On Multiple Interpolation Functions of the Nörlund-Type 𝑞-Euler Polynomials

Abstract

In (2006) and (2009), Kim defined new generating functions of the Genocchi, Nörlund-type 𝑞-Euler polynomials and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz type 𝑞-zeta function. This function interpolates Nörlund-type 𝑞-Euler polynomials at negative integers. We also give some identities related to these polynomials and functions. Furthermore, we give some remarks about approximations of Bernoulli and Euler polynomials.

1. Introduction, Definitions and Notations

The classical Euler numbers and polynomials have been studied by many mathematicians, which are defined as follows, respectively, 2𝑒^𝑡=+1^∞_𝑛=0𝐸_𝑛𝑡^𝑛,𝑛!,|𝑡|<𝜋(1.1)2𝑒^𝑡𝑥𝑒^𝑡=+1^∞_𝑛=0𝐸_𝑛𝑡(𝑥)^𝑛𝑛!,|𝑡|<𝜋,(1.2) cf.[1–51]. Observe that 𝐸_𝑛(0)=𝐸_𝑛.

These numbers and polynomials are interpolates by the Euler zeta function and Hurwitz-type-zeta functions, respectively, 𝜁_𝐸(𝑠)=^∞_𝑛=1(−1)^𝑛𝑛^𝑠,𝑠∈ℂ,(1.3)𝜁_𝐸(𝑠,𝑥)=^∞_𝑛=0(−1)^𝑛(𝑛+𝑥)^𝑠,𝑠∈ℂ.(1.4) Let [𝑥]=[𝑥∶𝑞]=1−𝑞^𝑥/1−𝑞. Observe that lim_𝑞→1[𝑥]=𝑥, cf. [3, 47].

Various kinds of the 𝑞-analogue of the Euler numbers and polynomials, recently, have been studied by many mathematicians. In this paper, we use Kim's [13, 21] and Simsek's [43] methods. By using 𝑝-adic 𝑞-Volkenborn integral [12], Kim [13, 26] constructed many kinds of generating functions of the 𝑞-Euler numbers and polynomials and their interpolation functions. He also gave many applications of these numbers and functions. Simsek [40, 43] studied on the generating functions of the Euler numbers and Bernoulli numbers. By using these generating functions, Simsek constructed 𝑞-Dedekind-type sums and 𝑞-Hardy-type sums as well.

Recently, Cangul et al. gave higher-order 𝑞-Genocchi numbers and their interpolation functions. Applying 𝑝-adic 𝑞-fermionic integral on 𝑝-adic integers, they also gave Witt's formula of these numbers.

In [21], by using multivariate fermionic 𝑝-adic integral on ℤ_𝑝, Kim constructed generating function of the Nörlund-type 𝑞-Euler polynomials of higher order. Main motivation of this paper is to construct interpolation function of the Nörlund-type 𝑞-Euler polynomials. Therefore, we firstly give generating function of the Nörlund-type 𝑞-Euler polynomials.

Kim [21] defined Nörlund type 𝑞-extension Euler polynomials of higher order. He gave many applications and interesting identities. We give some of them in what follows.

Let 𝑞∈ℂ with |𝑞|<1: 𝐹_𝑞(𝑡,𝑥)=2^∞_𝑚=0(−1)^𝑚𝑒^{[𝑚+𝑥]𝑡}=^∞_𝑛=0𝐸_𝑛,𝑞𝑡(𝑥)^𝑛.𝑛!(1.5)

Observe that 𝐹_𝑞(𝑡)=𝐹_𝑞(𝑡,0). Hence, we have 𝐸_𝑛,𝑞(0)=𝐸_𝑛,𝑞. If 𝑞→1 into (1.5), then we easily obtain (1.2).

Higher-order 𝑞-Euler polynomials of the Nörlund type are defined by Kim [21]. He gave generating functions related to Euler numbers of higher-order. In this paper, we use generating functions in [21]. Especially, we can use the following generating function, which are proved by Kim [21, Theorem 2.3, page 5].

Theorem 1.1 ([21, Theorem 2.3, page 5]). For 𝑟∈𝑁, and 𝑛≥0, one has 𝐹_𝑞^(𝑟)(𝑡,𝑥)=2^𝑟∞_𝑚=0(−1)^𝑚𝑚𝑒𝑚+𝑟−1_{[𝑚+𝑥]𝑡}=^∞_𝑛=0𝐸^(𝑟)_𝑛,𝑞𝑡(𝑥)^𝑛.𝑛!(1.6) It is noted that if 𝑟=1, then (1.6) reduces to (1.5).

Remark 1.2. In (1.6); we easily see that lim_𝑞→1𝐹_𝑞^(𝑟)(𝑡,𝑥)=2^𝑟∞_𝑚=0(−1)^𝑚𝑚𝑒𝑚+𝑟−1_{(𝑚+𝑥)𝑡}=2^𝑟𝑒^∞𝑥𝑡_𝑚=0(−1)^𝑚𝑚𝑒𝑚+𝑟−1_𝑚𝑡=2^𝑟𝑒^𝑥𝑡(1+𝑒^𝑡)^𝑟=𝐹^(𝑟)(𝑡,𝑥).(1.7) From the above, we obtain generating function of the Nörlund Euler numbers of higher order. That is 𝐹^(𝑟)2(𝑡,𝑥)=^𝑟𝑒^𝑥𝑡(1+𝑒^𝑡)^𝑟=^∞_𝑛=0𝐸_𝑛^(𝑟)𝑡(𝑥)^𝑛.𝑛!(1.8) Thus, we have lim_𝑞→1𝐸^(𝑟)_𝑛,𝑞(𝑥)=𝐸_𝑛^(𝑟)(𝑥).(1.9) cf.[21].
Hence, we have
𝐹^(𝑟)2(𝑡,𝑥)=(𝑒^𝑡2+1)(𝑒^𝑡2+1)⋯(𝑒^𝑡+1)𝑒^𝑡𝑥=2^𝑟𝑒^∞𝑡𝑥_𝑛1=0𝑒^𝑡𝑛1(−1)^𝑛1∞_𝑛2=0𝑒^𝑡𝑛2(−1)^𝑛2⋯^∞_𝑛𝑟=0𝑒^𝑡𝑛𝑟(−1)^𝑛𝑟=2^𝑟𝑒^∞𝑡𝑥_{𝑛1,𝑛2,…,𝑛𝑟=0}(−1)^{𝑛1+𝑛2+⋯+𝑛𝑟}𝑒^{𝑡(𝑛1+𝑛2+⋯+𝑛𝑟)}=^∞_𝑛=0𝐸_𝑛^(𝑟)𝑡(𝑥)^𝑛.𝑛!(1.10)

We now summarize the results of this paper.

In Section 2, we study on modified generating functions of higher-order Nörlund-type 𝑞-Euler polynomials and numbers. We obtain some relations related to these numbers and polynomials.

In Section 3, we give interpolation functions of the higher order Nörlund-type 𝑞-Euler polynomials.

In Section 4, we obtain some relations related to he higher order Nörlund-type 𝑞-Euler polynomials.

In Section 5, we give remarks and observations on an Approximation theory related to Bernoulli and Euler polynomials.

2. Modified Generating Functions of Higher-Order Nörlund-Type 𝑞-Euler Polynomials and Numbers

In this section we define generating function of modified higher order Nörlund type 𝑞-Euler polynomials and numbers, which are denoted by 𝐄^(𝑟)_𝑛,𝑞(𝑥), and 𝐸^(𝑟)_𝑛,𝑞 respectively. We give relations between these numbers and polynomials.

We modify (1.6) as follows: ℱ_𝑞^(𝑟)(𝑡,𝑥)=𝐹_𝑞^(𝑟)(𝑞^−𝑥𝑡,𝑥),(2.1) where 𝐹_𝑞^(𝑟)(𝑡,𝑥) is defined in (1.6). From the above we find that ℱ_𝑞^(𝑟)(𝑡,𝑥)=^∞_𝑛=0𝑞^−𝑛𝑥𝐸^(𝑟)_𝑛,𝑞𝑡(𝑥)^𝑛.𝑛!(2.2) After some elementary calculations, we obtain ℱ_𝑞^(𝑟)(𝑡,𝑥)=exp([𝑥]𝑞^−𝑥𝑡)𝔣_𝑞^(𝑟)(𝑡),(2.3) where 𝔣_𝑞^(𝑟)(𝑡)=(2^𝑟∞_𝑚=0(−1)^𝑚𝑚𝑒𝑚+𝑟−1_[𝑚]𝑡)=^∞_𝑛=0𝐸^(𝑟)_𝑛,𝑞𝑡^𝑛.𝑛!(2.4) From the above we have ℱ_𝑞^(𝑟)(𝑡,𝑥)=^∞_𝑛=0𝜀^(𝑟)_𝑛,𝑞𝑡(𝑥)^𝑛,𝑛!(2.5) where 𝜀^(𝑟)_𝑛,𝑞(𝑥)=𝑞^−𝑛𝑥𝐸^(𝑟)_𝑛,𝑞(𝑥).(2.6) By using Cauchy product in (2.3), we arrive at the following theorem.

Theorem 2.1. For 𝑟∈ℕ, and 𝑛≥0, one has 𝜀^(𝑟)_𝑛,𝑞(𝑥)=^𝑛_𝑗=0𝑛𝑗𝑞^𝑗𝑥[𝑥]^𝑛−𝑘𝐸^(𝑟)_𝑗,𝑞.(2.7)

By using (2.7), we easily obtain the following result.

Corollary 2.2. For 𝑟∈ℕ, and 𝑛≥0, one has 𝜀^(𝑟)_𝑛,𝑞(𝑥)=^∞𝑛_𝑚=0𝑗=0^𝑛−𝑗_𝑙=0𝑛𝑗,𝑙,𝑛−𝑗−𝑙𝑚𝑛−𝑗+𝑚−1(−1)^𝑙𝑞^{𝑚+𝑥(𝑙+𝑗)}𝐸^(𝑟)_𝑗,𝑞.(2.8)

We now give some identity related to Nörlund type Euler polynomials and numbers of higher-order.

Substituting 𝑥=0 into (1.10), we find that 𝐸_𝑛^(𝑟)=2^𝑟∞_{𝑛1,𝑛2,…,𝑛𝑟=0𝑗1,…,𝑗𝑟𝑗=01+⋯+𝑗𝑟=𝑛}𝑛𝑗₁,…,𝑗_𝑟(−1)^{𝑛1+𝑛2+⋯+𝑛𝑟𝑟}_𝑘=0𝑛^𝑗𝑘_𝑘.(2.9) By (1.10) and (2.9), we arrive at the following theorem.

Theorem 2.3. For 𝑟∈ℕ, and 𝑛≥0, one has 𝐸_𝑛^(𝑟)=^𝑛_𝑗=0𝑛𝑗(−𝑥)^𝑛−𝑗𝐸_𝑗^(𝑟)(𝑥).(2.10)By using (1.10) and [35, Theorem  3.6, page 7] , we easily arrive at the following result.

Corollary 2.4. For 𝑟, 𝑣∈ℕ, and 𝑛≥0, one has (𝐸^(𝑟)(𝑥)+𝐸^(𝑣)(𝑦))^𝑛=^𝑛_𝑗=0𝑛𝑗𝑥^𝑛−𝑗𝐸_𝑗^(𝑟+𝑣)(𝑦),(2.11) where (𝐸^(𝑟)(𝑥))^𝑛 is replaced by 𝐸_𝑛^(𝑟)(𝑥).

3. Interpolation Function of Higher-Order Nörlund-Type 𝑞-Euler Polynomials

Recently, higher-order Bernoulli polynomials and Euler polynomials have studied by many mathematicians. Especially, in this paper, we study on higher-order Euler polynomials which are constructed by Kim (see, e.g., [14, 17, 21, 24, 27, 28, 33]) and see also the references cited in each of these earlier works.

In [20], by using the fermionic 𝑝-adic invariant integral on ℤ_𝑝, the set of 𝑝-adic integers, Kim gave a new construction of 𝑞-Genocchi numbers, Euler numbers of higher order. By using 𝑞-Genoucchi, Euler numbers of higher order, he investigated the interesting relationship between 𝑤-𝑞-Euler polynomials and 𝑤-𝑞-Genocchi polynomials. He also defined the multiple 𝑤-𝑞-zeta functions which interpolate 𝑞-Genoucchi, Euler numbers of higher order.

By using similar method of the papers given by Kim [20, 21], in this section, applying derivative operator 𝑑^𝑘/𝑑𝑡^𝑘|_𝑡=0 and Mellin Transformation to the generating functions of the higher-order Nörlund-type 𝑞-Euler polynomials, we give interpolation function of these polynomials.

By applying operator 𝑑^𝑘/𝑑𝑡^𝑘|_𝑡=0 to (1.6), we obtain the following theorem.

Theorem 3.1. Let 𝑟,𝑘∈ℤ⁺ and 𝑥∈ℝ with 0<𝑥≤1. Then one has 𝐸^(𝑟)_𝑘,𝑞(𝑥)=2^𝑟∞_𝑚=0(−1)^𝑚𝑚𝑚+𝑟−1[𝑚+𝑥]^𝑘.(3.1)Let us define interpolation function of higher-order Nörlund-type 𝑞-Euler numbers as follows. Definition 3.2. Let 𝑞,𝑠∈ℂ with |𝑞|<1, and 0<𝑥≤1. Then we define 𝜁_𝑞^(𝑟)(𝑠,𝑥)=2^𝑟∞_𝑛=0(−1)^𝑛(_𝑛^{𝑛+𝑟−1})[𝑛+𝑥]^𝑠.(3.2)Remark 3.3. It holds that lim_𝑞→1𝜁_𝑞^(𝑟)(𝑠,𝑥)=2^𝑟∞_𝑛=0(−1)^𝑛(_𝑛^{𝑛+𝑟−1})[𝑛+𝑥]^𝑠.(3.3) For detail about the above function (see [5–38, 44, 47]). By applying 𝑑^𝑘/𝑑𝑡^𝑘|_𝑡=0 derivative operator to (1.10), we easily see that 𝜁^(𝑟)(𝑠,𝑥)=2^𝑟∞_{𝑛1,…,𝑛𝑟=0}(−1)^{𝑛1+⋯+𝑛𝑟}(∑^𝑟_𝑗=1𝑛_𝑗+𝑥)^𝑠,(3.4) where 𝑠∈ℂ.

The functions in (3.3) and (3.4) interpolate same numbers at negative integers. That is, these functions interpolate higher-order Nörlund-type Euler numbers at negative integers. So, by (3.3), we modify (3.4) in sense of 𝑞-analogue.

In [3–51], many authors extensively have studied on similar type of (3.4).

In (3.3) and (3.4), setting 𝑟=1, we have 𝜁⁽¹⁾(𝑠,𝑥)=2^∞_𝑛=0(−1)^𝑛(𝑛+𝑥)^𝑠=𝜁_𝐸(𝑠,𝑥)(3.5) where 𝜁_𝐸(𝑠,𝑥) denotes Hurwitz type Euler zeta function, which interpolates classical Euler polynomials at negative integers.Theorem 3.4. Let 𝑛∈ℤ⁺. Then one has 𝜁_𝑞^(𝑟)(−𝑛,𝑥)=𝐸^(𝑟)_𝑛,𝑞(𝑥).(3.6)

Proof. Substituting 𝑠=−𝑘, 𝑘∈ℤ⁺ into (3.2). Then we have 𝜁_𝑞^(𝑟)(−𝑘,𝑥)=2^𝑟∞_𝑛=0(−1)^𝑛𝑛𝑛+𝑟−1[𝑛+𝑥]^𝑘.(3.7) Setting (3.1) into the above, and after some elementary calculations, we easily arrive at the desired result.

By applying the Mellin transformation to (2.5), we find that 1Γ(𝑠)^∞₀𝑡^𝑠−1ℱ_𝑞^(𝑟)(−𝑡,𝑥)𝑑𝑡=2^𝑟∞_𝑚=0(−1)^𝑚(_𝑚^{𝑚+𝑟−1})𝑞^𝑛𝑥𝑠[𝑚+𝑥]^𝑠.(3.8) From the above we define the following function, which interpolate 𝐄^(𝑟)_𝑛,𝑞(𝑥) at negative integers.

Definition 3.5. Let 𝑞,𝑠∈ℂ with |𝑞|<1, and 0<𝑥≤1. Then we define 𝒵_𝑞^(𝑟)(𝑠,𝑥)=2^𝑟∞∞_𝑚=0𝑗=0(−1)^𝑚𝑚𝑗𝑞𝑟+𝑚−1𝑠+𝑗−1_{𝑥(𝑛𝑠+𝑗)+𝑚𝑗}.(3.9)Theorem. Let 𝑛∈ℤ⁺. Then one has 𝒵_𝑞^(𝑟)(−𝑛,𝑥)=𝜀^(𝑟)_𝑛,𝑞(𝑥).(3.10)

By Theorems 5, 6, a relation between the functions 𝜻_𝑞^(𝑟)(−𝑛,𝑥) and 𝜁_𝑞^(𝑟)(−𝑛,𝑥) is given by the following corollary.

Corollary 3.7. 𝒵_𝑞^(𝑟)(−𝑛,𝑥)=𝑞^−𝑛𝑥𝜁_𝑞^(𝑟)(−𝑛,𝑥).(3.11)

Remark 3.8. Recently many authors have studied on the Riemann zeta function, Hurwitz zeta function, Lerch zeta function, Dirichlet series for the polylogarithm function and Dirichlet's eta function and the other functions. The Lerch trancendentΦ(𝑧,𝑠,𝑎) is the analytic continuation of the series 1Φ(𝑧,𝑠,𝑎)=𝑎^𝑠+𝑧(𝑎+1)^𝑠+𝑧(𝑎+2)^𝑠+⋯=^∞_𝑛=0𝑧^𝑛(𝑛+𝑎)^𝑠,(3.12) which converges for (𝑎∈ℂ⧵,ℤ⁻₀, 𝑠∈ℂ when |𝑧|<1; ℜ(𝑠)>1 when |𝑧|=1), where as usual ℤ⁻₀=ℤ⁻∪{0},ℤ⁻={−1,−2,−3,…}.(3.13)
However, Φ denotes the familiar Hurwitz-Lerch Zeta function (cf. e. g., [8], [49, page 121 et seq.] ). Some special cases of the function Φ(𝑧,𝑠,𝑎) are given by the following relations (e.g., and details see [8], [49, page 121 et seq.] ):
(1)the Riemann zeta function Φ(1,𝑠,1)=𝜁(𝑠)=^∞_𝑛=11𝑛^𝑠,ℜ(𝑠)>1,(3.14)(2) the Hurwitz zeta function Φ(1,𝑠,𝑎)=𝜁(𝑠,𝑎)=^∞_𝑛=01(𝑛+𝑎)^𝑠,ℜ(𝑠)>1,(3.15)(3) the Dirichlet's eta function Φ(−1,𝑠,1)=𝜁^∗(𝑠)=^∞_𝑛=1(−1)^𝑛−1𝑛^𝑠,(3.16)(4) the Dirichlet beta function Φ(−1,𝑠,1/2)2^𝑠=𝛽(𝑠)=^∞_𝑛=0(−1)^𝑛(2𝑛+1)^𝑠,(3.17)(5) the Legendre chi function 𝑧Φ(𝑧²,𝑠,1/2)2^𝑠=𝜒_𝑠(𝑧)=^∞_𝑛=0𝑧^2𝑛+1(2𝑛+1)^𝑠,(|𝑧|≤1;ℜ(𝑠)>1),(3.18)(6) the polylogarithm 𝑧Φ(𝑧,𝑛,1)=𝐿𝑖_𝑚(𝑧)=^∞_𝑛=0𝑧^𝑘𝑛^𝑚,(3.19)(7) the Lerch zeta function (sometimes called the Hurwitz-Lerch zeta function) 𝐿(𝜆,𝛼,𝑠)=Φ(𝑒^{2𝜋𝑖𝜆},𝑠,𝛼),(3.20) which is a special function and generalizes the Hurwitz zeta function and polylogarithm cf. [6, 8, 20, 46, 49] and see also the references cited in each of these earlier works. Consequently, the functions 𝒵_𝑞^(𝑟)(−𝑛,𝑥) and 𝜁_𝑞^(𝑟)(−𝑛,𝑥) are related to the Hurwitz-Lerch zeta function and the other special functions, which are defined above:2Φ(−1,𝑠,𝑥)=𝜁⁽¹⁾(𝑠,𝑥)=𝜁_𝐸(𝑠,𝑥).(3.21)

4. Some Relations Related to Higher-Order Nörlund 𝑞-Euler Polynomials

In this section, by using generating function of higher-order Nörlund 𝑞-Euler polynomials, which is defined by Kim [20, 21], we obtain the following identities.

Theorem 4.1. Let 𝑞∈ℂ with |𝑞|<1. Let 𝑟 be a positive integer. Then one has 𝐸^(𝑟)_𝑘,𝑞(𝑥)=2^𝑟𝑘𝑗_𝑗=0𝑎=0(−1)^𝑎𝑘𝑞𝑎,𝑗−𝑎,𝑘−𝑗^𝑗𝑎[𝑥]^𝑘−𝑗(1−𝑞)^𝑗(1+𝑞^𝑘−𝑗)^𝑟−1.(4.1)Proof. By using (3.1), we have 𝐸^(𝑟)_𝑘,𝑞(𝑥)=2^𝑟∞_𝑚=0(−1)^𝑚𝑚𝑚+𝑟−1([𝑚]+𝑞^𝑚[𝑥])^𝑘=2^𝑟∞_𝑚=0(−1)^𝑚𝑚𝑚+𝑟−1^𝑘_𝑗=0𝑘𝑗[𝑚]^𝑗𝑞^{𝑚(𝑘−𝑗)}[𝑥]^𝑘−𝑗=2^𝑟∞_𝑚=0(−1)^𝑚𝑚𝑚+𝑟−1^𝑘_𝑗=0𝑘𝑗(1−𝑞^𝑚)^𝑗(1−𝑞)^𝑗𝑞^{𝑚(𝑘−𝑗)}⋅[𝑥]^𝑘−𝑗=2^𝑟∞_𝑚=0(−1)^𝑚𝑚𝑚+𝑟−1^𝑘𝑗_𝑗=0𝑎=0(^𝑘_𝑗)(^𝑗_𝑎)(−1)^𝑎𝑞^{𝑎𝑗+𝑚(𝑘−𝑗)}⋅[𝑥]^𝑘−𝑗(1−𝑞)^𝑗=2^𝑟𝑘𝑗_𝑗=0𝑎=0(^𝑘_𝑗)(^𝑗_𝑎)(−1)^𝑎𝑞^𝑗𝑎⋅[𝑥]^𝑘−𝑗(1−𝑞)^𝑗∞_𝑚=0(−1)^𝑚𝑚𝑞𝑚+𝑟−1_{𝑚(𝑘−𝑗)}=2^𝑟𝑘𝑗_𝑗=0𝑎=0(^𝑘_𝑗)(^𝑗_𝑎)(−1)^𝑎𝑞^𝑗𝑎⋅[𝑥]^𝑘−𝑗(1−𝑞)^𝑗(1+𝑞^𝑘−𝑗)^𝑟−1=2^𝑟𝑘𝑗_𝑗=0𝑎=0(−1)^𝑎𝑘𝑞𝑎,𝑗−𝑎,𝑘−𝑗^𝑗𝑎[𝑥]^𝑘−𝑗(1−𝑞)^𝑗(1+𝑞^𝑘−𝑗)^𝑟−1.(4.2) Thus, we complete the proof.Theorem 4.2. Let 𝑞∈ℂ with |𝑞|<1. Let 𝑟 be a positive integer. Then one has 𝐸^(𝑟)_𝑛,𝑞(𝑥)=^𝑛_𝑗=0(−1)^𝑗𝑛𝑗𝑞^𝑗𝑥(1−𝑞^𝑗)^−𝑟(1−𝑞)^−𝑛.(4.3)Proof. By using (1.6) 𝐹_𝑞^(𝑟)(𝑡,𝑥)=2^𝑟∞_𝑚=0𝑚𝑚+𝑟−1(−1)^𝑚𝑒^{[𝑚+𝑥]𝑡}=2^𝑟∞∞_𝑚=0𝑛=0𝑚𝑚+𝑟−1(−1)^𝑚(1−𝑞^𝑚+𝑥)1−𝑞^𝑛𝑡^𝑛𝑛!=2^𝑟∞∞_𝑚=0𝑛=0(_𝑚^{𝑚+𝑟−1})(−1)^𝑚(1−𝑞)^𝑛(𝑛!^𝑛_𝑗=0𝑛𝑗(−1)^𝑗⋅𝑞^{𝑗𝑥+𝑗𝑚})𝑡^𝑛=2^𝑟∞∞_𝑛=0𝑗=0(^𝑛_𝑗)(−1)^𝑗.𝑞^𝑗𝑥(1−𝑞)^𝑛⋅𝑡^𝑛𝑛!^∞_𝑚=0𝑚𝑚+𝑟−1(−1)^𝑚⋅𝑞^𝑗𝑚(4.4) Thus we have; ^∞_𝑛=0𝐸^(𝑟)_𝑛,𝑞𝑡(𝑥)^𝑛𝑛!=2^𝑟∞_𝑛=0(^𝑛_𝑗=0𝑛𝑗(−1)^𝑗𝑞^𝑗𝑥(1−𝑞^𝑗)^−𝑟(𝑡)1−𝑞^𝑛1𝑛!).(4.5) By comparing the coefficients 𝑡^𝑛/𝑛! both sides in the above, we arrive at the desired result.Theorem 4.3. Let 𝑟,𝑦∈ℤ⁺. Then one has ^𝑘_𝑗=0𝑘𝑗𝐸^(𝑟)_𝑗,𝑞(𝑥)𝐸^(𝑦)_{𝑘−𝑗,𝑞}(𝑥)=2^{∞𝑟+𝑦𝑛}_𝑛=0𝑗=0(−1)^𝑛𝑗𝑗+𝑟−1𝑛−𝑗+𝑦−1𝑛−𝑗([𝑥+𝑦]+[𝑛−𝑗+𝑥])^𝑘.(4.6)Proof. By using (1.6), we have ^∞_𝑛=0𝐸^(𝑟)_𝑛,𝑞𝑡(𝑥)^𝑛𝑛!^∞_𝑛=0𝐸^(𝑦)_𝑛,𝑞𝑡(𝑥)^𝑛𝑛!=2^∞𝑟+𝑦_𝑛=0(−1)^𝑛𝑛𝑒𝑛+𝑟−1^∞_{[𝑛+𝑥]𝑡𝑛=0}(−1)^𝑛𝑛𝑒𝑛+𝑦−1_{[𝑛+𝑥]𝑡}.(4.7) By using Cauchy product into the above, we obtain ^∞_𝑛=0(^𝑛_𝑗=0𝐸^(𝑟)_𝑗,𝑞(𝑥)𝐸^(𝑦)_{𝑛−𝑗,𝑞}1(𝑥)𝑗!(𝑛−𝑗)!)𝑡^𝑛=2^∞𝑟+𝑦_𝑛=0(^𝑛_𝑗=0𝑗𝑗+𝑟−1𝑛−𝑗+𝑦−1𝑛−𝑗(−1)^𝑛𝑒^{[𝑗+𝑥]𝑡}𝑒^{[𝑛−𝑗+𝑥]𝑡}),(4.8) From the above, we have ^∞_𝑚=0(^𝑚_𝑗=0𝐸^(𝑟)_𝑗,𝑞(𝑥)𝐸^(𝑦)_{𝑚−𝑗,𝑞}1(𝑥)𝑗!(𝑚−𝑗)!)𝑡^𝑚=^∞_𝑚=0(2^{∞𝑟+𝑦𝑛}_𝑛=0𝑗=0𝑗𝑗+𝑟−1𝑛−𝑗+𝑦−1𝑛−𝑗([𝑗+𝑥]+[𝑛−𝑗+𝑥])^𝑚)𝑡^𝑚(4.9) By comparing the coefficients of both sides of 𝑡^𝑛 in the above we arrive at the desired result.Remark 4.4. In (4.1) setting 𝑦=1, we have ^𝑚_𝑗=0𝑚𝑗𝐸^(𝑟)_𝑗,𝑞(𝑥)𝐸_{𝑚−𝑗,𝑞}(𝑥)=2^∞𝑟+1_𝑛=0(−1)^𝑛𝑛_𝑗=0𝑗𝑗+𝑟−1([𝑗+𝑥]+[𝑛−𝑗+𝑥])^𝑚.(4.10)

The above relations give us (3.1) related to (4.1).

5. Further Remarks and Observations on Approximation

Apostol [1, page 481] gave Weierstrass theorem as follows.

Theorem 5.1. Let 𝑓 be real valued and continuous on a closed interval [𝑎,𝑏]. Then, given any 𝜀>0, there exists a polynomial 𝑝 (which may be depend on 𝜀) such that |𝑓(𝑥)−𝑝(𝑥)|<𝜀,(5.1) for every 𝑥∈[𝑎,𝑏].

According to Apostol [1]; the above theorem is described by saying that every continuous function can be “uniformly approximated” by a polynomial.

We now give, more useful, and more interesting result concerning the approximation by polynomials which is related to the Bernstein polynomials (cf. [1, 2, 4, 34, 39]).

Definition 5.2. ([2]) Let 𝑓 be a function with domain 𝐼=[0,1] and range 𝑅. The 𝑛th Bernstein polynomial for 𝑓 is defined to be 𝐵_𝑛(𝑥)=𝐵_𝑛(𝑓;𝑥)=^𝑛_𝑘=0𝑘𝑓(𝑛)𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘.(5.2)

These Bernstein polynomials are not only used with probability (the Binomial Distribution) but also used in the approximation theory.

Let 𝑓 be continuous on 𝐼 with values in ℝ. Then the sequence of Bernstein polynomials for 𝑓, defined in (5.2) converges uniformly on 𝐼 to 𝑓 (cf.[2]).

In [4], Costabile and Dell'Accio collected classical and more recent results on polynomial approximation of sufficiently regular real functions defined in bounded closed intervals by means of boundary values only. Their problem is considered from the point of view of the existence of explicit formulas, interpolation to boundary data, bounds for the remainder and convergence of the polynomial series. Applications to some problems of numerical analysis are pointed out, such as nonlinear equations, numerical differentiation and integration formulas, specially associated differential boundary value problems. Some polynomial expansions for smooth enough functions defined in rectangles or in triangles of ℝ² as well as in cuboids or in tetrahedrons in ℝ³ and their applications are also discussed. They also used Bernoulli and Euler polynomials for the polynomial approximation cf. see for detail [4].

Lopez and Temme [34] studied on uniform approximations of the Bernoulli and Euler polynomials for large values of the order in terms of hyperbolic functions. They obtained convergent expansions for 𝐵_𝑛1(𝑛𝑧+2),𝐸_𝑛1(𝑛𝑧+2)(5.3) in powers of 1/𝑛, and coefficients are rational functions of 𝑧 and hyperbolic functions of argument 1/2𝑧, here 𝐵_𝑛(𝑥) and 𝐸_𝑛(𝑥) denote Bernoulli and Euler polynomials, respectively. Their expansions are uniformly valid for |𝑧±𝑖/2𝜋|>1/2𝜋 and |𝑧±𝑖/𝜋|>1/𝜋, respectively. For a real argument, the accuracy of these approximations is restricted to the monotonic region cf. see for detail [34].

Recently, many authors studied on very different type of the approximation theory. Consequently, by using the above motivations, we conclude this section by the following questions:

Bernoulli functions and Euler functions are related to trigonometric polynomials cf. [46]. Approximation by 𝑞-analogue of these functions may be possible.

(1)whether or not define better uniform approximations for the Nörlund 𝑞-Euler polynomials higher order;(2)is it possible to define uniform expansions of the Nörlund 𝑞-Euler polynomials higher order?

Acknowledgment

Y. Simsek is supported by the research fund of Akdeniz University.