# Calculating Zeros of theq-Genocchi Polynomials Associated with p-Adic q-Integral on ℤp

Authors: C. S. Ryoo
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

In this paper we construct the new analogues of Genocchi the numbers and polynomials. We also observe the behavior of complex roots of theq-Genocchi polynomials Gn,q(x), using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the q-Genocchi polynomials Gn,q(x). Finally, we give a table for the solutions of the q-Genocchi polynomials Gn,q(x).


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## Body

## 1. Introduction

Many mathematicians have the studied Bernoulli numbers and polynomials, the Euler numbers and polynomials, and the Genocchi numbers and the Genocchi polynomials. The Bernoulli numbers and polynomials, the Euler numbers and polynomials, and the Genocchi polynomials posses many interesting properties and arising in many areas of mathematics and physics (see [1–12]). We introduce the new analogs of  the Genocchi numbers and polynomials. In the 21st century, the computing environment would make more and more rapid progress. Using computer, a realistic study for new analogs of Genocchi numbers and polynomials is very interesting. It is the aim of this paper to observe an interesting phenomenon of “scattering” of the zeros of q-Genocchi polynomials Gn,q(x). The outline of this paper is as follows. In Section 2, we study the q-Genocchi polynomials Gn,q(x). In Section 3, we describe the beautiful zeros of q-Genocchi polynomials Gn,q(x) using a numerical investigation. Also we display distribution and structure of the zeros of the q-Genocchi polynomials Gn,q(x) by using computer. By using the results of our paper, the readers can observe the regular behaviour of the roots of q-Genocchi polynomials Gn,q(x). Finally, we carried out computer experiments that demonstrate a remarkably regular structure of the complex roots of q-Genocchi polynomials Gn,q(x). Throughout this paper we use the following notations. By ℤp we denote the ring of p-adic rational integers, ℚ denotes the field of rational numbers, ℚp denotes the field of p-adic rational numbers, ℂ denotes the complex number field, and ℂp denotes the completion of algebraic closure of ℚp. 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:(1.1)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q.
Compare [1, 2, 4, 10, 11, 13–16]. 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=limN←(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, cf. [1, 2, 4, 5, 9, 10, 13]. We say that g is a uniformly differentiable function at a point a∈ℤp and denote this property by g∈UD(ℤp), if the difference quotients Fg(x,y)=f(x)-f(y)/(x-y) have a limit l=g'(a) as (x,y)→(a,a). For
(1.4)g∈UD(ℤp)={g∣g:ℤp→ℂpis uniformly differentiable function},
the q-deformed bosonic p-adic integral of the function g is defined by Kim:
(1.5)Iq(g)=∫ℤp‍g(x)dμq(x)=limN→∞1[pN]q∑0≤x<pN‍g(x)qx.
Note that
(1.6)qIq(f1)=Iq(f)+(q-1)f(0)+q-1logqf′(0),
where f1(x)=f(x+1),f'(0)=df(0)/dx. Now, the fermionic p-adic invariant q-integral on ℤp is defined as
(1.7)I-q(g)=∫ℤp‍g(x)dμ-q(x)=limN→∞1[pN]-q∑0≤x<pN‍g(x)(-q)x.
If we take g1(x)=g(x+1) in (1.7), then we easily see that
(1.8)qI-q(g1)+I-q(g)=[2]qg(0).
From (1.8), we obtain
(1.9)qnI-q(gn)+(-1)n-1I-q(g)=[2]q∑l=0n-1‍(-1)n-1-lg(l),
where gn(x)=g(x+n). First, we introduce the Genocchi numbers and the Genocchi polynomials. The Genocchi numbers Gn are defined by the generating function:
(1.10)F(t)=2tet+1=∑n=0∞‍Gntnn!,(|t|<π).
Compare [4, 9–11, 17], where we use the technique method notation by replacing Gn by Gn(n≥0) symbolically. We consider the Genocchi polynomials Gn(x) as follows:
(1.11)F(x,t)=2tet+1ext=∑n=0∞‍Gn(x)tnn!.
Note that Gn(x)=∑k=0n‍(nk)Gkxn-k. In the special case x=0, we define Gn(0)=Gn.


## 2. An Analogue of the Genocchi Numbers and Polynomials

The versions ofq-Genocchi numbers and polynomials, which were derived from different considerations and different formulas, were defined by Kim [13, 14]. Kim [14] treated analogue of the Genocchi numbers, which is called q-analogue of the Genocchi numbers. Kim defined the q-extension of the Genocchi numbers and polynomials as follows:
(2.1)Fq(t)=∑n=0∞‍cn,qtnn!=et/(1-q)∑n=0∞‍(2n+1)[2n+1]q[n]q(1q-1)n-1tnn!,Fq(x,t)=∑n=0∞‍cn,q(x)tnn!=et/(1-q)∑n=0∞‍(2n+1)[2n+1]q[n]q(1q-1)n-1qnxtnn!.
In [14], Kim introduced the q-analogue of the Genocchi polynomials as follows:
(2.2)Gq(x,t)=[2]qt∑n=0∞‍(-1)nqn+xe[n+x]qt=∑n=0∞‍Gn,q(x)tnn!.
We now consider another construction q-Genocchi numbers and polynomials. In (1.8), if we take g(x)=ext, then one has
(2.3)(logq+t)∫ℤp‍extdμ-q(x)=[2]q(logq+t)qet+1.
Let us define the q-Genocchi numbers and polynomials as follows:
(2.4)(logq+t)∫ℤp‍eytdμ-q(y)=∑n=0∞‍Gn,qtnn!,(2.5)(logq+t)∫ℤp‍e(x+y)tdμ-q(y)=∑n=0∞‍Gn,q(x)tnn!.
Note that Gn,q(0)=Gn,q, limq→1Gn,q=Gn, where Gn are the nth Genocchi numbers. By (2.4) and (2.5), we obtain the following Witt's formula.Theorem 2.1.
Forq∈ℂp with |1-q|p≤p-1/(p-1), we have
(2.6)n∫ℤp‍xn-1dμ-q(x)+logq∫ℤp‍xndμ-q(x)=Gn,q,n∫ℤp‍(x+y)n-1dμ-q(y)+logq∫ℤp‍(x+y)ndμ-q(y)=Gn,q(x).By the above theorem, easily see that(2.7)Gn,q(x)=∑k=0n‍(nk)xn-kGk,q.
Let q be a complex number with |q|<1. By the meaning of (1.10) and (1.11), let us define the q-Genocchi numbers Gn,q and polynomials Gn,q(x) as follows:
(2.8)Fq(t)=[2]q(logq+t)qet+1=∑n=0∞‍Gn,qtnn!,Fq(x,t)=[2]q(logq+t)qet+1ext=∑n=0∞‍Gn,q(x)tnn!.
For q-Euler numbers, Kim constructed q-Euler numbers which can be uniquely determined by
(2.9)q(qEq+1)n+En,q={[2]q,ifn=0,0,ifn>0,
with the usual convention of symbolically replacing Eqn by En,q, where En,q denotes the q-Euler numbers. For q-Genocchi numbers, we have the following theorem.Theorem 2.2.
q-Genocchi numbers Gn,q are defined inductively by
(2.10)G0,q=[2]qlogq1+q,q(Gq+1)n+Gn,q={[2]q,ifn=1,0,ifn>1,
with the usual convention about replacing (Gq)n by Gn,q in the binomial expansion.Proof.
From (2.4), we obtain
(2.11)[2]q(logq+t)qet+1=∑n=0∞‍Gn,qtnn!=∑n=0∞‍(Gq)ntnn!=eGqt,
which yields
(2.12)[2]q(logq+t)=(qet+1)eGqt=qe(Gq+1)t+eGqt.
Using the Taylor expansion of exponential function, we have
(2.13)[2]qlogq+[2]qt=∑n=0∞‍{q(Gq+1)n+(Gq)n}tnn!=q(Gq+1)0+(Gq)0+q(Gq+1)1+(Gq)1+∑n=2∞‍{q(Gq+1)n+(Gq)n}tnn!.
The result follows by comparing the coefficients.
Here is the list of the firstq-Genocchi numbers Gn,q:(2.14)G0,q=logq,G1,q=--1-q+qlogq(1+q),G2,q=q(-2-2q-logq+qlogq)(1+q)2,G3,q=-q(3-3q2+logq-4qlogq+q2logq)(1+q)3,⋮
We display the shapes of the q-Genocchi numbers Gn,q. For n=1,…,10, we can draw a curve of Gn,q,1/10≤q≤9/10, respectively. This shows the ten curves combined into one. We display the shape of Gn,q: (Figure 1).
Because(2.15)∂∂xFq(t,x)=tFq(t,x)=∑n=0∞‍ddxGn,q(x)tnn!,
it follows the important relation
(2.16)ddxGn,q(x)=nGn-1,q(x).
Here is the list of the first the q-Genocchi polynomials Gn,q(x):
(2.17)G0,q(x)=logq,G1,q(x)=(1+q-qlogq+xlogq+qxlogq)(1+q),⋮
Since
(2.18)∑l=0∞‍Gl,q(x+y)tll!=[2]qlogq+[2]qtqet+1e(x+y)t=∑n=0∞‍Gn,q(x)tnn!∑m=0∞‍ymtmm!=∑l=0∞‍(∑n=0l‍Gn,q(x)tnn!yl-ntl-n(l-n)!)=∑l=0∞‍(∑n=0l‍(ln)Gn,q(x)yl-n)tll!,
we have the following theorem.Figure 1
Curves ofGn,q.Theorem 2.3.
q-Genocchi polynomials Gn,q(x) satisfy the following relation:
(2.19)Gl,q(x+y)=∑n=0l‍(ln)Gn,q(x)yl-n.
It is easy to see that
(2.20)∑n=0∞‍Gn,q(x)tnn!=[2]qlogq+[2]qtqet+1ext=[2]qm[2]qm∑a=0m-1‍(-1)aqa[2]qmlogqm+[2]qmmtqmemt+1e(a/m+x/m)(mt)=[2]qm[2]qm∑a=0m-1‍(-1)aqa∑n=0∞‍Gn,qm(a+xm)(mt)nn!=∑n=0∞‍(mn-1[2]q[2]qm∑a=0m-1‍(-1)aqaGn,qm(a+xm))tnn!.
Hence we have the following theorem.Theorem 2.4.
For any positive integerm(=odd), one obtains
(2.21)Gn,q(x)=mn-1[2]q[2]qm∑i=0m-1‍(-1)iqiGn,qm(i+xm),forn≥0.


## 3. Distribution and Structure of the Zeros

In this section, we investigate the zeros of theq-Genocchi polynomials Gn,q(x) by using computer. We display the shapes of the q-Genocchi polynomials Gn,q(x). For n=1,…,10, we can draw a curve of Gn,q(x),-2≤x≤2, respectively. This shows the ten curves combined into one. We display the shape of Gn,q(x) (Figures 2, 3, 4, and 5).Figure 2
Curves ofGn,1/5(x).Figure 3
Curves ofGn,1/4(x).Figure 4
Curves ofGn,1/3(x).Figure 5
Curves ofGn,1/2(x).We plot the zeros ofGn,q(x),x∈ℂ for n=10,20,25,30,q=1/3 (Figures 6, 7, 8, and 9).Figure 6
Zero ofG10,1/3(x).Figure 7
Zero ofG20,1/3(x).Figure 8
Zero ofG25,1/3(x).Figure 9
Zero ofG30,1/3(x).Next, we plot the zeros ofGn,q(x),x∈ℂ for n=30,q=1/2,1/3,1/4,1/5. (Figures 10, 11, 12, and 13).Figure 10
Zero ofG30,1/2(x).Figure 11
Zero ofG30,1/3(x).Figure 12
Zero ofG30,1/4(x).Figure 13
Zero ofG30,1/5(x).In Figures6, 7, 8, 9, 10, 11, 12, and 13, Gn,q(x),x∈ℂ, has Im(x)=0 reflection symmetry. This translates to the following open problem: prove or disprove: Gn,q(x),x∈ℂ, has Im(x)=0 reflection symmetry. Our numerical results for numbers of real and complex zeros of Gn,q(x),q=1/2,1/3, are displayed in Table 1.Table 1
Numbers of real and complex zeros ofGn,q(x).
Degreen
q
=
1
/
2
q
=
1
/
3
Real zeros
Complex zeros
Real zeros
Complex zeros
1
1
0
1
0
2
2
0
2
0
3
3
0
3
0
4
4
0
4
0
5
3
2
3
2
6
4
2
4
2
7
5
2
5
2
8
4
4
6
2
9
5
4
5
4
10
4
6
4
6Figure15 shows the distribution of real zeros of Gn,q(x) for 1≤n≤20.In Figure15(a), we choose q=1/10. In Figure 15(b), we choose q=3/10. In Figure 15(c), we choose q=5/10. In Figure 15(d), we choose q=6/10.We calculated an approximate solution satisfyingGn,q(x),q=1/2,1/3,x∈ℝ. The results are given in Tables 2 and 3.Table 2
Approximate solutions ofGn,1/2(x)=0,x∈ℝ.
Degreen
x
1
1.7760
2
0.2583, 3.294
3
−0.1698, 0.7313, 4.767
4
−0.4188, 0.1527, 1.145, 6.225
5
0.5848, 1.492, 7.677
6
0.01656, 1.017, 1.772, 9.126
7
−0.5269, 0.4468, 1.452, 1.974, 10.573
8
−0.8536, −0.1221, 0.8779, 12.019
9
−0.969, −0.707, 0.3088, 1.309, 13.46
10
−0.2604, 0.7396, 1.738, 14.91Table 3
Approximate solutions ofGn,1/3(x)=0,x∈ℝ.
Degreen
x
1
1.1602
2
0.1523, 2.168
3
−0.2107, 0.5657, 3.126
4
−0.3621, −0.02976, 0.9703, 4.062
5
0.3561, 1.333, 4.989
6
−0.2472, 0.7504, 1.652, 5.910
7
−0.6547, 0.1435, 1.144, 1.923, 6.828
8
−0.798, −0.4682, 0.5362, 1.540, 2.139, 7.744
9
−0.07080, 0.9292, 1.98, 2.25, 8.659
10
−0.673, 0.3221, 1.322, 9.573The plot above showsGn,q(x) for real 1/10≤q≤9/10 and -3≤x≤3, with the zero contour indicated in black (Figure 16). In Figure 16(a), we choose n=2. In Figure 16(b), we choose n=3. In Figure 16(c), we choose n=4. In Figure 16(d), we choose n=5.We will consider the more general open problem. In general, how many roots doesGn,q(x) have? Prove or disprove: Gn,q(x) has n distinct solutions. Find the numbers of complex zeros CGn,q(x) of Gn,q(x),Im(x)≠0. Prove or give a counterexample: Conjecture: since n is the degree of the polynomial Gn,q(x), the number of real zeros RGn,q(x) lying on the real plane Im(x)=0 is then RGn,q(x)=n-CGn,q(x), where CGn,q(x) denotes complex zeros. See Table 1 for tabulated values of RGn,q(x) and CGn,q(x). Find the equation of envelope curves bounding the real zeros lying on the plane and the equation of a trajectory curve running through the complex zeros on any one of the arcs. For n=1,…,10, we can draw a plot of the Gn,q(x), respectively. This shows the ten curves combined into one. These figures give mathematicians an unbounded capacity to create visual mathematical investigations of the behavior of the Gn,q(x) and roots of the Gn,q(x) (Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16). Moreover, it is possible to create new mathematical ideas and analyze them in ways that generally are not possible by hand. The author has no doubt that investigation along this line will lead to a new approach employing numerical method in the field of research of the q-Genocchi polynomials Gn,q(x) to appear in mathematics and physics. For related topics the interested reader is referred to [15–19].Figure 14
Stacks of zerosGn,1/3(x)for1≤n≤30.Plot of real zeros ofGn,q(x)for1≤n≤20.
(a)
(b)
(c)
(d)Zero contour ofGn,q(x).
(a)
(b)
(c)
(d)



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