# Generalizedw-Euler Numbers and Polynomials

Authors: H. Y. Lee; N. S. Jung; C. S. Ryoo
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

We generalize the Euler numbers and polynomials by the generalizedw-Euler numbers En,w(a) and polynomials En,w(x:a). For the complement theorem, En,w(x:a) have interesting different properties from the Euler polynomials and we observe an interesting phenomenon of “scattering” of the zeros of the the generalized Euler polynomials En,w(x:a) in complex plane.


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## Body

## 1. Introduction

The Euler numbers and polynomials possess many interesting properties and arising in  many areas of mathematics and physics. Recently, many mathematicians have studied in the area of the Euler numbers and polynomials (see [1–15]). In [14], we introduced that Euler equation En(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 the generalized w-Euler polynomials En,w(x:a) in complex plane. Throughout this paper we use the following notations. By ℤp 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 assumes 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.For(1.1)g∈UD(Zp)={gg:Zp⟶Cpisuniformlydifferentiablefunction},
Kim defined the fermionic p-adic q-integral on ℤp:(1.2)I-1(g)=∫Zpg(x)dμ-1(x)=lim⁡N→∞∑x=0pN-1g(x)(-1)x
(cf. [5–7]).If we takeg1(x)=g(x+1) in (1.2), then we easily see that(1.3)I-1(g1)+I-1(g)=2g(0).
From (1.3), we obtain(1.4)I-1(gn)+(-1)n-1I-q(g)=2∑l=0n-1(-1)n-1-lg(l),
where gn(x)=g(x+n) (cf. [1–15]).As a well-known definition, the Euler polynomials are defined by(1.5)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. [1–15]).Our aim in this paper is to define the generalizedw-Euler numbers En,w(a) and polynomials En,w(x:a). We investigate some properties which are related to the generalized w-Euler numbers En,w(a) and polynomials En,w(x:a). In particular, distribution of roots for En,w(x:a)=0 is different from En(x)=0’s. We also derive the existence of a specific interpolation function which interpolate the generalized w-Euler numbers En,w(a) and polynomialsEn,w(x:a).


## 2. The Generalizedw-Euler Numbers and Polynomials

Our primary goal of this section is to define the generalizedw-Euler numbers En,w(a) and polynomials En,w(x:a). We also find generating functions of the generalized w-Euler numbers En,w(a) and polynomials En,w(x:a). Let a be strictly positive real number.The generalizedw-Euler numbers and polynomials En,w(a),En,w(x:a) are defined by(2.1)∑n=0∞En,w(a)tnn!=∫Zpwaxeaxtdμ-1(x),(2.2)∑n=0∞En,w(x:a)tnn!=∫Zpwaye(ay+x)tdμ-1(y),fort∈R,w∈C,
respectively.From above definition, we obtain(2.3)En,w(a)=∫Zpwax(ax)ndμ-1(x),En,w(x:a)=∫Zpway(ay+x)ndμ-1(y).Letg(x)=waxeaxt. By (1.3) and using p-adic integral on ℤp, we have(2.4)I-1(g1)+I-1(g)=∫Zpwa(x+1)ea(x+1)tdμ-1(x)+∫Zpwaxeaxtdμ-1(x)=(waeat+1)∫Zpwaxeaxtdμ-1(x)=2.Hence, by (2.1), we obtain(2.5)∑n=0∞En,w(a)tnn!=2waeat+1.By (1.3), (2.2) and g(y)=waye(ay+x)t, we have(2.6)∑n=0∞En,w(x:a)tnn!=2waeat+1ext.After some elementary calculations, we obtain(2.7)∑n=0∞En,w(x:a)tnn!=2∑n=0∞(-1)nwaeantext.From (2.6), we have(2.8)En,w(x:a)=∑n=0n(nk)xn-kEk,w(a)=(x+Ew(a))n,
with the usual convention of replacing (Ew(a))n by En,w(a).


## 3. Basic Properties for the Generalizedw-Euler Numbers and Polynomials

By (2.5), we have(3.1)∂∂x∑n=0∞En,w(x:a)tnn!=∂∂x(2waeat+1ext)=t∑n=0∞En,w(x:a)tnn!=∑n=0∞nEn-1,w(x:a)tnn!.By (3.1), we have the following differential relation.Theorem 3.1.
For positive integersn, one has
(3.2)∂∂xEn,w(x:a)=nEn-1,w(x:a).By Theorem3.1, we easily obtain the following corollary.Corollary 3.2 (Integral formula).
One has(3.3)∫pqEn-1,w(x:a)dx=1n(En,w(q:a)-En,w(p:a)).By (2.5), we obtain(3.4)∑n=0∞En,w(x+y:a)tnn!=2waeat+1e(x+y)t=∑n=0∞En,w(x:a)tnn!∑k=0∞yktkk!=∑n=0∞(∑k=0n(nk)Ek,w(x:a)yn-k)tnn!.By comparing coefficients oftn/n! in the above equation, we arrive at the following addition theorem.Theorem 3.3 (Addition theorem).
Forn∈ℤ+,
(3.5)En,w(x+y:a)=∑k=0n(nk)Ek,w(x:a)yn-k.By (2.5), for m≡1(mod2), we have(3.6)∑n=0∞(mn∑k=0m-1(-1)kwakEn,wm(x+akm:a))tnn!=∑k=0m-1(-1)kwak(∑n=0∞En,wm(x+akm:a))(mt)nn!=∑k=0m-1((-1)kwak2wmaemate(x+ak)t)=2waeat+1ext=∑n=0∞En,w(x:a)tnn!.
By comparing coefficients of tn/n! in the above equation, we arrive at the following multiplication theorem.Theorem 3.4 (Multiplication theorem).
Form,n∈ℕ(3.7)En,w(x:a)=mn∑k=0m-1(-1)kwakEn,wm(x+akm:a).From (1.3), we note that(3.8)2=∫Zpwax+ae(ax+a)tdμ-1(x)+∫Zpwaxeaxtdμ-1(x)=∑n=0∞(wa∫Zpwax(ax+a)ndμ-1(x)+∫Zpwax(ax)ndμ-1(x))tnn!=∑n=0∞(waEn,w(a:a)+En,w(a))tnn!.
From the above, we obtain the following theorem.Theorem 3.5.
Forn∈ℤ+, one has
(3.9)waEn,w(a:a)+En,w(a)={2,ifn=0,0,ifn>0.By (2.8) in the above, we arrive at the following corollary.Corollary 3.6.
Forn∈ℤ+, one has
(3.10)wa(a+Ew(a))n+En,w(a)={2,ifn=0,0,ifn>0,
with the usual convention of replacing (Ew(a))n by En,w(a).From (1.4), we note that(3.11)∑m=0∞(2∑l=0n-1(-1)n-1-lwal(al)m)tnm!=∫Zpwax+ane(ax+an)tdμ-1(x)+(-1)n-1∫Zpwaxeaxtdμ-1(x)=∑m=0∞(wan∫Zpwax(ax+an)mdμ-1(x)+(-1)n∫Zpwax(ax)mdμ-1(x))tmm!=∑m=0∞(wanEm,w(an:a)+(-1)n-1Em,w(a))tmm!.By comparing coefficients oftn/n! in the above equation, we arrive at the following theorem.Theorem 3.7.
Forn∈ℤ+, one has
(3.12)wanEm,w(na:a)+(-1)n-1Em,w(a)=2∑l=0n-1(-1)n-1-lwal(al)m.


## 4. The Analogue of the Euler Zeta Function

By using the generalizedw-Euler numbers and polynomials, the generalized w-Euler zeta function and the generalized Hurwitz w-Euler zeta functions are defined. These functions interpolate the generalized w-Euler numbers and w-Euler polynomials, respectively. Let(4.1)Fw(x:a)(t)=2∑n=0∞(-1)nwaeantext=∑n=0∞En,w(x:a)tnn!.By applying derivative operator,dk/dtk|t=0 to the above equation, we have(4.2)dkdtkFw(x:a)(t)|t=0=2∑n=0∞(-1)nwan(an+x)k,(k∈N),(4.3)Ek,w(x:a)=2∑n=0∞(-1)nwan(an+x)k.By using the above equation, we are now ready to define the generalizedw-Euler zeta functions.Definition 4.1.
Fors∈ℂ, one defines
(4.4)ζw(a)(x:s)=2∑n=1∞(-1)nwan(an+x)s.Note thatζw(a)(x,s) is a meromorphic function on ℂ. Note that if w→1 and a=1, then ζw(a)(x:s)=ζ(x:s) which is the Hurwitz Euler zeta functions. Relation between ζw(a)(x:s) and Ek,w(x:a) is given by the following theorem.Theorem 4.2.
Fork∈ℕ, one has
(4.5)ζw(a)(x:-s)=Es,w(x:a).Observe thatζw(a)(x:s) function interpolates Ew(x:s) numbers at nonnegative integers.By using (4.2), we note that(4.6)dkdtkFw(0:a)(t)|t=0=2∑n=0∞(-1)nwan(an)k,(k∈N).
Hence, we obtain(4.7)Ek,w(a)=2∑n=0∞(-1)nwan(an)k.By using the above equation, we are now ready to define the generalized Hurwitzw-Euler zeta functions.Definition 4.3.
Lets∈ℂ. One defines
(4.8)ζw(a)(s)=2∑n=1∞(-1)nwan(an)s.
Note that ζw(a)(s) is a meromorphic function on ℂ. Obverse that, if w→1 and a=1, then ζw(a)(s)=ζ(s) which is the Euler zeta functions. Relation between ζw(a)(s) and Ek,w(s) is given by the following theorem.Theorem 4.4.
Fork∈ℕ, one has
(4.9)ζw(a)(-k)=Ek,w(a).
Observe that ζw(a)(-k) function interpolates Ek,w(a) numbers at nonnegative integers.


## 5. Zeros of the Generalizedw-Euler Polynomials En,w(x:a)

In this section, we investigate the reflection symmetry of the zeros of the generalizedw-Euler polynomials En,w(x:a).In the special case,w=1,En,w(x:a) are called generalized Euler polynomials En(x:a). Since(5.1)∑n=0∞En(a-x:a)(-t)nn!=2e-at+1e(a-x)(-t)=2eat+1ext=∑n=0∞En(x:a)tnn!,
we have(5.2)En(x:a)=(-1)nEn(a-x:a),forn∈N.
We observe that En(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(5.3)Fw,a(x:t)=2waeat+1ext=∑n=0∞En,w(x:a)tnn!.
Then we have(5.4)Fw-1,a(a-x:-t)=2w-ae-at+1e(a-x)(-t)=wa2waeat+1ext=wa∑n=0∞En,w(x:a)tnn!.
Hence, we arrive at the following complement theorem.Theorem 5.1 (Complement theorem).
Forn∈ℕ,
(5.5)En,w-1(a-x:a)=(-1)nwaEn,w(x:a).Throughout the numerical experiments, we can finally conclude thatEn,w(x:a),x∈ℂ has not Re(x)=a/2 reflection symmetry analytic complex functions. However, we observe that En,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 En,w(x:a) will also inherit these symmetries:(5.6)ifEn,w(x0:a)=0,thenEn,w(x0*:a)=0,
where * denotes complex conjugation (see Figures 1, 2 and 3).Zeros ofEn,w(x:a) for a=1,2,3,4.
(a)(b)(c)(d)Real zeros ofEn,w(x:a) for 1≤n≤20.
(a)(b)(c)(d)Zeros ofEn,w(x:a) for w=1,2,3,4.
(a)(b)(c)(d)We investigate the beautiful zeros of the generalizedw-Euler polynomials En,w(x:a) by using a computer. We plot the zeros of the generalized Euler polynomials En,w(x:a) for n=30,a=1,2,3,4, and x∈ℂ(Figure 1).In Figure1(a), we choose n=30,w=1, and a=1. In Figure 1(b), we choose n=30,w=1, and a=2. In Figure 1(c), we choose n=30,w=3, and a=3. In Figure 1(d), we choose n=30,w=4, and a=4.Plots of real zeros ofEn,w(x:a) for 1≤n≤20 structure are presented (Figure 2).In Figure2(a), we choose w=1 and a=1. In Figure 2(b), we choose w=1 and a=2. In Figure 2(c), we choose w=3 and a=3. In Figure 2(d), we choose w=4 and a=4.We investigate the beautiful zeros of the generalizedEn,w(x:a) by using a computer. We plot the zeros of the generalized w-Euler polynomials En,w(x:a) for n=30 and x∈ℂ(Figure 3).In Figure3(a), we choose a=3 and w=1. In Figure 3(b), we choose a=3 and w=2. In Figure 3(c), we choose a=3 and w=3. In Figure 3(d), we choose a=3 and w=4.Stacks of zeros ofEn,w(x:a) for 1≤n≤30,w=4,a=3 from a 3D structure are presented (Figure 4).Figure 4
Stacks of zeros ofEn,w(x:a) for 1≤n≤30.Our numerical results for approximate solutions of real zeros of the generalizedEn,w(x:a) are displayed (Tables 1 and 2).Table 1
Numbers of real and complex zeros ofEn,w(x:a).
w=2,a=2w=3,a=3nReal zerosComplex zerosReal zerosComplex zeros110102202033012422225323264224752348444493636104646113856124848135858Table 2
Approximate solutions ofEn,w(x:a)=0,x∈ℝ.
nx11.600020.8000,2.40030.04621,2.046,2.7074−0.6337,1.3265−1.2404,0.5878,2.586−1.7731,−0.14859,1.851,3.317−2.2231,−0.8856,1.117,3.2,3.48−1.756,0.244,2.26,3.09−0.3544,1.646,3.610−1.0898,0.9102,2.91,4.111−1.8231,−2.34,0.17473,2.17We observe a remarkably regular structure of the complex roots of the generalizedw-Euler polynomials En,w(x:a). We hope to verify a remarkably regular structure of the complex roots of the generalized w-Euler polynomials En,w(x:a) (Table 1). Next, we calculated an approximate solution satisfying En,w(x:a),w=2,a=2,x∈ℝ. The results are given in Table 2.The plot above shows the generalizedw-Euler polynomials En,w(x:a) for real 1≤a≤7 and -5≤x≤5, with the zero contour indicated in black (Figure 5). In Figure 5(a), we choose n=2 and w=2. In Figure 5(b), we choose n=3 and w=3. In Figure 5(c), we choose n=4 and w=4. In Figure 5(d), we choose n=5 and w=5.Zero contour ofEn,w(x:a).
(a)(b)(c)(d)Finally, we will consider the more general problems. How many roots doesEn,w(x:a) have? This is an open problem. Prove or disprove: En,w(x:a)=0 has n distinct solutions. Find the numbers of complex zeros CEn,w(x:a) of En,w(x:a),Im(x:a)≠0. Since n is the degree of the polynomial En,w(x:a), the number of real zeros REn,w(x:a) lying on the real plane Im(x:a)=0 is then REn,w(x:a)=n-CEn,w(x:a), where CEn,w(x:a) denotes complex zeros. See Table 1 for tabulated values of REn,w(x:a) and CEn,w(x:a). We plot the zeros of En,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 En,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 w-Euler polynomials En,w(x:a) to appear in mathematics and physics.



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