We observe the behavior of roots of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x). By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the q-extension of Bernoulli polynomials Bn,q(h)(x). The main purpose of this paper is also to investigate
the zeros of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x). Furthermore, we give a table for the zeros of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x).

1. Introduction

Throughout this paper ℤ,ℤp,ℚp, and ℂp will be denoted by the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of ℚp, respectively, compare with [1–6]. 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 variously considered as an indeterminate, a complex number q∈ℂ, or p-adic number q∈ℂp. If q∈ℂp, then we normally assume that |q-1|p<p-1/(p-1), so that qx=exp(xlogq) for |x|p≤1. If q∈ℂ, then we normally assume that |q|<1. For f∈UD(ℤp,ℂp)={f∣f:ℤp→ℂpisuniformlydifferentiablefunction}, the p-adic q-integral (or q-Volkenborn integration) was defined as Iq(f)=∫ℤpf(x)dμq(x)=limN→∞1[pN]q∑x=0pN-1f(x)qx,
where [x]q=(1-qx)/(1-q), compare with [1–8]. Thus, we note that I1(f)=limq→1Iq(f)=∫ℤpf(x)dμ1(x)=limN→∞1pN∑0≤x<pNf(x),comparewith[1,2,3,4,5,6].
By (1.2), we easily see that I1(f1)=I1(f)+f′(0),comparewith[1,2,3,4,5,6],
where f1(x)=f(x+1),f′(0)=(d/dx)f(x)|x=0.

In (1.3), if we take f(x)=qhxext, then we have ∫ℤpqhxextdμ1(x)=hlogq+tqhet-1,comparewith[6],
for |t|≤p-1/(p-1),h∈ℤ.

Recently, many mathematicians have studied Bernoulli numbers and Bernoulli polynomials. Bernoulli polynomials possess many interesting properties and arising in many areas of mathematics and physics. For more studies in this subject we may see references [1–8]. The motivation for this study comes from the following papers. Some interesting analogues of the Bernoulli numbers and polynomials were investigated by Ryoo and Kim [6]. We begin by recalling here definitions of (h,q)-extension of Bernoulli numbers and polynomials as follows.

Definition 1.1 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

The (h,q)-extension of Bernoulli numbers Bn,q(h) and polynomials Bn,q(h)(x) is defined by means of the generating functions as follows:
Fq(h)(t)=hlogq+tqhet-1=∑n=0∞Bn,q(h)tnn!,Fq(h)(t,x)=hlogq+tqhet-1ext=∑n=0∞Bn,q(h)(x)tnn!.

Note that Bn,q(h)(0)=Bn,q(h),limq→1Bn,q(h)(x)=Bn(x), and Bn,q(0)(x)=Bn(x), where Bn are the nth Bernoulli numbers.

By (1.4) and (1.5), we have the following Witt formula. For h∈ℤ,q∈ℂp with |1-q|p≤p-1/(p-1), we have ∫ℤpqhxxndμ1(x)=Bn,q(h),∫ℤpqhy(x+y)ndμ1(y)=Bn,q(h)(x).
In this paper, we investigate the (h,q)-extension of Bernoulli numbers and Bernoulli polynomials in order to obtain some interesting results and explicit relationships. The aim of this paper to observe an interesting phenomenon of “scattering” of the zeros of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x). The outline of this paper is as follows. In Section 2, we study the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x). In Section 3, we describe the beautiful zeros of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x) using a numerical investigation. Also we display distribution and structure of the zeros of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x) by using computer. By using the results of our paper, the readers can observe the regular behaviour of the roots of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x). Finally, we carried out computer experiments for demonstrating a remarkably regular structure of the complex roots of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x).

2. Basic Properties for the <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M65"><mml:mo stretchy="false">(</mml:mo><mml:mi>h</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>-Extension of Bernoulli Numbers and Bernoulli Polynomials

Let q be a complex number with |q|<1 and h∈ℤ. By the meaning of (1.5), the (h,q)-extension of Bernoulli numbers Bn,q(h) and Bernoulli polynomials Bn,q(h)(x) is defined by means of the following generating function: Fq(h)(t)=hlogq+tqhet-1=∑n=0∞Bn,q(h)tnn!,Fq(h)(x,t)=hlogq+tqhet-1ext=∑n=0∞Bn,q(h)(x)tnn!,
respectively.

Here is the list of the first (h,q)-extension of Bernoulli numbers Bn,q(h). B0,q(h)=hlogq-1+qh,B1,q=1-1+qh-hqhlogq(-1+qh)2,B2,qh(h)=-2qh(-1+qh)2-hqhlogq(-1+qh)2+2hq2hlogq(-1+qh)3,B3,qh(h)=-3qh(-1+qh)2+6q2h(-1+qh)3-hqhlogq(-1+qh)2+6hq2hlogq(-1+qh)3-6hq3hlogq(-1+qh)4,…,

because ∂∂xFq(h)(x,t)=tFq(h)(x,t)=∑n=0∞ddxBn,q(h)(x)tnn!,
it follows the important relation ddxBn,q(h)(x)=nBn-1,q(h)(x).
We have the integral formula as follows: ∫abBn-1,q(h)(x)dx=1n(Bn,q(h)(b)-Bn,q(h)(a)).

Here is the list of the first (h,q)-extension of Bernoulli Polynomials Bn,q(h)(x). B0,q(h)=hlogq-1+qh,B1,q(h)=1(-1+qh)-hqhlogq(-1+qh)2+hxlogq(-1+qh),B2,q(h)=2qh(-1+qh)2+2x(-1+qh)+2hq2hlogq(-1+qh)3-hqhlogq(-1+qh)2-2hqhxlogq(-1+qh)2+hx2logq(-1+qh),….

Since ∑l=0∞Bl,q(h)(x+y)tll!=hlogq+tqhet-1e(x+y)t=∑n=0∞Bn,q(h)(x)tnn!∑m=0∞ymtmm!=∑l=0∞(∑n=0lBn,q(h)(x)tnn!yl-ntl-n(l-n)!)=∑l=0∞(∑n=0l(ln)Bn,q(h)(x)yl-n)tll!,
we have the following theorem.

Theorem 2.1.

(h,q)-extension of Bernoulli polynomials Bn,q(h)(x) satisfies the following relation:
Bl,q(h)(x+y)=∑n=0l(ln)Bn,q(h)(x)yl-n.

From (2.2), we can derive the following equality:
∑n=0∞(qhBn,q(h)(x+1)-Bn,q(h)(x))tnn!=∑n=0∞(xnhlogq+nxn-1)tnn!.

Hence, we obtain the following difference equation.

Theorem 2.2.

For any positive integer n, we obtain
qhBn,q(h)(x+1)-Bn,q(h)(x)=xnhlogq+nxn-1.

3. Distribution and Structure of the Zeros

In this section, we assume that q∈ℂ, with |q|<1. We observed the behavior of real roots of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x). We display the shapes of the (h,q)-extension of Bernoulli polynomials Bn,q(x) and we investigate the zeros of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x). For n=1,…,10, we can draw a plot of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x), respectively. This shows the ten plots combined into one. We display the shape of Bn,q(h)(x),-1≤x≤1,q=1/2 (Figure 1). We investigate the beautiful zeros of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x) by using a computer. We plot the zeros of the (h,q)-extension of Bernoulli polynomials Bn,q(3)(x) for n=15,20,25,30 and x∈ℂ (Figure 2).

CurveofBn,1/2(3)(x).

ZerosofBn,1/2(3)(x)forn=15,20,25,30.

Our numerical results for approximate solutions of real zeros of Bn,1/2(h)(x) are displayed (Tables 1 and 2).

Numbers of real and complex zeros of Bn,q(h)(x).

degree n

h=3

h=5

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

2

2

5

3

2

3

2

6

4

2

4

2

7

5

2

5

2

8

6

2

4

4

9

3

6

5

4

10

4

6

6

4

11

5

6

5

6

12

6

6

4

8

Approximate solutions of Bn,q(3)(x)=0,q=1/2,x∈ℝ.

degree n

x

1

0.338041204

2

0.077277108,0.598805301

3

-0.079078401,0.27548817,0.81771384

4

-0.14859649,0.00639164,0.48798387,1.00638579

5

0.1922804,0.6960320,1.16991832

6

-0.1019335,0.3908601,0.8972294,1.3100253

7

-0.300737,0.094132,0.592060,1.094167,1.4250444

⋮

⋮

We plot the zeros of (h,q)-extension of Bernoulli polynomials Bn,q(h)(x) for n=30,q=1/2,h=5,7,9,11, and x∈ℂ (Figure 3). We plot the zeros of (h,q)-extension of Bernoulli polynomials Bn,q(h)(x) for n=30,q=9/10,99/100, and x∈ℂ (Figure 4).

ZerosofB30,1/2(x)forh=5,7,9,11.

ZerosofBn,30(3)(x)forq=9/10,99/100.

We observe a remarkably regular structure of the complex roots of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x). We hope to verify a remarkably regular structure of the complex roots of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x) (Table 1). This numerical investigation is especially exciting because we can obtain an interesting phenomenon of scattering of the zeros of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x). These results are used not only in pure mathematics and applied mathematics, but also used in mathematical physics and other areas. Next, we calculated an approximate solution satisfying the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x). The results are given in Table 2.

Stacks of zeros of Bn,q(h)(x) for q=1/3,1≤n≤30 from a 3D structure are presented (in Figure 5).

StacksofzerosofBn,q(h)(x),1≤n≤30.

Figure 6 presents the distribution of real zeros of the (h,q)-extension of Bernoulli polynomials Bn,q(3)(x) for q=1/2,1≤n≤30.

RealofzerosofBn,q(3)(x),q=1/2,1≤n≤30.

Figure 7 presents the distribution of real zeros of the (h,q)-extension of Bernoulli polynomials Bn,q(3)(x) for q=9/10,1≤n≤30.

RealofzerosofBn,q(3)(x),q=9/10,1≤n≤30.

Figure 8 presents the distribution of real zeros of the Bernoulli polynomials Bn(x) for 1≤n≤30.

RealofzerosofBn(x),1≤n≤30.

4. Direction for Further Research

In [7], we observed the behavior of complex roots of the Bernoulli polynomials Bn(x), using numerical investigation. Prove that Bn(x),x∈ℂ, has Re(x)=1/2 reflection symmetry in addition to the usual Im(x)=0 reflection symmetry analytic complex functions. The obvious corollary is that the zeros of Bn(x) will also inherit these symmetries. IfBn(x0)=0,thenBn(1-x0)=0=Bn(x0*)=Bn(1-x0*),
where * denotes complex conjugation (see [7]). Finally, we shall consider the more general problems. Prove that Bn(x)=0 has n distinct solutions. If B2n+1(x) has Re(x)=1/2 and Im(x)=0 reflection symmetries, and 2n+1 nondegenerate zeros, then 2n of the distinct zeros will satisfy (4.1). If the remaining one zero is to satisfy (4.1) too, it must reflect into itself, and therefore it must lie at 1/2, the center of the structure of the zeros, that is, Bn(12)=0∀oddn.
Prove that Bn,q(h)(x)=0 has n distinct solutions, that is, all the zeros are nondegenerate. Find the numbers of complex zeros CBn,q(h)(x) of Bn,q(h)(x),Im(x)≠0. Since n is the degree of the polynomial Bn,q(h)(x), the number of real zeros RBn,q(h)(x) lying on the real plane Im(x)=0 is then RBn,q(h)(x)=n-CBn,q(h)(x), where CBn,q(h)(x) denotes complex zeros. See Table 1 for tabulated values of RBn,q(h)(x) and CBn,q(h)(x). Find the equation of envelope curves bounding the real zeros lying on the plane. We prove that Bn,q(h)(x),x∈ℂ, has Im(x)=0 reflection symmetry analytic complex functions. If Bn,q(h)(x)=0, then Bn,q(h)(x*)=0, where * denotes complex conjugate (see Figures 2, 3, and 4). Observe that the structure of the zeros of the Bernoulli polynomials Bn(x) resembles the structure of the zeros of the q-Bernoulli polynomials Bn,q(h)(x) as q→1 (see Figures 3, 4, and 5). In order to study the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x), we must understand the structure of the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x). Therefore, using computer, a realistic study for the (h,q)-extension of Bernoulli polynomials Bn,q(h)(x) plays an important part. 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 (h,q)-extension of Bernoulli polynomials Bn,q(h)(x) to appear in mathematics and physics. For related topics, the interested reader is referred to [3–8].

KimT.Note on the Euler q-zeta functionsKimT.q-Euler numbers and polynomials associated with p-adic q-integralsKimT.On p-adic interpolating function for q-Euler numbers and its derivativesKimT.q-Volkenborn integrationKimT.RimSeog-HoonGeneralized Carlitz's q-Bernoulli numbers in the p-adic number fieldRyooC. S.KimT.An analogue of the zeta function and its applicationsRyooC. S.A numerical computation on the structure of the roots of q-extension of Genocchi polynomialsRyooC. S.Calculating zeros of the twisted Genocchi polynomials