# Zeros of Analytic Continuedq-Euler Polynomials and q-Euler Zeta Function

Authors: C. S. Ryoo
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We study that theq-Euler numbers En,q and q-Euler polynomials En,q(x) are analytic continued to Eq(s) and Eq(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 Eq(s,w).


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## Body

## 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 En,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 En,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 En,q and polynomials En,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 En,q and polynomials En,q(x) as follows (see [11]):
(1)Fq(t)=2qet+1=∑n=0∞‍En,qtnn!,(2)Fq(x,t)=(2qet+1)ext=∑n=0∞‍En,q(x)tnn!.
Observe that if q→1, then En,q(x)=En(x) and En,q=En.By using computer, theq-Euler numbers En,q can be determined explicitly. A few of them are
(3)E0,q=21+q,E1,q=-2q(1+q)2,E2,q=-2q(1+q)2+4q2(1+q)3,E3,q=-2q(1+q)2+12q2(1+q)3-12q3(1+q)4.Theorem 1.
Forn∈N0, one has
(4)En,q(x)=∑l=0n‍(nl)El,qxn-l.By Theorem1, after some elementary calculations, we have
(5)∫ab‍En,q(x)dx=∑l=0n‍(nl)El,q∫ab‍xn-ldx=1n+1∑l=0n+1‍(n+1l)El,qxn-l+1|ab=En+1,q(b)-En+1,q(a)n+1.SinceEn,q(0)=En,q, by (5), we obtain
(6)En,q(x)=En,q+n∫0x‍En-1,q(t)dt,forn∈N.Then, it is easy to deduce thatEn,q(x) are polynomials of degree n. Here is the list of the first q-Euler’s polynomials:
(7)E0,q(x)=21+q,E1,q(x)=-2q(1+q)2+2x1+q,E2,q(x)=-2q(1+q)2+4q2(1+q)3-4qx(1+q)2+2x21+q,E3,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+2x31+q.


## 2. Analytic Continuation of Euler NumbersEn,q

In this section, we introduced theq-Euler zeta function and Hurwitz q-Euler zeta function. By q-Euler zeta function, we consider the function Eq(s) as the analytic continuation of q-Euler numbers.From (1), we note that
(8)dkdtkFq(t)|t=0=2∑m=0∞‍(-1)mqmmk=Ek,q,(k∈N).
By using the above equation, we are now ready to define q-Euler zeta functions.Definition 2.
Lets∈C with Re(s)>0. Consider
(9)ζE,q(s)=2∑n=1∞‍(-1)nqnns.
Observe thatζE,q(s) is a meromorphic function on C. Clearly, lim⁡q→1ζE,q(s)=ζE(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.
Fork∈N, one has
(10)ζE,q(-k)=Ek,q.
Observe thatζE,q(s) function interpolates Ek,q numbers at nonnegative integers.
By using (2), we note that
(11)dkdtkFq(x,t)|t=0=2∑m=0∞‍(-1)mqm(x+m)k=Ek,q(x),5555555555555555555555555555555551(k∈N),(12)(ddt)k(∑n=0∞‍En,q(x)tnn!)|t=0=Ek,q(x),fork∈N.
By (12), we are now ready to define the Hurwitz-type q-Euler zeta functions.Definition 4.
Lets∈C with Re(s)>0. Consider
(13)ζE,q(s,x)=2∑n=0∞‍(-1)nqn(n+x)s.
Note thatζE,q(s,x) is a meromorphic function on C. The relation between ζE,q(s,x) and Ek,q(x) is given by the following theorem.Theorem 5.
Fork∈N, one has
(14)ζE,q(-k,x)=Ek,q(x).We now consider the functionEq(s) as the analytic continuation of q-Euler numbers. From the above analytic continuation of q-Euler numbers, we consider
(15)En,q⟼Eq(s),ζE,q(-n)=En,q⟼ζE,q(-s)=Eq(s).All theq-Euler numbers En,q agree with Eq(n), the analytic continuation of q-Euler numbers evaluated at n (see Figure 1). Consider
(16)En,q=Eq(n),forn≥1exceptEq(0)=-2q1+q,butE0,q=21+q.
In Figure 1, we choose q=1/2. In fact, we can express Eq′(s) in terms of ζE,q′(s), the derivative of ζE(s). Consider
(17)Eq(s)=ζE,q(-s),Eq′(s)=-ζE,q′(-s),Eq′(2n+1)=-ζE,q′(-2n-1),forn∈N0.
From the relation (17), we can define the other analytic continued half of q-Euler numbers:
(18)Eq(s)=ζE,q(-s),Eq(-s)=ζE,q(s)⟹Eq(-n)=ζE,q(n),n∈N.
By (18), we obtain
(19)lim⁡n→∞E-n,q=ζE,q(n)=-2q.
The curve Eq(s) runs through the points E-n,q=Eq(-n) and grows ~-2q asymptotically as -n→∞ (see Figure 2).Figure 1
The curveEq(s) runs through the points of all En,q except E0,q.Figure 2
The curveEq(s) runs through the points E-n,q for q=1/2.


## 3. Analytic Continuation of Euler PolynomialsEn,q(x)

In this section, we observe the analytic continuedq-Euler polynomials. Looking back at (9) and (18), for consistency with the definition of En,q(x)=Eq(n,x), q-Euler polynomials should be analogously redefined as
(20)Eq(0,x)=-qE0,q(x),Eq(n,x)=∑l=0n‍(nl)El,qxn-l.LetΓ(s) be the gamma function. The analytic continuation can be then obtained as
(21)n⟼s∈R,x⟼w∈C,E0,q⟼Eq(0)=-1qζE,q(0),Ek,q⟼Eq(k+s-[s])=ζE,q(-(k+(s-[s]))),(nk)⟼Γ(1+s)Γ(1+k+(s-[s]))Γ(1+[s]-k)⟹En,q(w)⟼Eq(s,w)=∑k=-1[s]‍Γ(1+s)Eq(k+s-[s])w[s]-kΓ(1+k+(s-[s]))Γ(1+[s]-k)=∑k=0[s]+1‍Γ(1+s)Eq((k-1)+s-[s])w[s]+1-kΓ(k+(s-[s]))Γ(2+[s]-k),
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
(22)Eq(0,w)≈1.33333,Eq(1,w)≈-0.44444+1.33333w,Eq(2,w)≈-0.14814-0.88888w+1.33333w2,Eq(2.1,w)≈-0.11635-0.87711w-0.74865w2-0.03078w3,Eq(2.3,w)≈-0.05297-0.83227w-0.91011w2-0.11583w3,Eq(2.5,w)≈0.00907-0.75822w-1.06129w2-0.23359w3,Eq(2.7,w)≈0.06832-0.65450w-1.19317w2-0.38417w3,Eq(2.9,w)≈0.123069-0.521512w-1.29600w2-0.56561w3,Eq(3,w)≈0.14814-0.444444w-1.33333w2+1.33333w3.By using (22), we plot the deformation of the curve Eq(2,w) into the curve of Eq(3,w) via the real analytic continuation Eq(s,w), 2≤s≤3, w∈R (see Figure 3).Figure 3
The curve ofEq(s,w), 1≤s≤2, and -0.1≤w≤0.1.Next, we investigate the beautiful zeros of theEq(n,w) by using a computer. We plot the zeros of Eq(n,w) for n∈N, q=1/2, and w∈C (Figure 4).Stacks of zeros ofEq(n,x) for 1≤n≤30.
(a)
(b)
(c)
(d)In Figure4, we observe that Eq(n,w), w∈C, has Im⁡(w)=0 reflection symmetry analytic complex functions (Figure 4). The obvious corollary is that the zeros of Eq(n,w) will also inherit these symmetries. Consider
(23)ifEq(n,w0)=0,thenEq(n,w0*)=0,
where * denotes complex conjugation.Finally, we investigate the beautiful zeros of theEq(s,w) by using a computer. We plot the zeros of Eq(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.Zeros ofEq(s,w) for s=20,20.6,20.8,21.
(a)
(b)
(c)
(d)Since(24)∑n=0∞‍En,q-1(1-x)(-1)ntnn!=2q-1e-t+1e(1-x)(-t)=q(2qet+1)ext=∑n=0∞‍qEn(x)tnn!,
we obtain
(25)qEn,q(x)=(-1)nEn,q(1-x).The question is as follows: what happens with the reflexive symmetry (25), when one considers q-Euler polynomials? Prove that Eq(n,w), w∈C, has not Re(w)=1/2 reflection symmetry analytic complex functions (Figure 4). However, we observe that Eq(s,w), w∈C, has Im⁡(x)=0 reflection symmetry analytic complex functions (Figure 5).Stacks of zeros ofEq(s,w) for s=n+1/2, 1≤n≤30, forming a 3D structure are presented (Figure 6).Stacks of zeros ofEq(s,w) for 1≤n≤30.
(a)
(b)
(c)
(d)Our numerical results for approximate solutions of real zeros ofEq(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 ofEq(s,w).
s
Real zeros
Complex zeros
1.5
2
0
2.5
3
0
3.5
4
0
4.5
3
2
5.5
4
2
6.5
5
2
7.5
4
4
8.5
5
4
9.5
4
6
10
4
6
10.6
5
6
10.8
5
6
11
5
6Next, we calculated an approximate solution satisfyingEq(s,w), q=1/2, w∈R. The results are given in Table 2.Table 2
Approximate solutions ofEq(s,w)=0,w∈R.
s
w
6
−0.471899, 0.51601, 1.52785, and 1.95754
6.5
−7.71893, −1.22075, −0.268389, 0.814309, and 0.947618
7
−0.842392, −0.0529263, and 0.947074
7.5
−8.73262, −1.55376, −0.838007, and 0.16246
8
−1.00171, −0.627539, 0.377875, and 1.37783
8.5
−9.74602, −1.62998, −1.46812, −0.406677, and 0.592636
9
−0.191227, 0.808773, and 1.80456
9.5
−10.7592, −0.975608, 0.0242117, and 1.0081
10
−0.758565, 0.239647, 1.23965, and 2.19578
10.6
−8.91801, −1.48957, −0.501829, 0.498175, and 1.39252
10.8
−5.27281, −1.41108, −0.415653, 0.58436, and 1.4511
11
−1.22287, −0.329477, 0.670523, 1.67056, and 2.5067In Figure7, we plot the real zeros of the q-Euler polynomials Eq(s,w) for s=n+(1/2), q=1/2, and w∈C (Figure 7).Figure 7
Real zeros ofEq(s,w).q-Euler polynomials Eq(n,w) are polynomials of degree n. Thus, Eq(n,w) has n zeros and Eq(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 doesEq(s,w), the analytic continuation of Eq(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 thew complex plane as we vary the parameter s.To have a physical picture of the motion of the zeros in the complexw 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].



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