Zeros of Analytic Continued 𝑞-Euler Polynomials and 𝑞-Euler Zeta Function

Abstract

We study that the 𝑞-Euler numbers 𝐸_𝑛,𝑞 and 𝑞-Euler polynomials 𝐸_𝑛,𝑞(𝑥) are analytic continued to 𝐸_𝑞(𝑠) and 𝐸_𝑞(𝑠,𝑤). We investigate the new concept of dynamics of the zeros of analytic continued polynomials. Finally, we observe an interesting phenomenon of ‘‘scattering’’ of the zeros of 𝐸_𝑞(𝑠,𝑤).

1. Introduction

Recently, the computing environment would make more and more rapid progress and there has been increasing interest in solving mathematical problems with the aid of computers. By using software, many mathematicians can explore concepts much easier than in the past. The ability to create and manipulate figures on the computer screen enables mathematicians to quickly visualize and produce many problems, examine properties of the figures, look for patterns, and make conjectures. This capability is especially exciting because these steps are essential for most mathematicians to truly understand even basic concept. Mathematicians have studied different kinds of the Euler, Bernoulli, Tangent, and Genocchi numbers and polynomials (see [1–14]). Numerical experiments of Bernoulli polynomials, Euler polynomials, Genocchi polynomials, and Tangent polynomials have been the subject of extensive study in recent years and much progress have been made both mathematically and computationally. Using computer, a realistic study for 𝑞-Euler polynomials 𝐸_𝑛,𝑞(𝑥) is very interesting. It is the aim of this paper to observe an interesting phenomenon of “scattering” of the zeros of the 𝑞-Euler polynomials 𝐸_𝑛,𝑞(𝑥) in complex plane. Throughout this paper, we always make use of the following notations: ℕ denotes the set of natural numbers, ℕ₀ denotes the set of nonnegative integers, ℝ denotes the set of real numbers, and ℂ denotes the set of complex numbers. In [11], we introduce the 𝑞-Euler numbers 𝐸_𝑛,𝑞 and polynomials 𝐸_𝑛,𝑞(𝑥) and investigate their properties. Let 𝑞 be a complex number with |𝑞|<1. By the meaning of (1) and (2), let us define the 𝑞-Euler numbers 𝐸_𝑛,𝑞 and polynomials 𝐸_𝑛,𝑞(𝑥) as follows (see [11]): 𝐹_𝑞(𝑡)=2𝑞𝑒^𝑡=+1^∞_𝑛=0𝐸_𝑛,𝑞𝑡^𝑛,𝑛!𝐹_𝑞()=2𝑥,𝑡𝑞𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛! (1)(2) Observe that if 𝑞→1, then 𝐸_𝑛,𝑞(𝑥)=𝐸_𝑛(𝑥) and 𝐸_𝑛,𝑞=𝐸_𝑛.

By using computer, the 𝑞-Euler numbers 𝐸_𝑛,𝑞 can be determined explicitly. A few of them are 𝐸_0,𝑞=21+𝑞,𝐸_1,𝑞=−2𝑞(1+𝑞)²,𝐸_2,𝑞=−2𝑞(1+𝑞)²+4𝑞²(1+𝑞)³,𝐸_3,𝑞=−2𝑞(1+𝑞)²+12𝑞²(1+𝑞)³−12𝑞³(1+𝑞)⁴. (3)

Theorem 1. For 𝑛∈ℕ₀, one has 𝐸_𝑛,𝑞(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐸_𝑙,𝑞𝑥^𝑛−𝑙. (4)

By Theorem 1, after some elementary calculations, we have ^𝑏_𝑎𝐸_𝑛,𝑞(𝑥)𝑑𝑥=^𝑛_𝑙=0𝑛𝑙𝐸_𝑙,𝑞^𝑏_𝑎𝑥^𝑛−𝑙=1𝑑𝑥𝑛+1^𝑛+1_𝑙=0𝑙𝐸𝑛+1_𝑙,𝑞𝑥^{𝑛−𝑙+1𝑏}_𝑎=𝐸_𝑛+1,𝑞(𝑏)−𝐸_𝑛+1,𝑞(𝑎).𝑛+1 (5)

Since 𝐸_𝑛,𝑞(0)=𝐸_𝑛,𝑞, by (5), we obtain 𝐸_𝑛,𝑞(𝑥)=𝐸_𝑛,𝑞+𝑛^𝑥₀𝐸_{𝑛−1,𝑞}(𝑡)𝑑𝑡,for𝑛∈ℕ. (6)

Then, it is easy to deduce that 𝐸_𝑛,𝑞(𝑥) are polynomials of degree 𝑛. Here is the list of the first 𝑞-Euler’s polynomials: 𝐸_0,𝑞(𝑥)=2,𝐸1+𝑞_1,𝑞(𝑥)=−2𝑞(1+𝑞)²+2𝑥,𝐸1+𝑞_2,𝑞(𝑥)=−2𝑞(1+𝑞)²+4𝑞²(1+𝑞)³−4𝑞𝑥(1+𝑞)²+2𝑥²,𝐸1+𝑞_3,𝑞(𝑥)=−2𝑞(1+𝑞)²+12𝑞²(1+𝑞)³−12𝑞³(1+𝑞)⁴+12𝑞²𝑥(1+𝑞)³−6𝑞𝑥(1+𝑞)²−6𝑞𝑥²(1+𝑞)²+2𝑥³.1+𝑞 (7)

2. Analytic Continuation of Euler Numbers 𝐸_𝑛,𝑞

In this section, we introduced the 𝑞-Euler zeta function and Hurwitz 𝑞-Euler zeta function. By 𝑞-Euler zeta function, we consider the function 𝐸_𝑞(𝑠) as the analytic continuation of 𝑞-Euler numbers.

From (1), we note that 𝑑^𝑘𝑑𝑡^𝑘𝐹_𝑞(𝑡)_𝑡=0=2^∞_𝑚=0(−1)^𝑚𝑞^𝑚𝑚^𝑘=𝐸_𝑘,𝑞,(𝑘∈ℕ). (8) By using the above equation, we are now ready to define 𝑞-Euler zeta functions.

Definition 2. Let 𝑠∈ℂ with Re(𝑠)>0. Consider 𝜁_𝐸,𝑞(𝑠)=2^∞_𝑛=1(−1)^𝑛𝑞^𝑛𝑛^𝑠. (9)
Observe that 𝜁_𝐸,𝑞(𝑠) is a meromorphic function on ℂ. Clearly, lim_𝑞→1𝜁_𝐸,𝑞(𝑠)=𝜁_𝐸(𝑠) (see [4, 7, 12]). Notice that the 𝑞-Euler zeta function can be analytically continued to the whole complex plane, and these zeta function has the values of the 𝑞-Euler numbers at negative integers.

Theorem 3. For 𝑘∈ℕ, one has 𝜁_𝐸,𝑞(−𝑘)=𝐸_𝑘,𝑞. (10)
Observe that 𝜁_𝐸,𝑞(𝑠) function interpolates 𝐸_𝑘,𝑞 numbers at nonnegative integers.
By using (2), we note that 𝑑^𝑘𝑑𝑡^𝑘𝐹_𝑞(𝑥,𝑡)_𝑡=0=2^∞_𝑚=0(−1)^𝑚𝑞^𝑚(𝑥+𝑚)^𝑘=𝐸_𝑘,𝑞(𝑥),(𝑘∈ℕ),𝑑𝑑𝑡^𝑘∞_𝑛=0𝐸_𝑛,𝑞𝑡(𝑥)^𝑛𝑛!_𝑡=0=𝐸_𝑘,𝑞(𝑥),𝑓𝑜𝑟𝑘∈ℕ. (11)(12) By (12), we are now ready to define the Hurwitz-type 𝑞-Euler zeta functions.

Definition 4. Let 𝑠∈ℂ with Re(𝑠)>0. Consider 𝜁_𝐸,𝑞(𝑠,𝑥)=2^∞_𝑛=0(−1)^𝑛𝑞^𝑛(𝑛+𝑥)^𝑠. (13)
Note that 𝜁_𝐸,𝑞(𝑠,𝑥) is a meromorphic function on ℂ. The relation between 𝜁_𝐸,𝑞(𝑠,𝑥) and 𝐸_𝑘,𝑞(𝑥) is given by the following theorem.

Theorem 5. For 𝑘∈ℕ, one has 𝜁_𝐸,𝑞(−𝑘,𝑥)=𝐸_𝑘,𝑞(𝑥). (14)

We now consider the function 𝐸_𝑞(𝑠) as the analytic continuation of 𝑞-Euler numbers. From the above analytic continuation of 𝑞-Euler numbers, we consider 𝐸_𝑛,𝑞⟼𝐸_𝑞(𝑠),𝜁_𝐸,𝑞(−𝑛)=𝐸_𝑛,𝑞⟼𝜁_𝐸,𝑞(−𝑠)=𝐸_𝑞(𝑠). (15)

All the 𝑞-Euler numbers 𝐸_𝑛,𝑞 agree with 𝐸_𝑞(𝑛), the analytic continuation of 𝑞-Euler numbers evaluated at 𝑛 (see Figure 1). Consider 𝐸_𝑛,𝑞=𝐸_𝑞(𝑛),for𝑛≥1except𝐸_𝑞(0)=−2𝑞,1+𝑞but𝐸_0,𝑞=2.1+𝑞 (16) In Figure 1, we choose 𝑞=1/2. In fact, we can express 𝐸^′_𝑞(𝑠) in terms of 𝜁^′_𝐸,𝑞(𝑠), the derivative of 𝜁_𝐸(𝑠). Consider 𝐸_𝑞(𝑠)=𝜁_𝐸,𝑞(−𝑠),𝐸^′_𝑞(𝑠)=−𝜁^′_𝐸,𝑞(),𝐸−𝑠_′𝑞(2𝑛+1)=−𝜁^′_𝐸,𝑞(−2𝑛−1),for𝑛∈ℕ₀. (17) From the relation (17), we can define the other analytic continued half of 𝑞-Euler numbers: 𝐸_𝑞(𝑠)=𝜁_𝐸,𝑞(−𝑠),𝐸_𝑞(−𝑠)=𝜁_𝐸,𝑞(𝑠)⟹𝐸_𝑞(−𝑛)=𝜁_𝐸,𝑞(𝑛),𝑛∈ℕ. (18) By (18), we obtain lim_𝑛→∞𝐸_−𝑛,𝑞=𝜁_𝐸,𝑞(𝑛)=−2𝑞. (19) The curve 𝐸_𝑞(𝑠) runs through the points 𝐸_−𝑛,𝑞=𝐸_𝑞(−𝑛) and grows ∼−2𝑞 asymptotically as −𝑛→∞ (see Figure 2).

Figure 1 

The curve 𝐸𝑞(𝑠) runs through the points of all 𝐸𝑛,𝑞 except 𝐸0,𝑞.

Figure 2 

The curve 𝐸𝑞(𝑠) runs through the points 𝐸−𝑛,𝑞 for 𝑞=1/2.

3. Analytic Continuation of Euler Polynomials 𝐸_𝑛,𝑞(𝑥)

In this section, we observe the analytic continued 𝑞-Euler polynomials. Looking back at (9) and (18), for consistency with the definition of 𝐸_𝑛,𝑞(𝑥)=𝐸_𝑞(𝑛,𝑥), 𝑞-Euler polynomials should be analogously redefined as 𝐸_𝑞(0,𝑥)=−𝑞𝐸_0,𝑞(𝑥),𝐸_𝑞(𝑛,𝑥)=^𝑛_𝑙=0𝑛𝑙𝐸_𝑙,𝑞𝑥^𝑛−𝑙. (20)

Let Γ(𝑠) be the gamma function. The analytic continuation can be then obtained as 𝐸𝑛⟼𝑠∈ℝ,𝑥⟼𝑤∈ℂ,_0,𝑞⟼𝐸_𝑞(0)1=−𝑞𝜁_𝐸,𝑞(0),𝐸_𝑘,𝑞⟼𝐸_𝑞(𝑘+𝑠−[𝑠])=𝜁_𝐸,𝑞(−(([𝑠,𝑛𝑘⟼Γ()𝑘+𝑠−])))1+𝑠Γ(1+𝑘+(𝑠−[𝑠]))Γ(1+[𝑠]−𝑘)⟹𝐸_𝑛,𝑞(𝑤)⟼𝐸_𝑞()=𝑠,𝑤_{[𝑠]𝑘=−1}Γ(1+𝑠)𝐸_𝑞(𝑘+𝑠−[𝑠])𝑤^{[𝑠]−𝑘}Γ(([𝑠Γ([𝑠])=1+𝑘+𝑠−]))1+−𝑘_{[𝑠]+1𝑘=0}Γ(1+𝑠)𝐸_𝑞((𝑘−1)+𝑠−[𝑠])𝑤^{[𝑠]+1−𝑘}Γ(([𝑠Γ([𝑠]),𝑘+𝑠−]))2+−𝑘 (21) where [𝑠] gives the integer part of 𝑠, and so 𝑠−[𝑠] gives the fractional part.

By (21), we have analytic continuation of 𝑞-Euler polynomials for 𝑞=1/2. Consider 𝐸_𝑞()𝐸0,𝑤≈1.33333,_𝑞()𝐸1,𝑤≈−0.44444+1.33333𝑤,_𝑞(2,𝑤)≈−0.14814−0.88888𝑤+1.33333𝑤²,𝐸_𝑞(2.1,𝑤)≈−0.11635−0.87711𝑤−0.74865𝑤²−0.03078𝑤³,𝐸_𝑞(2.3,𝑤)≈−0.05297−0.83227𝑤−0.91011𝑤²−0.11583𝑤³,𝐸_𝑞(2.5,𝑤)≈0.00907−0.75822𝑤−1.06129𝑤²−0.23359𝑤³,𝐸_𝑞(2.7,𝑤)≈0.06832−0.65450𝑤−1.19317𝑤²−0.38417𝑤³,𝐸_𝑞(2.9,𝑤)≈0.123069−0.521512𝑤−1.29600𝑤²−0.56561𝑤³,𝐸_𝑞(3,𝑤)≈0.14814−0.444444𝑤−1.33333𝑤²+1.33333𝑤³. (22)

By using (22), we plot the deformation of the curve 𝐸_𝑞(2,𝑤) into the curve of 𝐸_𝑞(3,𝑤) via the real analytic continuation 𝐸_𝑞(𝑠,𝑤), 2≤𝑠≤3, 𝑤∈ℝ (see Figure 3).

Figure 3 

The curve of 𝐸𝑞(𝑠,𝑤), 1≤𝑠≤2, and −0.1≤𝑤≤0.1.

Next, we investigate the beautiful zeros of the 𝐸_𝑞(𝑛,𝑤) by using a computer. We plot the zeros of 𝐸_𝑞(𝑛,𝑤) for 𝑛∈ℕ, 𝑞=1/2, and 𝑤∈ℂ (Figure 4).

(a)

(b)

(c)

(d)

Figure 4 

Stacks of zeros of 𝐸𝑞(𝑛,𝑥) for 1≤𝑛≤30.

In Figure 4, we observe that 𝐸_𝑞(𝑛,𝑤), 𝑤∈ℂ, has Im(𝑤)=0 reflection symmetry analytic complex functions (Figure 4). The obvious corollary is that the zeros of 𝐸_𝑞(𝑛,𝑤) will also inherit these symmetries. Consider if𝐸_𝑞𝑛,𝑤₀=0,then𝐸_𝑞𝑛,𝑤^∗₀=0, (23) where ∗ denotes complex conjugation.

Finally, we investigate the beautiful zeros of the 𝐸_𝑞(𝑠,𝑤) by using a computer. We plot the zeros of 𝐸_𝑞(𝑠,𝑤) for 𝑠=20,20.6,20.8,21, 𝑞=1/2, and 𝑤∈ℂ (Figure 5). In Figure 5(a), we choose 𝑠=20. In Figure 5(b), we choose 𝑠=20.6. In Figure 5(c), we choose 𝑠=20.8. In Figure 5(d), we choose 𝑠=21.

(a)

(b)

(c)

(d)

Figure 5 

Zeros of 𝐸𝑞(𝑠,𝑤) for 𝑠=20,20.6,20.8,21.

Since ^∞_𝑛=0𝐸_{𝑛,𝑞−1}(1−𝑥)(−1)^𝑛𝑡^𝑛=2𝑛!𝑞⁻¹𝑒^−𝑡𝑒+1^{(1−𝑥)(−𝑡)}2=𝑞𝑞𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝑞𝐸_𝑛(𝑥)𝑡^𝑛,𝑛! (24) we obtain 𝑞𝐸_𝑛,𝑞(𝑥)=(−1)^𝑛𝐸_𝑛,𝑞(1−𝑥). (25)

The question is as follows: what happens with the reflexive symmetry (25), when one considers 𝑞-Euler polynomials? Prove that 𝐸_𝑞(𝑛,𝑤), 𝑤∈ℂ, has not Re(𝑤)=1/2 reflection symmetry analytic complex functions (Figure 4). However, we observe that 𝐸_𝑞(𝑠,𝑤), 𝑤∈ℂ, has Im(𝑥)=0 reflection symmetry analytic complex functions (Figure 5).

Stacks of zeros of 𝐸_𝑞(𝑠,𝑤) for 𝑠=𝑛+1/2, 1≤𝑛≤30, forming a 3D structure are presented (Figure 6).

(a)

(b)

(c)

(d)

Figure 6 

Stacks of zeros of 𝐸𝑞(𝑠,𝑤) for 1≤𝑛≤30.

Our numerical results for approximate solutions of real zeros of 𝐸_𝑞(𝑠,𝑤), 𝑞=1/2, are displayed. We observe a remarkably regular structure of the complex roots of Euler polynomials. We hope to verify a remarkably regular structure of the complex roots of Euler polynomials (Table 1).

Table 1 

Numbers of real and complex zeros of 𝐸_𝑞(𝑠,𝑤).

Next, we calculated an approximate solution satisfying 𝐸_𝑞(𝑠,𝑤), 𝑞=1/2, 𝑤∈ℝ. The results are given in Table 2.

Table 2 

Approximate solutions of 𝐸_𝑞(𝑠,𝑤)=0,𝑤∈ℝ.

In Figure 7, we plot the real zeros of the 𝑞-Euler polynomials 𝐸_𝑞(𝑠,𝑤) for 𝑠=𝑛+(1/2), 𝑞=1/2, and 𝑤∈ℂ (Figure 7).

Figure 7 

Real zeros of 𝐸𝑞(𝑠,𝑤).

𝑞-Euler polynomials 𝐸_𝑞(𝑛,𝑤) are polynomials of degree 𝑛. Thus, 𝐸_𝑞(𝑛,𝑤) has 𝑛 zeros and 𝐸_𝑞(𝑛+1,𝑤) has 𝑛+1 zeros. When discrete 𝑛 is analytic continued to continuous parameter 𝑠, it naturally leads to the following question.

How does 𝐸_𝑞(𝑠,𝑤), the analytic continuation of 𝐸_𝑞(𝑛,𝑤), pick up an additional zero as 𝑠 increases continuously by one?

This introduces the exciting concept of the dynamics of the zeros of analytic continued Euler polynomials—the idea of looking at how the zeros move about in the 𝑤 complex plane as we vary the parameter 𝑠.

To have a physical picture of the motion of the zeros in the complex 𝑤 plane, imagine that each time, as 𝑠 increases gradually and continuously by one, an additional real zero flies in from positive infinity along the real positive axis, gradually slowing down as if “it is flying through a viscous medium.”

For more studies and results on this subject you may see [5, 10–12].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.