# A Note on theq-Euler Numbers and Polynomials with Weak Weight α

Authors: H. Y. Lee; N. S. Jung; C. S. Ryoo
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

We construct a new type ofq-Euler numbers and polynomials with weak weight α : En,q(α), En,q(α)(x), respectively. Some interesting results and relationships are obtained. Also, we observe the behavior of roots of the q-Euler numbers En,q(α) and polynomials En,q(α)(x) with weak weight α. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of q-Euler polynomials En,q(α) with weak weight α.


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## Body

## 1. Introduction

The Euler numbers and polynomials possess many interesting properties are arising in many areas of mathematics and physics. Recently, many mathematicians have studied the area of theq-Euler numbers and polynomials (see [1–19]). In this paper, we construct a new type of q-Euler numbers En,q(α) and polynomials En,q(α)(x) with weak weight α. The main purpose of this paper is also to investigate the zeros of the q-Euler polynomials En,q(α)(x) with weak weight α. Furthermore, we give a table for the zeros of the q-Euler numbers and polynomials En,q(α)(x) with weak weight α.Throughout this paper we use the following notations. Byℤp we denote the ring of p-adic rational integers, ℚp denotes the field of p-adic rational numbers, ℂp denotes the completion of algebraic closure of ℚp, 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 νp be the normalized exponential valuation of ℂp 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∈ℂp. If q∈ℂ one normally assume that |q|<1. If q∈ℂp, we normally assume that |q-1|p<p-1/(p-1) so that qx=exp(xlogq) for |x|p≤1. Throughout this paper we use the notation (1.1)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q.
(cf. [1–11, 15–18]). Hence, limq→1[x]q=x for any x with |x|p≤1 in the present p-adic case. For (1.2)g∈UD(Zp)={g∣g:Zp⟶Cpisuniformlydifferentiablefunction},
the fermionic p-adic q-integral on ℤp is defined by Kim as follows: (1.3)I-q(g)=∫Zpg(x)dμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1g(x)(-q)x.(cf. [3–6]). If we take g1(x)=g(x+1) in (1.3), then we easily see that (1.4)qI-q(g1)+I-q(g)=[2]qg(0).
From (1.4), we obtain (1.5)qnI-q(gn)+(-1)n-1I-q(g)=[2]q∑l=0n-1(-1)n-1-lqlg(l),
where gn(x)=g(x+n) (cf. [3–6]).As well-known definition, the Euler polynomials are defined by(1.6)F(t)=2et+1=eEt=∑n=0∞Entnn!,F(t,x)=2et+1ext=eE(x)t=∑n=0∞En(x)tnn!,
with the usual convention of replacing En(x) by En(x). In the special case, x=0, En(0)=En are called the nth Euler numbers (cf. [1–11]).Our aim in this paper is to defineq-Euler numbers En,q(α) and polynomials En,q(α)(x) with weak weight α. We investigate some properties which are related to q-Euler numbers En,q(α) and polynomials En,q(α)(x) with weak weight α. We also derive the existence of a specific interpolation function which interpolates q-Euler numbers En,q(α) and polynomials En,q(α)(x) with weak weight α at negative integers. Finally, we investigate the behavior of roots of the q-Euler polynomials En,q(α) with weak weight α.


## 2. Basic Properties forq-Euler Numbers and Polynomials with Weak Weight α

Our primary goal of this section is to defineq-Euler numbers En,q(α) and polynomials En,q(α)(x) with weak weight α. We also find generating functions of q-Euler numbers En,q(α) and polynomials En,q(α)(x) with weak weight α.Forα∈ℤ and q∈ℂp with |1-q|p≤1, q-Euler numbers En,q(α) are defined by (2.1)En,q(α)=∫Zp[x]qndμ-qα(x).
By using p-adic q-integral on ℤp, we obtain (2.2)∫Zp[x]qndμ-qα(x)=lim⁡N→∞1[pN]-qα∑x=0pN-1[x]qn(-qα)x=[2]qα(11-q)n∑l=0n(nl)(-1)l11+qα+l=[2]qα∑m=0∞(-1)mqαm[m]qn.
By (2.1), we have (2.3)En,q(α)=[2]qα(11-q)n∑l=0n(nl)(-1)l11+qα+l=[2]qα∑m=0∞(-1)mqαm[m]qn.
We set (2.4)Fq(α)(t)=∑n=0∞En,q(α)tnn!.
By using above equation and (2.2), we have (2.5)Fq(α)(t)=∑n=0∞En,q(α)tnn!=[2]qα∑n=0∞((11-q)n∑l=0n(nl)(-1)l11+qα+l)tnn!=[2]qα∑m=0∞(-1)mqαme[m]qt.
Thus q-Euler numbers with weak weight α, En,q(α) are defined by means of the generating function (2.6)Fq(α)(t)=[2]qα∑m=0∞(-1)mqαme[m]qt.
By using (2.1), we have (2.7)∑n=0∞En,q(α)tnn!=∑n=0∞∫Zp[x]qndμ-qα(x)tnn!=∫Zpe[x]qtdμ-qα(x).
By (2.5), (2.7), we have (2.8)∫Zpe[x]qtdμ-qα(x)=[2]qα∑m=0∞(-1)mqαme[m]qt.
Next, we introduce q-Euler polynomials En,q(α)(x) with weak weight α. The q-Euler polynomials En,q(α)(x) with weak weight α are defined by (2.9)En,q(α)(x)=∫Zp[x+y]qndμ-qα(y).By usingp-adic q-integral, we obtain (2.10)En,q(α)(x)=[2]qα(11-q)n∑l=0n(nl)(-1)lqxl11+qα+l.
We set (2.11)Fq(α)(t,x)=∑n=0∞En,q(α)(x)tnn!.
By using (2.10) and (2.11), we obtain (2.12)Fq(α)(t,x)=∑n=0∞En,q(α)(x)tnn!=[2]qα∑m=0∞(-1)mqαme[m+x]qt.
Obverse that if q→1, then Fq(α)(t,x)→F(t,x) and Fq(α)(t)→F(t).Since[x+y]q=[x]q+qx[y]q, we easily obtain that (2.13)En,q(α)(x)=∫Zp[x+y]qndμ-qα(y)=∑l=0n(nl)[x]qn-lqxlEl,q(α)=([x]q+qxEq(α))n=[2]qα∑m=0∞(-1)mqαm[x+m]qn.Observe that ifq→1, then En,q(α)→En and En,q(α)(x)→En(x).By (2.10), we have the following complement relation.Theorem 2.1 (property of complement).
One has(2.14)En,q-1(α)(1-x)=(-1)nqnEn,q(α)(x).
By (2.10), we have the following distribution relation.Theorem 2.2 (distribution relation).
For any positive integerm(=odd), one has
(2.15)En,q(α)(x)=[2]qα[2]qαm[m]qn∑i=0m-1(-1)iqαiEn,qm(α)(i+xm),n∈Z+.
By (1.5), (2.1), and (2.9), we easily see that
(2.16)[2]qα∑l=0n-1(-1)n-1-lqαl[l]qm=qαnEm,q(α)(n)+(-1)n-1Em,q(α).
Hence, we have the following theorem.Theorem 2.3.
Letm∈ℤ+. If n≡0(mod2), then
(2.17)qαnEm,q(α)(n)-Em,q(α)=[2]qα∑l=0n-1(-1)l+1qαl[l]qm.
If n≡1(mod2), then
(2.18)qαnEm,q(α)(n)+Em,q(α)=[2]qα∑l=0n-1(-1)lqαl[l]qm.
From (1.4), one notes that
(2.19)[2]qα=qα∫Zpe[x+1]qtdμ-qα(x)+∫Zpe[x]qtdμ-qα(x)=∑n=0∞(qα∫Zp[x+1]qndμ-qα(x)+∫Zp[x]qndμ-qα(x))tnn!=∑n=0∞(qαEn,q(α)(1)+En,q(α))tnn!.
Therefore, we obtain the following theorem.Theorem 2.4.
Forn∈ℤ+, one has
(2.20)qαEn,q(α)(1)+En,q(α)={[2]qα,ifn=0,0,ifn>0.
By Theorem 2.4 and (2.13), we have the following corollary.Corollary 2.5.
Forn∈ℤ+, one has
(2.21)qα(qEq(α)+1)n+En,q(α)={[2]qα,ifn=0,0,ifn>0,
with the usual convention of replacing (Eq(α))n by En,q(α).
By (2.12), one has
(2.22)∑n=0∞(qαEn,q(α)(x+1)+En,q(α)(x))tnn!=[2]qαqα∑m=0∞(-1)mqαme[m+1+x]qt+[2]qα∑m=0∞(-1)mqαme[m+x]qt=[2]qαe[x]qt=[2]qα∑n=0∞[x]qntnn!.
Hence we have the following difference equation.Theorem 2.6 (difference equation).
Forn∈ℤ+, one has
(2.23)qαEn,q(α)(x+1)+En,q(α)(x)=[2]qα[x]qn.Usingq-Euler numbers and polynomials with weak weight α, q-Euler zeta function with weak weight α and Hurwitz q-Euler zeta functions with weak weight α are defined. These functions interpolate the q-Euler numbers and q-Euler polynomials with weak weight α, respectively. In this section we assume that q∈ℂ with |q|<1. From (2.6), we note that
(2.24)dkdtkFq(α)(t)|t=0=[2]qα∑n=1∞(-1)nqαn[n]qk,(k∈N).
Using the above equation, we are now ready to define q-Euler zeta functions.Definition 2.7.
Lets∈ℂ.
(2.25)ζq(α)(s)=[2]qα∑n=1∞(-1)nqαn[n]qs.
Note thatζq(α)(s) is a meromorphic function on ℂ. Note that, if q→1, then ζq(α)(s)=ζ(s) which is the Euler zeta functions. Relation between ζq(α)(s) and Ek,q(α) is given by the following theorem.Theorem 2.8.
Fork∈ℕ, one has
(2.26)ζq(α)(-k)=Ek,q(α).
Observe thatζq(α)(s) function interpolates Ek,q(α) numbers at nonnegative integers. By using (2.12), we note that
(2.27)dkdtkFq(α)(t,x)|t=0=[2]qα∑n=0∞(-1)nqαn[n+x]qk,(k∈N),(2.28)(ddt)k(∑n=0∞En,q(α)(x)tnn!)|t=0=Ek,q(α)(x),fork∈N.
By (2.27) and (2.28), we are now ready to define the Hurwitz q-Euler zeta functions.Definition 2.9.
Lets∈ℂ. Then, one has
(2.29)ζq(α)(s,x)=[2]qα∑n=0∞(-1)nqαn[n+x]qs.
Note thatζq(α)(s,x) is a meromorphic function on ℂ. Obverse that, if q→1, then ζq(α)(s,x)=ζ(s,x) which is the Hurwitz Euler zeta functions. Relation between ζq(α)(s,x) and Ek,q(α)(x) is given by the following theorem.Theorem 2.10.
Fork∈ℕ, one has
(2.30)ζq(α)(-k,x)=Ek,q(α)(x).
Observe that ζq(α)(-k,x) function interpolates Ek,q(α)(x) numbers at nonnegative integers.


## 3. Distribution and Structure of the Zeros

In this section, we assume thatα∈ℕ and q∈ℂ, with |q|<1. We observe the behavior of roots of the q-Euler polynomials En,q(α)(x). We display the shapes of the q-Euler polynomials En,q(α)(x), and we investigate the zeros of the q-Euler polynomials En,q(α)(x). We plot the zeros of the q-Euler polynomials En,q(α)(x) for n=10,20,30,40 and x∈ℂ (Figure 1). In Figure 1 (top-left), we choose n=10,q=1/2, and α=3. In Figure 1 (top-right), we choose n=20,q=1/2, and α=3. In Figure 1 (bottom-left), we choose n=30,q=1/2, and α=3. In Figure 1 (bottom-right), we choose n=40,q=1/2, and α=3.ZerosofEn,1/2(3)(x).In order to understand zeros behavior better, we present Figures2 and 3. We plot the zeros of En,q(α)(x) (Figure 2).ZerosofEn,q(α)(x).ZerosofE30,1/2(x)forα=5,7,9,11.In Figure2 (top-left), we choose n=30,q=1/5, and α=3. In Figure 2 (top-right), we choose n=30,q=1/4, and α=3. In Figure 2 (bottom-left), we choose n=30,q=1/3, and α=3. In Figure 2 (bottom-right), we choose n=30,q=1/2, and α=3.We plot the zeros of theq-Euler polynomials En,q(α)(x) for n=30,q=1/2,α=5,7,9,11 and x∈ℂ  (Figure 3).In Figure3 (top-left), we choose n=30,q=1/2, and α=5. In Figure 3 (top-right), we choose n=30,q=1/2, and α=7. In Figure 3 (bottom-left), we choose n=30,q=1/2, and α=9. In Figure 3 (bottom-right), we choose n=30,q=1/2, and α=11.Our numerical results for approximate solutions of real zeros of theq-Euler polynomials En,q(α)(x),q=1/2, are displayed (Tables 1 and 2).Table 1
Numbers of real and complex zeros ofEn,q(α)(x).
α=3α=5DegreenReal zerosComplex zerosReal zerosComplex zeros110102202031212422225321462424734348262693636102828113838124821013310310Table 2
Approximate solutions ofEn,q(3)(x)=0,q=1/2,x∈ℝ.
Degreenx10.08246222−0.176174,  0.30170430.5130124−0.220226,  0.7013015−0.306596,  −0.132473,  0.86883960.0191767,  1.019187−0.41178,  0.155365,  1.1553480.279948,  1.27971⋮⋮Next, we calculated an approximate solution satisfying theq-Euler polynomials En,q(α)(x). The results are given in Table 2.We observe a remarkably regular structure of the complex roots of theq-Euler polynomials En,q(α)(x). We hope to verify a remarkably regular structure of the complex roots of the q-Euler polynomials En,q(α)(x) (Table 1). This numerical investigation is especially exciting because we can obtain an interesting phenomenon of scattering of the zeros of the q-Euler polynomials En,q(α)(x). These results are used not only in pure mathematics and applied mathematics, but also in mathematical physics and other areas.Stacks of zeros ofEn,q(3)(x) for q=1/2,1≤n≤30 from a 3D structure are presented (Figure 4).Figure 4
StacksofzerosofEn,q(3)(x),1≤n≤40.We present the distribution of real zeros of theq-Euler polynomials En,q(α)(x) for q=1/2,1≤n≤30 (Figure 5).ZerosofEn,30(3)(x).In Figure5 (left), we choose α=3. In Figure 3 (right), we choose α=5.The plot above showsEn,q(α)(x) for real 1/10≤q≤9/10 and -2≤x≤2, with the zero contour indicated in black (Figure 6). In Figure 6 (top-left), we choose n=1 and α=3. In Figure 6 (top-right), we choose n=2 and α=3. In Figure 6 (bottom-left), we choose n=3 and α=3. In Figure 6 (bottom-right), we choose n=4 and α=3.ZerocontourofEn,q(α)(x).


## 4. Direction for Further Research

We observe the behavior of complex roots of theq-Euler polynomials En,q(α)(x), using numerical investigation. How many roots does En,q(α)(x) have in general? This is an open problem. Prove or disprove: En,q(α)(x) has n distinct solutions, that is, all the zeros are nondegenerate. Find the numbers of complex zeros CEn,q(α)(x) of En,q(α)(x),Im(x)≠0. Since n is the degree of the polynomial En,q(α)(x), the number of real zeros REn,q(α)(x) lying on the real plane Im(x)=0 is then REn,q(α)(x)=n-CEn,q(α)(x), where CEn,q(α)(x) denotes complex zeros. See Table 1 for tabulated values of REn,q(α)(x) and CEn,q(α)(x). We prove that En,q(α)(x),x∈ℂ, has Im(x)=0 reflection symmetry analytic complex functions. If En,q(α)(x)=0, then En,q(h)(x*)=0, where * denotes complex conjugate (see Figures 1, 2, and 3). The theoretical prediction on the zeros of En,q(α)(x) requires further study. In order to study the q-Euler polynomials En,q(α)(x), we must understand the structure of the q-Euler polynomials En,q(α)(x). Therefore, using computer, in a realistic study for the q-Euler polynomials En,q(α)(x) play an important part. 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 the q-Euler polynomials En,q(α)(x) to appear in mathematics and physics. For related topics the interested reader is referred to [16].



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