Generalized (𝑞,𝑤)-Euler Numbers and Polynomials Associated with 𝑝-Adic 𝑞-Integral on ℤ_𝑝

Abstract

We generalize the Euler numbers and polynomials by the generalized (𝑞,𝑤)-Euler numbers 𝐸_{𝑛,𝑞,𝑤}(𝑎) and polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎). We observe an interesting phenomenon of “scattering” of the zeros of the generalized (𝑞,𝑤)-Euler polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) in complex plane.

1. Introduction

Recently, many mathematicians have studied in the area of the Euler numbers and polynomials (see [1–15]). The Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. In [14], we introduced that Euler equation 𝐸_𝑛(𝑥)=0 has symmetrical roots for 𝑥=1/2 (see [14]). It is the aim of this paper to observe an interesting phenomenon of “scattering” of the zeros of the generalized (𝑞,𝑤)-Euler polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) in complex plane. Throughout this paper, we use the following notations. By ℤ_𝑝, we denote the ring of 𝑝-adic rational integers, ℚ_𝑝 denotes the field of 𝑝-adic rational numbers, ℂ_𝑝 denotes the completion of algebraic closure of ℚ_𝑝, ℕ denotes the set of natural numbers, ℤ denotes the ring of rational integers, ℚ denotes the field of rational numbers, ℂ denotes the set of complex numbers, and ℤ₊=ℕ∪{0}. Let 𝜈_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝜈𝑝(𝑝)}=𝑝⁻¹. When one talks of 𝑞-extension, 𝑞 is considered in many ways such as an indeterminate, a complex number 𝑞∈ℂ, or 𝑝-adic number 𝑞∈ℂ_𝑝. If 𝑞∈ℂ one normally assume that |𝑞|<1. If 𝑞∈ℂ_𝑝, we normally assume that |𝑞−1|_𝑝<𝑝^{−1/(𝑝−1)} so that 𝑞^𝑥=exp(𝑥log𝑞) for |𝑥|_𝑝≤1[𝑥]_𝑞=1−𝑞^𝑥,[𝑥]1−𝑞^−𝑞=1−−𝑞^𝑥.1+𝑞 (1.1) Compared with [1, 4, 5]. Hence, lim_𝑞→1[𝑥]=𝑥 for any 𝑥 with |𝑥|_𝑝≤1 in the present 𝑝-adic case. Let 𝑑 be a fixed integer, and let 𝑝 be a fixed prime number. For any positive integer 𝑁, we set 𝑋=lim_←𝑁ℤ𝑑𝑝^𝑁ℤ,𝑋^∗=_{0<𝑎<𝑑𝑝𝑎,𝑝=1}𝑎+𝑑𝑝ℤ_𝑝,a+𝑑𝑝^𝑁ℤ_𝑝=𝑥∈𝑋∣𝑥≡𝑎mod𝑑𝑝^𝑁, (1.2) where 𝑎∈ℤ lies in 0≤𝑎<𝑑𝑝^𝑁. For any positive integer 𝑁, 𝜇_𝑞𝑎+𝑑𝑝^𝑁ℤ_𝑝=𝑞^𝑎𝑑𝑝^𝑁_𝑞 (1.3) is known to be a distribution on 𝑋, compared with [1–10, 14]. For 𝑔∈𝑈𝐷ℤ_𝑝=𝑔∣𝑔∶ℤ_𝑝→ℂ_𝑝isuniformlydifferentiablefunction. (1.4) Kim defined the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝𝐼_−𝑞𝑔=_ℤ𝑝𝑔(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁_{−𝑞0≤𝑥<𝑝𝑁}𝑔(𝑥)−𝑞^𝑥. (1.5) From (1.5), we also obtain 𝑞𝐼_−𝑞𝑔₁+𝐼_−𝑞𝑔=[2]_𝑞𝑔(0), (1.6) where 𝑔₁(𝑥)=𝑔(𝑥+1) (see [1–3]).

From (1.6), we obtain 𝑞^𝑛𝐼_−𝑞𝑔_𝑛+(−1)^𝑛−1𝐼_−𝑞𝑔=[2]_𝑞^𝑛−1_𝑙=0(−1)^{𝑛−1−𝑙}𝑞^𝑙𝑔(𝑙), (1.7) where 𝑔_𝑛(𝑥)=𝑔(𝑥+𝑛).

As well-known definition, the Euler polynomials are defined by 𝐹(𝑡)=2𝑒^𝑡+1=𝑒^𝐸𝑡=^∞_𝑛=0𝐸_𝑛𝑡^𝑛,𝐹()=2𝑛!𝑡,𝑥𝑒^𝑡𝑒+1^𝑥𝑡=𝑒^{𝐸(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛,𝑛! (1.8) with the usual convention of replacing 𝐸^𝑛(𝑥) by 𝐸_𝑛(𝑥). In the special case, 𝑥=0, 𝐸_𝑛(0)=𝐸_𝑛 are called the 𝑛-th Euler numbers (cf. [1–15]).

Our aim in this paper is to define the generalized (𝑞,𝑤)-Euler numbers 𝐸_{𝑛,𝑞,𝑤}(𝑎) and polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎). We investigate some properties which are related to the generalized (𝑞,𝑤)-Euler numbers 𝐸_{𝑛,𝑞,𝑤}(𝑎) and polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎). Especially, distribution of roots for 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎)=0 is different from 𝐸_𝑛(𝑥)=0^′s. We also derive the existence of a specific interpolation function which interpolate the generalized (𝑞,𝑤)-Euler numbers 𝐸_{𝑛,𝑞,𝑤}(𝑎) and polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎).

2. The Generalized (𝑞,𝑤)-Euler Numbers and Polynomials

Our primary goal of this section is to define the generalized (𝑞,𝑤)-Euler numbers 𝐸_{𝑛,𝑞,𝑤}(𝑎) and polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎). We also find generating functions of the generalized (𝑞,𝑤)-Euler numbers 𝐸_{𝑛,𝑞,𝑤}(𝑎) and polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎). Let 𝑎 be strictly positive real number.

The generalized (𝑞,𝑤)-Euler numbers and polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑎), 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) are defined by ^∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}(𝑎)𝑡^𝑛=𝑛!_ℤ𝑝𝑤^𝑎𝑥𝑒^𝑎𝑥𝑡𝑑𝜇_−𝑞(𝑥),_∞𝑛=0𝐸_{𝑛,𝑞,𝑤}()𝑡𝑥∶𝑎^𝑛=𝑛!_ℤ𝑝𝑤^𝑎𝑦𝑒^{(𝑎𝑦+𝑥)𝑡}𝑑𝜇_−𝑞𝑦,for𝑡,𝑤∈ℂ, (2.1)(2.2) respectively.

From above definition, we obtain 𝐸_{𝑛,𝑞,𝑤}(𝑎)=_ℤ𝑝𝑤^𝑎𝑥(𝑎𝑥)^𝑛𝑑𝜇_−𝑞(𝑥),𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎)=_ℤ𝑝𝑤^𝑎𝑦𝑥+𝑎𝑦^𝑛𝑑𝜇_−𝑞𝑦. (2.3)

Let 𝑔(𝑥)=𝑤^𝑎𝑥𝑒^𝑎𝑥𝑡. By (1.6) and using 𝑝-adic 𝑞-integral on ℤ_𝑝, we have 𝑞𝐼_−𝑞𝑔₁+𝐼_−𝑞𝑔=_ℤ𝑝𝑤^𝑎(𝑥+1)𝑒^{𝑎(𝑥+1)𝑡}𝑑𝜇_−𝑞(𝑥)+_ℤ𝑝𝑤^𝑎𝑥𝑒^𝑎𝑥𝑡𝑑𝜇_−𝑞(𝑥)=𝑞𝑤^𝑎𝑒^𝑎𝑡+1_ℤ𝑝𝑤^𝑎𝑥𝑒^𝑎𝑥𝑡𝑑𝜇_−𝑞(𝑥)=[2]_𝑞. (2.4)

Hence, by (2.1), we obtain ^∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}(𝑎)𝑡^𝑛=[2]𝑛!𝑞𝑤^𝑎𝑒^𝑞𝑎𝑡.+1 (2.5)

By (1.6), (2.2) and 𝑔(𝑦)=𝑤^𝑎𝑦𝑒^{(𝑎𝑦+𝑥)𝑡}, we have ^∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}()𝑡𝑥∶𝑎^𝑛=[2]𝑛!𝑞𝑤^𝑎𝑒^𝑞𝑎𝑡𝑒+1^𝑥𝑡. (2.6)

After some elementary calculations, we obtain ^∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}()𝑡𝑥∶𝑎^𝑛=[2]𝑛!^𝑞∞_𝑛=0(−1)^𝑛𝑞^𝑛𝑤^𝑎𝑛𝑒^𝑎𝑛𝑡𝑒^𝑥𝑡. (2.7)

From (2.6), we have 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎)=^𝑛_𝑘=0𝑛𝑘𝑥^𝑛−𝑘𝐸_{𝑘,𝑞,𝑤}(𝑎)=𝑥+𝐸_𝑞,𝑤(𝑎)^𝑛, (2.8) with the usual convention of replacing (𝐸_𝑞,𝑤(𝑎))^𝑛 by 𝐸_{𝑛,𝑞,𝑤}(𝑎).

3. Basic Properties for the Generalized (𝑞,𝑤)-Euler Numbers and Polynomials

By (2.5), we have 𝜕𝜕𝑥^∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}()𝑡𝑥∶𝑎^𝑛=𝜕𝑛![2]𝜕𝑥𝑞𝑤^𝑎𝑒^𝑞𝑎𝑡𝑒+1^𝑥𝑡=𝑡^∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}()𝑡𝑥∶𝑎^𝑛=𝑛!_∞𝑛=0𝑛𝐸_{𝑛−1,𝑞,𝑤}()𝑡𝑥∶𝑎^𝑛.𝑛! (3.1)

By (3.1), we have the following differential relation.

Theorem 3.1. For positive integers n, one has 𝜕𝐸𝜕𝑥^{𝑛,𝑞,𝑤}(𝑥∶𝑎)=𝑛𝐸_{𝑛−1,𝑞,𝑤}(𝑥∶𝑎). (3.2)

By Theorem 3.1, we easily obtain the following corollary.

Corollary 3.2 (integral formula). Consider that^𝑞_𝑝𝐸_{𝑛−1,𝑞,𝑤}()1𝑥∶𝑎𝑑𝑥=𝑛𝐸_{𝑛,𝑞,𝑤}𝑞∶𝑎−𝐸_{𝑛,𝑞,𝑤}𝑝∶𝑎. (3.3)
By (2.5), one obtains ^∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}𝑡𝑥+𝑦∶𝑎^𝑛=[2]𝑛!𝑞𝑤^𝑎𝑒^𝑞𝑎𝑡𝑒+1^{(𝑥+𝑦)𝑡}=^∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}()𝑡𝑥∶𝑎^𝑛𝑛!^∞_𝑘=0𝑦^𝑘𝑡^𝑘=𝑘!_{∞𝑛=0𝑛𝑘=0}𝑛𝑘𝐸_{𝑘,𝑞,𝑤}(𝑥∶𝑎)𝑦^𝑛−𝑘𝑡^𝑛.𝑛! (3.4)

By comparing coefficients of 𝑡^𝑛/𝑛! in the above equation, we arrive at the following addition theorem.

Theorem 3.3 (addition theorem). For 𝑛∈ℤ₊, 𝐸_{𝑛,𝑞,𝑤}𝑥+𝑦∶𝑎=^𝑛_𝑘=0𝑛𝑘𝐸_{𝑘,𝑞,𝑤}(𝑥∶𝑎)𝑦^𝑛−𝑘. (3.5)
By (2.5), for 𝑚≡1(mod2), one has ^∞_𝑛=0𝑚^𝑛[2]_𝑞[2]_𝑞𝑚^𝑚−1_𝑘=0(−1)^𝑘𝑞^𝑘𝑤^𝑎𝑘𝐸_{𝑛,𝑞𝑚,𝑤𝑚}𝑥+𝑎𝑘𝑚𝑡∶𝑎^𝑛=𝑛!_{𝑚−1𝑘=0}(−1)^𝑘𝑞^𝑘𝑤^𝑎𝑘∞_𝑛=0𝐸_{𝑛,𝑞𝑚,𝑤𝑚}𝑥+𝑎𝑘𝑚()∶𝑎𝑚𝑡^𝑛=𝑛!_{𝑚−1𝑘=0}(−1)^𝑘𝑞^𝑘𝑤^𝑎𝑘[2]_𝑞𝑞^𝑚𝑤^𝑚𝑎𝑒^𝑚𝑎𝑡𝑒+1^{(𝑥+𝑎𝑘)𝑡}=[2]_𝑞1+𝑞𝑤^𝑎𝑒^𝑎𝑡𝑒^𝑥𝑡=^∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}()𝑡𝑥∶𝑎^𝑛.𝑛! (3.6)

By comparing coefficients of 𝑡^𝑛/𝑛! in the above equation, we arrive at the following multiplication theorem.

Theorem 3.4 (multiplication theorem). For 𝑚,𝑛∈ℕ𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎)=𝑚^𝑛[2]_𝑞[2]_𝑞𝑚^𝑚−1_𝑘=0(−1)^𝑘𝑞^𝑘𝑤^𝑎𝑘𝐸_{𝑛,𝑞𝑚,𝑤𝑚}𝑥+𝑎𝑘𝑚∶𝑎. (3.7)
From (1.6), one notes that [2]_𝑞=_ℤ𝑝𝑞𝑤^{𝑎𝑥+𝑎}𝑒^{(𝑎𝑥+𝑎)𝑡}𝑑𝜇_−𝑞(𝑥)+_ℤ𝑝𝑤^𝑎𝑥𝑒^𝑎𝑥𝑡𝑑𝜇_−𝑞(𝑥)=^∞_𝑛=0𝑞𝑤^𝑎_ℤ𝑝𝑤^𝑎𝑥(𝑎𝑥+𝑎)^𝑛𝑑𝜇_−𝑞(𝑥)+_ℤ𝑝𝑤^𝑎𝑥(𝑎𝑥)^𝑛𝑑𝜇_−𝑞(𝑥)𝑡^𝑛=𝑛!_∞𝑛=0𝑞𝑤^𝑎𝐸_{𝑛,𝑞,𝑤}(𝑎∶𝑎)+𝐸_{𝑛,𝑞,𝑤}(𝑎)𝑡^𝑛.𝑛! (3.8)

From the above, we obtain the following theorem.

Theorem 3.5. For 𝑛∈ℤ₊, we have 𝑞𝑤^𝑎𝐸_{𝑛,𝑞,𝑤}(𝑎∶𝑎)+𝐸_{𝑛,𝑞,𝑤}(𝑎)=[2]_𝑞,if𝑛=0,0,if𝑛>0. (3.9)

By (2.8) in the above, we arrive at the following corollary.

Corollary 3.6. For 𝑛∈ℤ₊, one has 𝑞𝑤^𝑎𝑎+𝐸_𝑞,𝑤(𝑎)^𝑛+𝐸_{𝑛,𝑞,𝑤}(𝑎)=[2]_𝑞,if𝑛=0,0,if𝑛>0, (3.10) with the usual convention of replacing (𝐸_𝑞,𝑤(𝑎))^𝑛 by 𝐸_{𝑛,𝑞,𝑤}(𝑎).

From (1.7), one notes that ^∞_𝑚=0[2]_𝑞^𝑛−1_𝑙=0(−1)^{𝑛−1−𝑙}𝑞^𝑙𝑤^𝑎𝑙(𝑎𝑙)^𝑚𝑡^𝑛𝑚!=𝑞^𝑛_ℤ𝑝𝑤^{𝑎𝑥+𝑎𝑛}𝑒^{(𝑎𝑥+𝑎𝑛)𝑡}𝑑𝜇_−𝑞(𝑥)+(−1)^𝑛−1_ℤ𝑝𝑤^𝑎𝑥𝑒^𝑎𝑥𝑡𝑑𝜇_−𝑞(𝑥)=^∞_𝑚=0𝑞^𝑛𝑤^𝑎𝑛_ℤ𝑝𝑤^𝑎𝑥(𝑎𝑥+𝑎𝑛)^𝑚𝑑𝜇_−𝑞(𝑥)+(−1)^𝑛−1_ℤ𝑝𝑤^𝑎𝑥(𝑎𝑥)^𝑚𝑑𝜇_−𝑞(𝑥)𝑡^𝑚=𝑚!_∞𝑚=0𝑞^𝑛𝑤^𝑎𝑛𝐸_𝑚,𝑤(𝑎𝑛∶𝑎)+(−1)^𝑛−1𝐸_𝑚,𝑤(𝑎)𝑡^𝑚.𝑚! (3.11) By comparing coefficients of 𝑡^𝑛/𝑛! in the above equation, we arrive at the following theorem.

Theorem 3.7. For 𝑛∈ℤ₊, one has 𝑞^𝑛𝑤^𝑎𝑛𝐸_𝑚,𝑤(𝑛𝑎∶𝑎)+(−1)^𝑛−1𝐸_𝑚,𝑤(𝑎)=[2]_𝑞^𝑛−1_𝑙=0(−1)^{𝑛−1−𝑙}𝑤^𝑎𝑙𝑞^𝑙(𝑎𝑙)^𝑚. (3.12)

4. The Analogue of the 𝑞-Euler Zeta Function

By using the generalized (𝑞,𝑤)-Euler numbers and polynomials, the generalized (𝑞,𝑤)-Euler zeta function and the generalized Hurwitz (𝑞,𝑤)-Euler zeta functions are defined. These functions interpolate the generalized (𝑞,𝑤)-Euler numbers and (𝑞,𝑤)-Euler polynomials, respectively. Let 𝐹_𝑞,𝑤(𝑥∶𝑎)(𝑡)=[2]_𝑞^∞_𝑛=0(−1)^𝑛𝑞^𝑛𝑤^𝑎𝑛𝑒^𝑎𝑛𝑡𝑒^𝑥𝑡=^∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}()𝑡𝑥∶𝑎^𝑛.𝑛! (4.1) By applying derivative operator, 𝑑^𝑘/𝑑𝑡^𝑘|_𝑡=0 to the above equation, we have 𝑑^𝑘𝑑𝑡^𝑘𝐹_𝑞,𝑤(𝑥∶𝑎)(𝑡)_𝑡=0=[2]_𝑞^∞_𝑛=0(−1)^𝑛𝑞^𝑛𝑤^𝑎𝑛(𝑎𝑛+𝑥)^𝑘,(𝑘∈ℕ),𝐸_{𝑘,𝑞,𝑤}(𝑥∶𝑎)=[2]_𝑞^∞_𝑛=0(−1)^𝑛𝑞^𝑛𝑤^𝑎𝑛(𝑎𝑛+𝑥)^𝑘. (4.2)(4.3)

By using the above equation, we are now ready to define the generalized (𝑞,𝑤)-Euler zeta functions.

Definition 4.1. For 𝑠∈ℂ, one defines 𝜁^(𝑎)_𝑞,𝑤(𝑥∶𝑠)=2^∞_𝑛=1(−1)^𝑛𝑞^𝑛𝑤^𝑎𝑛(𝑎𝑛+𝑥)^𝑠. (4.4)
Note that 𝜁_𝑤^(𝑎)(𝑥,𝑠) is a meromorphic function on ℂ. Note that, if 𝑤→1, 𝑤→1, and 𝑎=1, then 𝜁^(𝑎)_𝑞,𝑤(𝑥∶𝑠)=𝜁(𝑥∶𝑠) which is the Hurwitz Euler zeta functions. Relation between 𝜁_𝑤^(𝑎)(𝑥∶𝑠) and 𝐸_𝑘,𝑤(𝑥∶𝑎) is given by the following theorem.

Theorem 4.2. For 𝑘∈ℕ, one has 𝜁^(𝑎)_𝑞,𝑤(𝑥∶−𝑘)=𝐸_𝑘,𝑤(𝑥∶𝑎). (4.5)
By using (4.2), one notes that 𝑑^𝑘𝑑𝑡^𝑘𝐹_𝑞,𝑤(0∶𝑎)(𝑡)_𝑡=0=[2]_𝑞^∞_𝑛=0(−1)^𝑛𝑞^𝑛𝑤^𝑎𝑛(𝑎𝑛)^𝑘,(𝑘∈ℕ). (4.6) Hence, one obtains 𝐸_{𝑘,𝑞,𝑤}(𝑎)=[2]_𝑞^∞_𝑛=0(−1)^𝑛𝑞^𝑛𝑤^𝑎𝑛(𝑎𝑛)^𝑘. (4.7)

By using the above equation, one is now ready to define the generalized Hurwitz (𝑞,𝑤)-Euler zeta functions.

Definition 4.3. Let 𝑠∈ℂ. One defines 𝜁^(𝑎)_𝑞,𝑤(𝑠)=2^∞_𝑛=1(−1)^𝑛𝑞^𝑛𝑤^𝑎𝑛(𝑎𝑛)^𝑠. (4.8) Note that 𝜁^(𝑎)_𝑞,𝑤(𝑠) is a meromorphic function on ℂ. Obverse that, if 𝑤→1, 𝑞→1, and 𝑎=1, then 𝜁_𝑤^(𝑎)(𝑠)=𝜁(𝑠) which is the Euler zeta functions. Relation between 𝜁_𝑤^(𝑎)(𝑠) and 𝐸_𝑘,𝑤(𝑠) is given by the following theorem.

Theorem 4.4. For 𝑘∈ℕ, one has 𝜁^(𝑎)_𝑞,𝑤(−𝑘)=𝐸_{𝑘,𝑞,𝑤}(𝑎). (4.9)

5. Zeros of the Generalized (𝑞,𝑤)-Euler Polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎)

In this section, we investigate the reflection symmetry of the zeros of the generalized (𝑞,𝑤)-Euler polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎).

In the special case, 𝑤=1 and 𝑞→1, 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) are called generalized Euler polynomials 𝐸_𝑛(𝑥∶𝑎). Since ^∞_𝑛=0𝐸_𝑛()()𝑎−𝑥∶𝑎−𝑡^𝑛=2𝑛!𝑒^−𝑎𝑡𝑒+1^{(𝑎−𝑥)(−𝑡)}=2𝑒^𝑎𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛()𝑡𝑥∶𝑎^𝑛,𝑛! (5.1) we have 𝐸_𝑛(𝑥∶𝑎)=(−1)^𝑛𝐸_𝑛(𝑎−𝑥∶𝑎)for𝑛∈ℕ. (5.2) We observe that 𝐸_𝑛(𝑥∶𝑎),𝑥∈ℂ has Re(𝑥)=𝑎/2 reflection symmetry in addition to the usual Im(𝑥)=0 reflection symmetry analytic complex functions.

Let 𝐹_𝑞,𝑤()=[2]𝑥∶𝑡^𝑞𝑞𝑤^𝑎𝑒^𝑎𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}()𝑡𝑥∶𝑎^𝑛.𝑛! (5.3) Then, we have 𝐹_{𝑞−1,𝑤−1}()=[2]𝑎−𝑥∶−𝑡^𝑞−1𝑞⁻¹𝑤^−𝑎𝑒^−𝑎𝑡𝑒+1^{(𝑎−𝑥)(−𝑡)}=𝑤^𝑎[2]_𝑞𝑞𝑤^𝑎𝑒^𝑎𝑡𝑒+1^𝑥𝑡=𝑤^𝑎∞_𝑛=0𝐸_{𝑛,𝑞,𝑤}()𝑡𝑥∶𝑎^𝑛.𝑛! (5.4) Hence, we arrive at the following complement theorem.

Theorem 5.1 (complement theorem). For 𝑛∈ℕ, 𝐸_{𝑛,𝑞−1,𝑤−1}(𝑎−𝑥∶𝑎)=(−1)^𝑛𝑤^𝑎𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎). (5.5)

Throughout the numerical experiments, we can finally conclude that 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎),𝑥∈ℂ has not Re(𝑥)=𝑎/2 reflection symmetry analytic complex functions. However, we observe that 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎),𝑥∈ℂ has Im(𝑥)=0 reflection symmetry (see Figures 1, 2, and 3). The obvious corollary is that the zeros of 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) will also inherit these symmetries. If𝐸_{𝑛,𝑞,𝑤}𝑥₀∶𝑎=0,then𝐸_{𝑛,𝑞,𝑤}𝑥^∗₀∶𝑎=0, (5.6) where ∗ denotes complex conjugation (see Figures 1, 2, and 3).

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Figure 1 

Zeros of 𝐸𝑛,𝑞,𝑤(𝑥∶𝑎) for 𝑎=1,2,3,4.

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Figure 2 

Zeros of 𝐸𝑛,𝑞,𝑤(𝑥∶𝑎) for 𝑞=1/10,3/10,7/10,9/10.

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Figure 3 

Real zeros of 𝐸𝑛,𝑞,𝑤(𝑥∶𝑎) for 1≤𝑛≤25.

We investigate the beautiful zeros of the generalized (𝑞,𝑤)-Euler polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) by using a computer. We plot the zeros of the generalized Euler polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) for 𝑛=30,𝑎=1,2,3,4, and 𝑥∈ℂ (Figure 1). In Figure 1 (top-left), we choose 𝑛=30,𝑞=1/2,𝑤=1, and 𝑎=1. In Figure 1 (top-right), we choose 𝑛=30,𝑞=1/2,𝑤=2, and 𝑎=2. In Figure 1 (bottom-left), we choose 𝑛=30,𝑞=1/2,𝑤=3, and 𝑎=3. In Figure 1 (bottom-right), we choose 𝑛=30,𝑞=1/2,𝑤=4, and 𝑎=4.

We plot the zeros of the generalized Euler polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) for 𝑛=30,𝑎=2,𝑤=2, and 𝑥∈ℂ (Figure 2).

In Figure 2 (top-left), we choose 𝑛=30,𝑞=1/10,𝑤=2, and 𝑎=2. In Figure 2 (top-right), we choose 𝑛=30,𝑞=3/10,𝑤=2, and 𝑎=2. In Figure 2 (bottom-left), we choose 𝑛=30,𝑞=7/10,𝑤=2, and 𝑎=2. In Figure 2 (bottom-right), we choose 𝑛=30,𝑞=9/10,𝑤=2 and 𝑎=2.

Plots of real zeros of 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) for 1≤𝑛≤25 structure are presented (Figure 3).

In Figure 3 (top-left), we choose 𝑞=1/2,𝑤=1, and 𝑎=2. In Figure 3 (top-right), we choose 𝑞=1/2,𝑤=2, and 𝑎=2. In Figure 3 (bottom-left), we choose 𝑞=1/2,𝑤=3, and 𝑎=2. In Figure 3 (bottom-right), we choose 𝑞=1/2,𝑤=4, and 𝑎=2.

Stacks of zeros of 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) for 1≤𝑛≤30,𝑞=1/2,𝑤=4,and𝑎=4 from a 3-D structure are presented (Figure 4).

Figure 4 

Stacks of zeros of 𝐸𝑛,𝑞,𝑤(𝑥∶𝑎) for 1≤𝑛≤30.

Our numerical results for approximate solutions of real zeros of the generalized 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) are displayed (Tables 1 and 2).

Table 1 

Numbers of real and complex zeros of 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎).

Table 2 

Approximate solutions of 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎)=0,𝑥∈ℝ.

We observe a remarkably regular structure of the complex roots of the generalized (𝑞,𝑤)-Euler polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎). We hope to verify a remarkably regular structure of the complex roots of the generalized (𝑞,𝑤)-Euler polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) (Table 1).

Next, we calculated an approximate solution satisfying 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎),𝑞=1/2,𝑤=2,𝑎=2,𝑥∈ℝ. The results are given in Table 2.

Figure 5 shows the generalized (𝑞,𝑤)-Euler polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) for real −9/10≤𝑞≤9/10 and −5≤𝑥≤5, with the zero contour indicated in black (Figure 5). In Figure 5 (top-left), we choose 𝑛=1, 𝑤=2, and 𝑎=2. In Figure 5 (top-right), we choose 𝑛=2, 𝑤=2, and 𝑎=2. In Figure 5 (bottom-left), we choose 𝑛=3, 𝑤=2, and 𝑎=2. In Figure 5 (bottom-right), we choose 𝑛=4,𝑤=2, and 𝑎=2.

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Figure 5 

Zero contour of 𝐸𝑛,𝑞,𝑤(𝑥∶𝑎).

Finally, we will consider the more general problems. How many roots does 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) have? This is an open problem. Prove or disprove: 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎)=0 has 𝑛 distinct solutions. Find the numbers of complex zeros 𝐶_{𝐸𝑛,𝑞,𝑤(𝑥∶𝑎)} of 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎),Im(𝑥∶𝑎)≠0. Since 𝑛 is the degree of the polynomial 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎), the number of real zeros 𝑅_{𝐸𝑛,𝑞,𝑤(𝑥∶𝑎)} lying on the real plane Im(𝑥∶𝑎)=0 is then 𝑅_{𝐸𝑛,𝑞,𝑤(𝑥∶𝑎)}=𝑛−𝐶_{𝐸𝑛,𝑞,𝑤(𝑥∶𝑎)}, where 𝐶_{𝐸𝑛,𝑞,𝑤(𝑥∶𝑎)} denotes complex zeros. See Table 1 for tabulated values of 𝑅_{𝐸𝑛,𝑞,𝑤(𝑥∶𝑎)} and 𝐶_{𝐸𝑛,𝑞,𝑤(𝑥∶𝑎)}. We plot the zeros of 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎), respectively (Figures 1–5). These figures give mathematicians an unbounded capacity to create visual mathematical investigations of the behavior of the roots of the 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎). Moreover, it is possible to create a new mathematical ideas and analyze them in ways that generally are not possible by hand. The authors have no doubt that investigation along this line will lead to a new approach employing numerical method in the field of research of (𝑞,𝑤)-Euler polynomials 𝐸_{𝑛,𝑞,𝑤}(𝑥∶𝑎) to appear in mathematics and physics.