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

Authors: H. Y. Lee; N. S. Jung; J. Y. Kang; C. S. Ryoo
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

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


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## Body

## 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 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 generalized (q,w)-Euler polynomials En,q,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 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(1.1)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q.
Compared with [1, 4, 5]. Hence, limq→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
(1.2)X=lim←N(ℤdpNℤ),X*=⋃0<a<dp(a,p)=1(a+dpℤp)‍,a+dpNℤp={x∈X∣x≡a(moddpN)},
where a∈ℤ lies in 0≤a<dpN. For any positive integer N,
(1.3)μq(a+dpNℤp)=qa[dpN]q
is known to be a distribution on X, compared with [1–10, 14]. For
(1.4)g∈UD(ℤp)={g∣g:ℤp→ℂpisuniformlydifferentiablefunction}.
Kim defined the fermionic p-adic q-integral on ℤp(1.5)I-q(g)=∫ℤp‍g(x)dμ-q(x)=limN→∞1[pN]-q∑0≤x<pN‍g(x)(-q)x.
From (1.5), we also obtain
(1.6)qI-q(g1)+I-q(g)=[2]qg(0),
where g1(x)=g(x+1) (see [1–3]).From (1.6), we obtain
(1.7)qnI-q(gn)+(-1)n-1I-q(g)=[2]q∑l=0n-1‍(-1)n-1-lqlg(l),
where gn(x)=g(x+n).As well-known definition, the Euler polynomials are defined by(1.8)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 n-th Euler numbers (cf. [1–15]).Our aim in this paper is to define the generalized(q,w)-Euler numbers En,q,w(a) and polynomials En,q,w(x:a).  We investigate some properties which are related to the generalized (q,w)-Euler numbers En,q,w(a) and polynomials En,q,w(x:a). Especially, distribution of roots for En,q,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 (q,w)-Euler numbers En,q,w(a) and polynomials En,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 En,q,w(a) and polynomials En,q,w(x:a). We also find generating functions of the generalized (q,w)-Euler numbers En,q,w(a) and polynomials En,q,w(x:a). Let a be strictly positive real number.The generalized(q,w)-Euler numbers and polynomials En,q,w(a), En,q,w(x:a) are defined by
(2.1)∑n=0∞‍En,q,w(a)tnn!=∫ℤp‍waxeaxtdμ-q(x),(2.2)∑n=0∞‍En,q,w(x:a)tnn!=∫ℤp‍waye(ay+x)tdμ-q(y),fort,w∈ℂ,
respectively.From above definition, we obtain(2.3)En,q,w(a)=∫ℤp‍wax(ax)ndμ-q(x),En,q,w(x:a)=∫ℤp‍way(x+ay)ndμ-q(y).Letg(x)=waxeaxt. By (1.6) and using p-adic q-integral on ℤp, we have
(2.4)qI-q(g1)+I-q(g)=∫ℤp‍wa(x+1)ea(x+1)tdμ-q(x)+∫ℤp‍waxeaxtdμ-q(x)=(qwaeat+1)∫ℤp‍waxeaxtdμ-q(x)=[2]q.Hence, by (2.1), we obtain
(2.5)∑n=0∞‍En,q,w(a)tnn!=[2]qqwaeat+1.By (1.6), (2.2) and g(y)=waye(ay+x)t, we have
(2.6)∑n=0∞‍En,q,w(x:a)tnn!=[2]qqwaeat+1ext.After some elementary calculations, we obtain(2.7)∑n=0∞‍En,q,w(x:a)tnn!=[2]q∑n=0∞‍(-1)nqnwaneantext.From (2.6), we have
(2.8)En,q,w(x:a)=∑k=0n‍(nk)xn-kEk,q,w(a)=(x+Eq,w(a))n,
with the usual convention of replacing (Eq,w(a))n by En,q,w(a).


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

By (2.5), we have
(3.1)∂∂x∑n=0∞‍En,q,w(x:a)tnn!=∂∂x([2]qqwaeat+1ext)=t∑n=0∞‍En,q,w(x:a)tnn!=∑n=0∞‍nEn-1,q,w(x:a)tnn!.By (3.1), we have the following differential relation.Theorem 3.1.
For positive integers n, one has(3.2)∂∂xEn,q,w(x:a)=nEn-1,q,w(x:a).By Theorem3.1, we easily obtain the following corollary.Corollary 3.2 (integral formula).
Consider that(3.3)∫pq‍En-1,q,w(x:a)dx=1n(En,q,w(q:a)-En,q,w(p:a)).
By (2.5), one obtains
(3.4)∑n=0∞‍En,q,w(x+y:a)tnn!=[2]qqwaeat+1e(x+y)t=∑n=0∞‍En,q,w(x:a)tnn!∑k=0∞‍yktkk!=∑n=0∞‍(∑k=0n‍(nk)Ek,q,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,q,w(x+y:a)=∑k=0n‍(nk)Ek,q,w(x:a)yn-k.
By (2.5), for m≡1(mod2), one has
(3.6)∑n=0∞‍(mn[2]q[2]qm∑k=0m-1‍(-1)kqkwakEn,qm,wm(x+akm:a))tnn!=∑k=0m-1‍(-1)kqkwak(∑n=0∞‍En,qm,wm(x+akm:a))(mt)nn!=∑k=0m-1‍((-1)kqkwak[2]qqmwmaemat+1e(x+ak)t)=[2]q1+qwaeatext=∑n=0∞‍En,q,w(x:a)tnn!.By comparing coefficients oftn/n! in the above equation, we arrive at the following multiplication theorem.Theorem 3.4 (multiplication theorem).
Form,n∈ℕ(3.7)En,q,w(x:a)=mn[2]q[2]qm∑k=0m-1‍(-1)kqkwakEn,qm,wm(x+akm:a).
From (1.6), one notes that
(3.8)[2]q=∫ℤp‍qwax+ae(ax+a)tdμ-q(x)+∫ℤp‍waxeaxtdμ-q(x)=∑n=0∞‍(qwa∫ℤp‍wax(ax+a)ndμ-q(x)+∫ℤp‍wax(ax)ndμ-q(x))tnn!=∑n=0∞‍(qwaEn,q,w(a:a)+En,q,w(a))tnn!.From the above, we obtain the following theorem.Theorem 3.5.
Forn∈ℤ+, we have
(3.9)qwaEn,q,w(a:a)+En,q,w(a)={[2]q,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)qwa(a+Eq,w(a))n+En,q,w(a)={[2]q,ifn=0,0,ifn>0,
with the usual convention of replacing (Eq,w(a))n by En,q,w(a).From (1.7), one notes that
(3.11)∑m=0∞‍([2]q∑l=0n-1‍(-1)n-1-lqlwal(al)m)tnm!=qn∫ℤp‍wax+ane(ax+an)tdμ-q(x)+(-1)n-1∫ℤp‍waxeaxtdμ-q(x)=∑m=0∞‍(qnwan∫ℤp‍wax(ax+an)mdμ-q(x)+(-1)n-1∫ℤp‍wax(ax)mdμ-q(x))tmm!=∑m=0∞‍(qnwanEm,w(an:a)+(-1)n-1Em,w(a))tmm!.
By comparing coefficients of tn/n! in the above equation, we arrive at the following theorem.Theorem 3.7.
Forn∈ℤ+, one has
(3.12)qnwanEm,w(na:a)+(-1)n-1Em,w(a)=[2]q∑l=0n-1‍(-1)n-1-lwalql(al)m.


## 4. The Analogue of theq-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
(4.1)Fq,w(x:a)(t)=[2]q∑n=0∞‍(-1)nqnwaneantext=∑n=0∞‍En,q,w(x:a)tnn!.
By applying derivative operator, dk/dtk|t=0 to the above equation, we have
(4.2)dkdtkFq,w(x:a)(t)|t=0=[2]q∑n=0∞‍(-1)nqnwan(an+x)k,(k∈ℕ),(4.3)Ek,q,w(x:a)=[2]q∑n=0∞‍(-1)nqnwan(an+x)k.By using the above equation, we are now ready to define the generalized(q,w)-Euler zeta functions.Definition 4.1.
Fors∈ℂ, one defines
(4.4)ζq,w(a)(x:s)=2∑n=1∞‍(-1)nqnwan(an+x)s.
Note thatζw(a)(x,s) is a meromorphic function on ℂ. Note that, if w→1, w→1, and a=1, then ζq,w(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)ζq,w(a)(x:-k)=Ek,w(x:a).
By using (4.2), one notes that
(4.6)dkdtkFq,w(0:a)(t)|t=0=[2]q∑n=0∞‍(-1)nqnwan(an)k,(k∈ℕ).
Hence, one obtains
(4.7)Ek,q,w(a)=[2]q∑n=0∞‍(-1)nqnwan(an)k.By using the above equation, one is now ready to define the generalized Hurwitz(q,w)-Euler zeta functions.Definition 4.3.
Lets∈ℂ. One defines
(4.8)ζq,w(a)(s)=2∑n=1∞‍(-1)nqnwan(an)s.
Note that ζq,w(a)(s) is a meromorphic function on ℂ. Obverse that, if w→1, q→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)ζq,w(a)(-k)=Ek,q,w(a).


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

In this section, we investigate the reflection symmetry of the zeros of the generalized(q,w)-Euler polynomials En,q,w(x:a).In the special case,w=1 and q→1, En,q,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)Fq,w(x:t)=[2]qqwaeat+1ext=∑n=0∞‍En,q,w(x:a)tnn!.
Then, we have
(5.4)Fq-1,w-1(a-x:-t)=[2]q-1q-1w-ae-at+1e(a-x)(-t)=wa[2]qqwaeat+1ext=wa∑n=0∞‍En,q,w(x:a)tnn!.
Hence, we arrive at the following complement theorem.Theorem 5.1 (complement theorem).
Forn∈ℕ,
(5.5)En,q-1,w-1(a-x:a)=(-1)nwaEn,q,w(x:a).Throughout the numerical experiments, we can finally conclude thatEn,q,w(x:a),x∈ℂ has not Re(x)=a/2 reflection symmetry analytic complex functions. However, we observe that En,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 En,q,w(x:a) will also inherit these symmetries.
(5.6)IfEn,q,w(x0:a)=0,thenEn,q,w(x0*:a)=0,
where * denotes complex conjugation (see Figures 1, 2, and 3).Zeros ofEn,q,w(x:a)  for  a=1,2,3,4.
(a)
(b)
(c)
(d)Zeros ofEn,q,w(x:a)  for  q=1/10,3/10,7/10,9/10.
(a)
(b)
(c)
(d)Real zeros ofEn,q,w(x:a)  for  1≤n≤25.
(a)
(b)
(c)
(d)We investigate the beautiful zeros of the generalized(q,w)-Euler polynomials En,q,w(x:a) by using a computer. We plot the zeros of the generalized Euler polynomials En,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 polynomialsEn,q,w(x:a) for n=30,a=2,w=2, and x∈ℂ (Figure 2).In Figure2 (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 ofEn,q,w(x:a) for 1≤n≤25 structure are presented (Figure 3).In Figure3 (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 ofEn,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 ofEn,q,w(x:a)  for  1≤n≤30.Our numerical results for approximate solutions of real zeros of the generalizedEn,q,w(x:a) are displayed (Tables 1 and 2).Table 1
Numbers of real and complex zeros ofEn,q,w(x:a).
n
q
=
1
/
2
,
w
=
2
,
a
=
2
q
=
1
/
2
,
w
=
4
,
a
=
4
Real zeros
Complex zeros
Real zeros
Complex zeros
1
1
0
1
0
2
2
0
2
0
3
3
0
1
2
4
2
2
2
2
5
3
2
1
4
6
4
2
2
4
7
3
4
3
4
8
4
4
2
6
9
3
6
3
6
10
4
6
2
8
11
5
6
3
8
12
6
6
4
8
13
5
8
3
10Table 2
Approximate solutions ofEn,q,w(x:a)=0,x∈ℝ.
n
x
1
1.3333
2
0.3905,2.2761
3
-
0.4011
,
1.560
,
2.841
4
-
1.0546
,
0.6907
5
-
1.5732
,
-
0.17085
,
1.829
6
-
1.9151
,
-
1.0557
,
0.9680,2.94
7
0.10585,2.106,3.68
8
-
0.7557
,
1.2442,3.26,4.00
9
-
1.6091
,
0.3825
,
2.382
10
-
2.392
,
-
0.4793
,
1.521,3.52
11
-
3.013
,
-
1.3411
,
0.6590,2.66
,
4.4We observe a remarkably regular structure of the complex roots of the generalized(q,w)-Euler polynomials En,q,w(x:a). We hope to verify a remarkably regular structure of the complex roots of the generalized (q,w)-Euler polynomials En,q,w(x:a) (Table 1).Next, we calculated an approximate solution satisfyingEn,q,w(x:a),q=1/2,w=2,a=2,x∈ℝ. The results are given in Table 2.Figure5 shows the generalized (q,w)-Euler polynomials En,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.Zero contour ofEn,q,w(x:a).
(a)
(b)
(c)
(d)Finally, we will consider the more general problems. How many roots doesEn,q,w(x:a) have? This is an open problem. Prove or disprove: En,q,w(x:a)=0 has n distinct solutions. Find the numbers of complex zeros CEn,q,w(x:a) of En,q,w(x:a),Im(x:a)≠0. Since n is the degree of the polynomial En,q,w(x:a), the number of real zeros REn,q,w(x:a) lying on the real plane Im(x:a)=0 is then REn,q,w(x:a)=n-CEn,q,w(x:a), where CEn,q,w(x:a) denotes complex zeros. See Table 1 for tabulated values of REn,q,w(x:a) and CEn,q,w(x:a). We plot the zeros of En,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 En,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 En,q,w(x:a) to appear in mathematics and physics.



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