_𝑝≤𝑝^−𝑁} (cf. [4, 12]). We will write 𝑑𝜇_−𝑞(𝑥) to remind ourselves that 𝑥 is the variable of integration. Let UD(ℤ_𝑝) be the space of uniformly differentiable function on ℤ_𝑝. Then 𝜇_−𝑞 yields the fermionic 𝑝-adic 𝑞-integral of a function 𝑓∈UD(ℤ_𝑝):𝐼_−𝑞(𝑓)=_Z𝑝𝑓(𝑥)d𝜇_−𝑞(𝑥)=lim_𝑁→∞1+𝑞1+𝑞^{𝑝𝑁𝑝𝑁−1𝑥=0}𝑓(𝑥)(−𝑞)^𝑥 (2.2) (cf. [12–15]). Many interesting properties of (2.2) were studied by many authors (see [12, 13] and the references given there). Using (2.2), we have the fermionic 𝑝-adic invariant integral on ℤ_𝑝 as follows:lim_𝑞→−1𝐼_𝑞(𝑓)=𝐼₋₁(𝑓)=_Z𝑝𝑓(𝑎)d𝜇₋₁(𝑥). (2.3) For 𝑛∈ℕ, write 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛). We have𝐼₋₁(𝑓_𝑛)=(−1)^𝑛𝐼₋₁(𝑓)+2^𝑛−1_𝑙=0(−1)^{𝑛−𝑙−1}𝑓(𝑙). (2.4) This identity is obtained by Kim in [12] to derive interesting properties and relationships involving 𝑞-Euler numbers and polynomials. For 𝑛∈ℤ₊, we note that𝐼₋₁([𝑥]^𝑛_𝑞)=_ℤ𝑝[𝑥]^𝑛_𝑞d𝜇₋₁(𝑥)=𝐸_𝑛,𝑞, (2.5) where 𝐸_𝑛,𝑞 are the 𝑞-Euler numbers (see [16]). It is easy to see that 𝐸_0,𝑞=1. For 𝑛∈ℕ, we have_𝑛𝑙=0𝑛𝑙𝑞^𝑙𝐸_𝑙,𝑞=^𝑛_𝑙=0𝑛𝑙𝑞^𝑙lim_𝑁→∞^𝑝𝑁−1_𝑥=0[𝑥]^𝑙_𝑞(−1)^𝑥=lim_𝑁→∞^𝑝𝑁−1_𝑥=0(−1)^𝑥(𝑞[𝑥]_𝑞+1)^𝑛=lim_𝑁→∞^𝑝𝑁−1_𝑥=0(−1)^𝑥[𝑥+1]^𝑛_𝑞=−lim_𝑁→∞^𝑝𝑁−1_𝑥=0(−1)^𝑥([𝑥]^𝑛_𝑞+[𝑝^𝑁]^𝑛_𝑞)=−𝐸_𝑛,𝑞. (2.6) From this formula, we have the following recurrence formula:𝐸_0,𝑞=1,(𝑞𝐸+1)^𝑛+𝐸_𝑛,𝑞=0if𝑛∈N, (2.7) with the usual convention of replacing 𝐸^𝑙 by 𝐸_𝑙,𝑞. By the simple calculation of the fermionic 𝑝-adic invariant integral on ℤ_𝑝, we see that𝐸_𝑛,𝑞=2(1−𝑞)^{𝑛𝑛𝑙=0}𝑛𝑙(−1)^𝑙11+𝑞^𝑙, (2.8) where (^𝑛_𝑙)=𝑛!/𝑙!(𝑛−𝑙)!=𝑛(𝑛−1)⋯(𝑛−𝑙+1)/𝑙!. Now, by introducing the following equations:[𝑥]^𝑛_1/𝑞=𝑞^𝑛𝑞^−𝑛𝑥[𝑥]^𝑛_𝑞,𝑞^−𝑛𝑥=^∞_𝑚=0(1−𝑞)^𝑚𝑚𝑛+𝑚−1[𝑥]^𝑚_𝑞 (2.9) into (2.5), we find that𝐸_𝑛,1/𝑞=𝑞^𝑛∞_𝑚=0(1−𝑞)^𝑚𝑚𝐸𝑛+𝑚−1_{𝑛+𝑚,𝑞}. (2.10) This identity is a peculiarity of the 𝑝-adic 𝑞-Euler numbers, and the classical Euler numbers do not seem to have a similar relation. Let 𝐹_𝑞(𝑡) be the generating function of the 𝑞-Euler numbers. Then we obtain that𝐹_𝑞(𝑡)=^∞_𝑛=0𝐸_𝑛,𝑞𝑡^𝑛𝑛!=^∞_𝑛=02(1−𝑞)^{𝑛𝑛𝑙=0}(−1)^𝑙𝑛𝑙11+𝑞^𝑙𝑡^𝑛𝑛!=2𝑒^{𝑡/(1−𝑞)∞}_𝑘=0(−1)^𝑘(1−𝑞)^𝑘11+𝑞^𝑘𝑡^𝑘.𝑘! (2.11) From (2.11), we note that𝐹_𝑞(𝑡)=2𝑒^{𝑡/(1−𝑞)∞}_𝑛=0(−1)^𝑛𝑒^{(−𝑞𝑛/(1−𝑞))𝑡}=2^∞_𝑛=0(−1)^𝑛𝑒^{[𝑛]𝑞𝑡}. (2.12) It is well known that𝐼₋₁([𝑥+𝑦]^𝑛)=_ℤ𝑝[𝑥+𝑦]^𝑛d𝜇₋₁(𝑦)=𝐸_𝑛,𝑞(𝑥), (2.13) where 𝐸_𝑛,𝑞(𝑥) are the 𝑞-Euler polynomials (see [16]). In the special case 𝑥=0, the numbers 𝐸_𝑛,𝑞(0)=𝐸_𝑛,𝑞 are referred to as the 𝑞-Euler numbers. Thus, we have_ℤ𝑝[𝑥+𝑦]^𝑛d𝜇₋₁(𝑦)=^𝑛_𝑘=0𝑛𝑘[𝑥]_𝑞^𝑛−𝑘𝑞^𝑘𝑥_ℤ𝑝[𝑦]^𝑘d𝜇₋₁(𝑦)=^𝑛_𝑘=0𝑛𝑘[𝑥]_𝑞^𝑛−𝑘𝑞^𝑘𝑥𝐸_𝑘,𝑞=(𝑞^𝑥𝐸+[𝑥]_𝑞)^𝑛. (2.14) It is easily verified, using (2.12) and (2.13), that the 𝑞-Euler polynomials 𝐸_𝑛,𝑞(𝑥) satisfy the following formula:^∞_𝑛=0𝐸_𝑛,𝑞𝑡(𝑥)^𝑛=𝑛!_ℤ𝑝𝑒^{[𝑥+𝑦]𝑞𝑡}d𝜇₋₁(𝑦)=^∞_𝑛=02(1−𝑞)^{𝑛𝑛𝑙=0}(−1)^𝑙𝑛𝑙𝑞^𝑙𝑥1+𝑞^𝑙𝑡^𝑛𝑛!=2^∞_𝑛=0(−1)^𝑛𝑒^{[𝑛+𝑥]𝑞𝑡}. (2.15) Using formula (2.15), when 𝑞 tends to 1, we can readily derive the Euler polynomials, 𝐸_𝑛(𝑥), namely, _ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}d𝜇₋₁(𝑦)=2𝑒^𝑥𝑡𝑒^𝑡+1=^∞_𝑛=0𝐸_𝑛𝑡(𝑥)^𝑛𝑛! (2.16) (see [12]). Note that 𝐸_𝑛(0)=𝐸_𝑛 are referred to as the 𝑛th Euler numbers. Comparing the coefficients of 𝑡^𝑛/𝑛! on both sides of (2.15), we have𝐸_𝑛,𝑞(𝑥)=2^∞_𝑚=0(−1)^𝑚[𝑚+𝑥]^𝑛_𝑞=2(1−𝑞)^{𝑛𝑛𝑙=0}(−1)^𝑙𝑛𝑙𝑞^𝑙𝑥1+𝑞^𝑙. (2.17)

We refer to [𝑛]_𝑞 as a 𝑞-integer and note that [𝑛]_𝑞 is a continuous function of 𝑞. In an obvious way we also define a 𝑞-factorial, [𝑛]_𝑞!=[𝑛]_𝑞[𝑛−1]_𝑞⋯[1]_𝑞,𝑛∈ℕ,1,𝑛=0, (2.18) and a 𝑞-analogue of binomial coefficient,𝑥𝑛_𝑞=[𝑥]_𝑞![𝑥−𝑛]_𝑞![𝑛]_𝑞!=[𝑥]_𝑞[𝑥−1]_𝑞⋯[𝑥−𝑛+1]_𝑞[𝑛]_𝑞! (2.19) (cf. [14, 16]). Note that lim_𝑞→1𝑥𝑛_𝑞=𝑥𝑛=𝑥(𝑥−1)⋯(𝑥−𝑛+1).𝑛! (2.20) It readily follows from (2.19) that𝑥𝑛_𝑞=(1−𝑞)^𝑛𝑞^𝑛2)−([𝑛]_𝑞!^𝑛𝑖=0𝑞^(𝑖2)𝑛𝑖_𝑞(−1)^𝑛+𝑖𝑞^{(𝑛−𝑖)𝑥} (2.21) (cf. [7, 16]). It can be readily seen that𝑞^𝑙𝑥=([𝑥]_𝑞(𝑞−1)+1)^𝑙=^𝑙_𝑚=0𝑙𝑚(𝑞−1)^𝑚[𝑥]^𝑚_𝑞. (2.22) Thus, by (2.13) and (2.22), we have_ℤ𝑝𝑥𝑛_𝑞d𝜇₋₁(𝑥)=(𝑞−1)^𝑛[𝑛]_𝑞!𝑞^{(𝑛2)𝑛𝑖=0}𝑞^(𝑖2)𝑛𝑖_𝑞(−1)^{𝑖𝑛−𝑖}_𝑗=0𝑗𝑛−𝑖(𝑞−1)^𝑗𝐸_𝑗,𝑞. (2.23) From now on, we use the following notation:[𝑥]_𝑞![𝑥−𝑘]_𝑞!=𝑞^{𝑘2)−(𝑘}_𝑙=0𝑠_1,𝑞(𝑘,𝑙)[𝑥]^𝑙_𝑞,𝑘∈ℤ₊,[𝑥]^𝑛_𝑞=^𝑛_𝑘=0𝑞^(𝑘2)𝑠_2,𝑞(𝑛,𝑘)[𝑥]_𝑞![𝑥−𝑘]_𝑞!,𝑛∈ℤ₊ (2.24) (see [7]). From (2.24), and (2.22), we calculate the following consequence:[𝑥]^𝑛_𝑞=^𝑛_𝑘=0𝑞^(𝑘2)𝑠_2,𝑞1(𝑛,𝑘)(1−𝑞)^{𝑘𝑘𝑙=0}𝑘𝑙_𝑞𝑞^(𝑙2)(−1)^𝑙𝑞^{𝑙(𝑥−𝑘+1)}=^𝑛_𝑘=0𝑞^(𝑘2)𝑠_2,𝑞1(𝑛,𝑘)(1−𝑞)^{𝑘𝑘𝑙=0}𝑘𝑙_𝑞𝑞^{(𝑙2)+𝑙(1−𝑘)}(−1)^𝑙×^𝑙_𝑚=0𝑙𝑚(𝑞−1)^𝑚[𝑥]^𝑚_𝑞=^𝑛_𝑘=0𝑞^(𝑘2)𝑠_2,𝑞1(𝑛,𝑘)(1−𝑞)^𝑘×^𝑘_𝑚=0(𝑞−1)^𝑚(^𝑘_𝑙=𝑚𝑘𝑙_𝑞𝑞^{(𝑙2)+𝑙(1−𝑘)}𝑙𝑚(−1)^𝑙)[𝑥]^𝑚_𝑞. (2.25) Therefore, we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊,𝐸_𝑛,𝑞=^𝑛_𝑘=0^𝑘_𝑚=0^𝑘_𝑙=𝑚𝑞^(𝑘2)𝑠_2,𝑞(𝑛,𝑘)(𝑞−1)^𝑚−𝑘𝑘𝑙_𝑞𝑞^{(𝑙2)+𝑙(1−𝑘)}𝑙𝑚(−1)^𝑙+𝑘𝐸_𝑚,𝑞. (2.26)

By (2.22) and simple calculation, we find that_𝑛𝑚=0𝑛𝑚(𝑞−1)^𝑚𝐸_𝑚,𝑞=_ℤ𝑝𝑞^𝑛𝑥d𝜇₋₁(𝑥)=^𝑛_𝑘=0(𝑞−1)^𝑘𝑞^(𝑘2)𝑛𝑘_𝑞^{ℤ𝑝𝑘−1𝑖=0}[𝑥−𝑖]_𝑞d𝜇₋₁(𝑥)=^𝑛_𝑘=0(𝑞−1)^𝑘𝑛𝑘_𝑞^𝑘𝑚=0𝑠_1,𝑞(𝑘,𝑚)_ℤ𝑝[𝑥]^𝑚_𝑞d𝜇₋₁(𝑥)=^𝑛_𝑚=0(^𝑛_𝑘=𝑚(𝑞−1)^𝑘𝑛𝑘_𝑞𝑠_1,𝑞(𝑘,𝑚))𝐸_𝑚,𝑞. (2.27) Therefore, we deduce the following theorem.

Theorem 2.2. For 𝑛∈ℤ₊,_𝑛𝑚=0𝑛𝑚(𝑞−1)^𝑚𝐸_𝑚,𝑞=^𝑛_𝑚=0^𝑛_𝑘=𝑚(𝑞−1)^𝑘𝑛𝑘_𝑞𝑠_1,𝑞(𝑘,𝑚)𝐸_𝑚,𝑞. (2.28)

Corollary 2.3. For 𝑚,𝑛∈ℤ₊ with 𝑚≤𝑛,𝑛𝑚(𝑞−1)^𝑚=^𝑛_𝑘=𝑚(𝑞−1)^𝑘𝑛𝑘_𝑞𝑠_1,𝑞(𝑘,𝑚). (2.29)

By (2.17) and Corollary 2.3, we obtain the following corollary.

Corollary 2.4. For 𝑛∈ℤ₊,𝐸_𝑛,𝑞2(𝑥)=(1−𝑞)^{𝑛𝑛𝑙=0𝑛𝑘=𝑙}(−1)^𝑙(𝑞−1)^𝑘−𝑙𝑛𝑘_𝑞𝑠_1,𝑞𝑞(𝑘,𝑙)^𝑙𝑥1+𝑞^𝑙. (2.30)

It is easy to see that𝑛𝑘_𝑞=_{𝑙0+⋯+𝑙𝑘=𝑛−𝑘}𝑞^{∑𝑘𝑖=0𝑖𝑙𝑖} (2.31) (cf. [7]). From (2.31) and Corollary 2.4, we can also derive the following interesting formula for 𝑞-Euler polynomials.

Theorem 2.5. For 𝑛∈ℤ₊,𝐸_𝑛,𝑞(𝑥)=2^𝑛_𝑙=0^𝑛_{𝑘=𝑙𝑙0+⋯+𝑙𝑘=𝑛−𝑘}𝑞^{∑𝑘𝑖=0𝑖𝑙𝑖}1(1−𝑞)^{𝑛+𝑙−𝑘}𝑠_1,𝑞(𝑘,𝑙)(−1)^𝑘𝑞^𝑙𝑥1+𝑞^𝑙. (2.32)

These polynomials are related to the many branches of Mathematics, for example, combinatorics, number theory, and discrete probability distributions for finding higher-order moments (cf. [14–16]). By substituting 𝑥=0 into the above, we have 𝐸_𝑛,𝑞=2^𝑛_𝑙=0^𝑛_{𝑘=𝑙𝑙0+⋯+𝑙𝑘=𝑛−𝑘}𝑞^{∑𝑘𝑖=0𝑖𝑙𝑖}1(1−𝑞)^{𝑛+𝑙−𝑘}𝑠_1,𝑞(𝑘,𝑙)(−1)^𝑘11+𝑞^𝑙, (2.33) where 𝐸_𝑛,𝑞 is the 𝑞-Euler numbers.

3. 𝑞-Euler Numbers, 𝑞-Stirling Numbers, and 𝑞-Bernstein Polynomials Related to the Fermionic 𝑝-Adic Integrals on ℤ_𝑝

First, we consider the 𝑞-extension of the generating function of Bernstein polynomials in (1.3).

For 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<𝑝^{−1/(𝑝−1)}, we obtain that𝐹_𝑞^(𝑘)𝑡(𝑡,𝑥)=^𝑘𝑒^{[1−𝑥]𝑞𝑡}[𝑥]^𝑘_𝑞𝑘!=[𝑥]^𝑘_𝑞^∞_𝑛=0𝑘𝑛+𝑘[1−𝑥]^𝑛_𝑞𝑡^𝑛+𝑘(𝑛+𝑘)!=^∞_𝑛=𝑘𝑛𝑘[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘𝑡^𝑛𝑛!=^∞_𝑛=0𝐵_𝑘,𝑛𝑡(𝑥,𝑞)^𝑛,𝑛! (3.1) which is the generating function of the modified 𝑞-Bernstein type polynomials (see [9]). Indeed, this generating function is also treated by Simsek and Acikgoz (see [1]). Note that lim_𝑞→1𝐹_𝑞^(𝑘)(𝑡,𝑥)=𝐹^(𝑘)(𝑡,𝑥). It is easy to show that[1−𝑥]_𝑞^𝑛−𝑘=^∞_𝑚=0^𝑛−𝑘_𝑙=0𝑚𝑙𝑙+𝑚−1𝑛−𝑘(−1)^𝑙+𝑚𝑞^𝑙[𝑥]_𝑞^𝑙+𝑚(𝑞−1)^𝑚. (3.2)

From (1.6), (2.3), (2.15), and (3.2), we derive the following theorem.

Theorem 3.1. For 𝑘,𝑛∈ℤ₊ with 𝑛≥𝑘,_ℤ𝑝𝐵_𝑘,𝑛(𝑥,𝑞)(^𝑛_𝑘)d𝜇₋₁(𝑦)=^∞_𝑚=0^𝑛−𝑘_𝑙=0𝑚𝑙𝑙+𝑚−1𝑛−𝑘(−1)^𝑙+𝑚𝑞^𝑙(𝑞−1)^𝑚𝐸_{𝑙+𝑚+𝑘,𝑞}, (3.3) where 𝐸_𝑛,𝑞 are the 𝑞-Euler numbers.

It is possible to write [𝑥]^𝑘_𝑞 as a linear combination of the modified 𝑞-Bernstein polynomials by using the degree evaluation formulae and mathematical induction. Therefore, we obtain the following theorem.

Theorem 3.2 (see [8, Theorem 7]). For 𝑘,𝑛∈ℤ₊,𝑖∈ℕ, and 𝑥∈[0,1],_{𝑛𝑘=𝑖−1}(^𝑘_𝑖)(^𝑛_𝑖)𝐵_𝑘,𝑛(𝑥,𝑞)=[𝑥]^𝑖_𝑞([𝑥]_𝑞+[1−𝑥]_𝑞)^𝑛−𝑖. (3.4)

Let 𝑖−1≤𝑛. Then from (1.7), (3.2), and Theorem 3.2, we have[𝑥]^𝑖_𝑞=∑^𝑛_{𝑘=𝑖−1}((^𝑘_𝑖)(^𝑛_𝑘)/(^𝑛_𝑖))[𝑥]^𝑘_𝑞[1−𝑥]_𝑞^𝑛−𝑘[𝑥]_𝑞^𝑛−𝑖(1+([1−𝑥]_𝑞/[𝑥]_𝑞))^𝑛−𝑘=^∞_𝑚=0^𝑛_{𝑘=𝑖−1}^{𝑚+𝑛−𝑘}_𝑙=0^∞_𝑝=0(^𝑘_𝑖)(^𝑛_𝑘)(^𝑛_𝑖)𝑝𝑙×𝑚𝑙+𝑝−1𝑚+𝑛−𝑘𝑛−𝑖+𝑚−1(−1)^{𝑙+𝑝+𝑚}𝑞^𝑙(𝑞−1)^𝑝[𝑥]_𝑞^{𝑖−𝑛−𝑚+𝑘+𝑝+𝑙}. (3.5) Using (2.13) and (3.5), we obtain the following theorem.

Theorem 3.3. For 𝑘,𝑛∈ℤ₊ and 𝑖∈ℕ with 𝑖−1≤𝑛,𝐸_𝑖,𝑞=^∞_𝑚=0^𝑛_{𝑘=𝑖−1}^{𝑚+𝑛−𝑘}_𝑙=0^∞_𝑝=0(^𝑘_𝑖)(^𝑛_𝑘)(^𝑛_𝑖)𝑝𝑙×𝑚𝑙+𝑝−1𝑚+𝑛−𝑘𝑛−𝑖+𝑚−1(−1)^{𝑙+𝑝+𝑚}𝑞^𝑙(𝑞−1)^𝑝𝐸_{𝑖−𝑛−𝑚+𝑘+𝑝+𝑙,𝑞}. (3.6)

The 𝑞-String numbers of the first kind is defined by^𝑛_𝑘=1(1+[𝑘]_𝑞𝑧)=^𝑛_𝑘=0𝑆₁(𝑛,𝑘;𝑞)𝑧^𝑘, (3.7) and the 𝑞-String number of the second kind is also defined by^𝑛_𝑘=1(1+[𝑘]_𝑞𝑧)⁻¹=^𝑛_𝑘=0𝑆₂(𝑛,𝑘;𝑞)𝑧^𝑘 (3.8) (see [9]). Therefore, we deduce the following theorem.

Theorem 3.4 (see [9, Theorem 4]). For 𝑘,𝑛∈Z₊ and 𝑖∈ℕ,∑^𝑛_{𝑘=𝑖−1}((^𝑘_𝑖)/(^𝑛_𝑖))𝐵_𝑘,𝑛(𝑥,𝑞)([𝑥]_𝑞+[1−𝑥]_𝑞)^𝑛−𝑖=^𝑖_𝑘=0^𝑘_𝑙=0𝑆₁(𝑘,𝑙;𝑞)𝑆₂(𝑘,𝑖−𝑘;𝑞)[𝑥]^𝑙_𝑞. (3.9)

By Theorems 3.2 and 3.4 and the definition of fermionic 𝑝-adic integrals on ℤ_𝑝, we obtain the following theorem.

Theorem 3.5. For 𝑘,𝑛∈ℤ₊ and 𝑖∈ℕ,𝐸_𝑖,𝑞=^𝑛_{𝑘=𝑖−1}(^𝑘_𝑖)(^𝑛_𝑖)_ℤ𝑝𝐵_𝑘,𝑛(𝑥,𝑞)([𝑥]_𝑞+[1−𝑥]_𝑞)^𝑛−𝑖d𝜇₋₁(𝑥)=^𝑖_𝑘=0^𝑘_𝑙=0𝑆₁(𝑘,𝑙;𝑞)𝑆₂(𝑘,𝑖−𝑘;𝑞)𝐸_𝑙,𝑞, (3.10) where 𝐸_𝑖,𝑞 is the 𝑞-Euler numbers.

Let 𝑖−1≤𝑛. It is easy to show that[𝑥]^𝑖_𝑞([𝑥]_𝑞+[1−𝑥]_𝑞)^𝑛−𝑖=^𝑛−𝑖_𝑙=0𝑙𝑛−𝑖[𝑥]_𝑞^𝑙+𝑖[1−𝑥]_𝑞^{𝑛−𝑖−𝑙}=^𝑛−𝑖_𝑙=0^{𝑛−𝑖−𝑙}_𝑚=0𝑙𝑚𝑛−𝑖𝑛−𝑖−𝑙(−1)^𝑚𝑞^𝑚[𝑥]_𝑞^{𝑚+𝑖+𝑙}𝑞^−𝑚𝑥=^𝑛−𝑖_𝑙=0^{𝑛−𝑖−𝑙}_𝑚=0^∞_𝑠=0𝑙𝑚𝑠𝑛−𝑖𝑛−𝑖−𝑙𝑚+𝑠−1(−1)^𝑚𝑞^𝑚(1−𝑞)^𝑠[𝑥]_𝑞^{𝑚+𝑖+𝑙+𝑠}. (3.11) From (3.11) and Theorem 3.2, we have the following theorem.

Theorem 3.6. For 𝑘,𝑛∈ℤ₊ and 𝑖∈ℕ,_{𝑛𝑘=𝑖−1}(^𝑘_𝑖)(^𝑛_𝑖)_ℤ𝑝𝐵_𝑘,𝑛(𝑥,𝑞)d𝜇₋₁(𝑥)=^𝑛−𝑖_𝑙=0^{𝑛−𝑖−𝑙}_𝑚=0^∞_𝑠=0𝑙𝑚𝑠𝑛−𝑖𝑛−𝑖−𝑙𝑚+𝑠−1×(−1)^𝑚𝑞^𝑚(1−𝑞)^𝑠𝐸_{𝑚+𝑖+𝑙+𝑠,𝑞}, (3.12) where 𝐸_𝑖,𝑞 are the 𝑞-Euler numbers.

In the same manner, we can obtain the following theorem.

Theorem 3.7. For 𝑘,𝑛∈ℤ₊ and 𝑖∈ℕ,_ℤ𝑝𝐵_𝑘,𝑛(𝑥,𝑞)d𝜇₋₁(𝑥)=^𝑛_𝑗=𝑘^∞_𝑚=0𝑗𝑘𝑛𝑗𝑚𝑗−𝑘+𝑚−1(−1)^{𝑗−𝑘+𝑚}𝑞^𝑗−𝑘(𝑞−1)^𝑚𝐸_{𝑚+𝑗,𝑞}, (3.13) where 𝐸_𝑖,𝑞 are the 𝑞-Euler numbers.

4. Further Remarks and Observations

The 𝑞-binomial formulas are known as(𝑎;𝑞)_𝑛=(1−𝑎)(1−𝑎𝑞)⋯(1−𝑎𝑞^𝑛−1)=^𝑛_𝑖=0𝑛𝑖_𝑞𝑞^(𝑖2)(−1)^𝑖𝑎^𝑖,1(𝑎;𝑞)_𝑛=1(1−𝑎)(1−𝑎𝑞)⋯(1−𝑎𝑞^𝑛−1)=^∞_𝑖=0𝑖𝑛+𝑖−1_𝑞𝑎^𝑖. (4.1) For ℎ∈ℤ,𝑛∈ℤ₊, and 𝑟∈ℕ, we introduce the extended higher-order 𝑞-Euler polynomials as follows [16]:𝐸^(ℎ,𝑟)_𝑛,𝑞(𝑥)=_ℤ𝑝⋯_ℤ𝑝𝑞^{∑𝑟𝑗=1(ℎ−𝑗)𝑥𝑗}[𝑥+𝑥₁+⋯+𝑥_𝑟]^𝑛_𝑞d𝜇₋₁(𝑥₁)⋯d𝜇₋₁(𝑥_𝑟). (4.2) Then,𝐸^(ℎ,𝑟)_𝑛,𝑞2(𝑥)=^𝑟(1−𝑞)^{𝑛𝑛𝑙=0}𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥(−𝑞^ℎ−1+𝑙;𝑞⁻¹)_𝑟=2^𝑟(1−𝑞)^{𝑛𝑛𝑙=0}𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥(−𝑞^{ℎ−𝑟+𝑙};𝑞)_𝑟. (4.3) Let us now define the extended higher-order Nörlund type 𝑞-Euler polynomials as follows [16]:𝐸^{(ℎ,−𝑟)}_𝑛,𝑞1(𝑥)=(1−𝑞)^{𝑛𝑛𝑙=0}𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥∫_ℤ𝑝⋯∫_ℤ𝑝𝑞^{𝑙(𝑥1+⋯+𝑥𝑟)}𝑞^{∑𝑟𝑗=1(ℎ−𝑗)𝑥𝑗}d𝜇₋₁(𝑥₁)⋯d𝜇₋₁(𝑥_𝑟). (4.4) In the special case 𝑥=0,𝐸^{(ℎ,−𝑟)}_𝑛,𝑞=𝐸^{(ℎ,−𝑟)}_𝑛,𝑞(0) are called the extended higher-order Nörlund type 𝑞-Euler numbers. From (4.4), we note that𝐸^{(ℎ,−𝑟)}_𝑛,𝑞1(𝑥)=2^𝑟(1−𝑞)^{𝑛𝑛𝑙=0}𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥(−𝑞^{ℎ−𝑟+𝑙};𝑞)_𝑟=12^{𝑟𝑟𝑚=0}𝑞^(𝑚2)𝑞^{(ℎ−𝑟)𝑚}𝑟𝑚_𝑞[𝑚+𝑥]^𝑛_𝑞. (4.5) A simple manipulation shows that 𝑞^(𝑚2)𝑟𝑚_𝑞=𝑞^(𝑚2)[𝑟]_𝑞⋯[𝑟−𝑚+1]_𝑞[𝑚]_𝑞!=1[𝑚]_𝑞!^𝑚−1_𝑘=0([𝑟]_𝑞−[𝑘]_𝑞),_{𝑛−1𝑘=0}(𝑧−[𝑘]_𝑞)=𝑧^𝑛𝑛−1_𝑘=0(1−[𝑘]_𝑞𝑧)=^𝑛_𝑘=0𝑆₁(𝑛−1,𝑘;𝑞)(−1)^𝑘𝑧^𝑛−𝑘. (4.6) Formula (4.5) implies the following lemma.

Lemma 4.1. For ℎ∈ℤ,𝑛∈ℤ₊, and 𝑟∈ℕ,𝐸^{(ℎ,−𝑟)}_𝑛,𝑞1(𝑥)=2^𝑟[𝑚]_𝑞!^{𝑟𝑚=0𝑚𝑘=0}𝑞^{(ℎ−𝑟)𝑚}𝑆₁(𝑚−1,𝑘;𝑞)(−1)^𝑘[𝑟]_𝑞^𝑚−𝑘[𝑥+𝑚]^𝑛_𝑞. (4.7)

From (2.22), we can easily see that[𝑥+𝑚]^𝑛_𝑞=1(1−𝑞)^{𝑛𝑛𝑗=0𝑗𝑙=0}𝑛𝑗𝑗𝑙(−1)^𝑗+𝑙(1−𝑞)^𝑙𝑞^𝑚𝑗[𝑥]^𝑙_𝑞. (4.8) Using (2.13) and (4.8), we obtain the following lemma.

Lemma 4.2. For 𝑚,𝑛∈ℤ₊,𝐸_𝑛,𝑞1(𝑚)=(1−𝑞)^{𝑛𝑛𝑗=0𝑗𝑙=0}𝑛𝑗𝑗𝑙(−1)^𝑗+𝑙(1−𝑞)^𝑙𝑞^𝑚𝑗𝐸_𝑙,𝑞. (4.9)

By Lemma 4.2, and the definition of fermionic 𝑝-adic integrals on ℤ_𝑝, we obtain the following theorem.

Theorem 4.3. For ℎ∈ℤ,𝑛∈ℤ₊, and 𝑟∈ℕ,_ℤ𝑝𝐸^{(ℎ,−𝑟)}_𝑛,𝑞(𝑥)d𝜇₋₁2(𝑥)=^−𝑟[𝑚]_𝑞!^{𝑟𝑚=0𝑚𝑘=0}𝑞^{(ℎ−𝑟)𝑚}𝑆₁(𝑚−1,𝑘;𝑞)(−1)^𝑘[𝑟]_𝑞^𝑚−𝑘𝐸_𝑛,𝑞=1(𝑚)2^𝑟[𝑚]_𝑞!^{𝑟𝑚=0𝑚𝑘=0}𝑞^{(ℎ−𝑟)𝑚}𝑆₁(𝑚−1,𝑘;𝑞)(−1)^𝑘[𝑟]_𝑞^𝑚−𝑘×1(1−𝑞)^{𝑛𝑛𝑗=0𝑗𝑙=0}𝑛𝑗𝑗𝑙(−1)^𝑗+𝑙(1−𝑞)^𝑙𝑞^𝑚𝑗𝐸_𝑙,𝑞. (4.10)

Put ℎ=0 in (4.4). We consider the following polynomials 𝐸^(0,−𝑟)_𝑛,𝑞(𝑥):𝐸^(0,−𝑟)_𝑛,𝑞(𝑥)=^𝑛_𝑙=0(1−𝑞)^−𝑛(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥∫_ℤ𝑝⋯∫_ℤ𝑝𝑞^{𝑙(𝑥1+⋯+𝑥𝑟)}𝑞^{−∑𝑟𝑗=1𝑗𝑥𝑗}d𝜇₋₁(𝑥₁)⋯d𝜇₋₁(𝑥_𝑟). (4.11) Then, 𝐸^(0,−𝑟)_𝑛,𝑞1(𝑥)=2^{𝑟𝑟𝑚=0}𝑟𝑚𝑞^{(𝑚2)−𝑟𝑚}[𝑚+𝑥]^𝑛_𝑞. (4.12) A simple calculation of the fermionic 𝑝-adic invariant integral on ℤ_𝑝 shows that _ℤ𝑝𝐸^(0,−𝑟)_𝑛,𝑞(𝑥)d𝜇₋₁1(𝑥)=2^{𝑟𝑟𝑚=0}𝑟𝑚𝑞^{(𝑚2)−𝑟𝑚}𝐸_𝑛,𝑞(𝑚). (4.13) Using Theorem 4.3, we can also prove that _ℤ𝑝𝐸^(0,−𝑟)_𝑛,𝑞(𝑥)d𝜇₋₁2(𝑥)=^−𝑟[𝑚]_𝑞!^{𝑟𝑚=0𝑚𝑘=0}𝑞^−𝑟𝑚𝑆₁(𝑚−1,𝑘;𝑞)(−1)^𝑘[𝑟]_𝑞^𝑚−𝑘𝐸_𝑛,𝑞(𝑚). (4.14) Therefore, we obtain the following theorem.

Theorem 4.4. For 𝑚∈ℤ₊,𝑟∈ℕ with 𝑚≤𝑟,𝑟𝑚𝑞^{(𝑚2)−𝑟𝑚}=1[𝑚]_𝑞!^𝑚𝑘=0𝑞^−𝑟𝑚𝑆₁(𝑚−1,𝑘;𝑞)(−1)^𝑘[𝑟]_𝑞^𝑚−𝑘. (4.15)

Acknowledgments

M.-S. Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2010-0001654). T. Kim was supported by the research grant of Kwangwoon University in 2010.