A Note on the (ℎ,𝑞)-Extension of Bernoulli Numbers and Bernoulli Polynomials

Abstract

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

1. Introduction

Throughout this paper ℤ,ℤ_𝑝,ℚ_𝑝, and ℂ_𝑝 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 ℚ_𝑝, respectively, compare with [1–6]. Let 𝜈_𝑝 be the normalized exponential valuation of ℂ_𝑝 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∈ℂ_𝑝. If 𝑞∈ℂ_𝑝, then we normally assume that |𝑞−1|_𝑝<𝑝^{−1/(𝑝−1)}, so that 𝑞^𝑥=exp(𝑥log𝑞) for |𝑥|_𝑝≤1. If 𝑞∈ℂ, then we normally assume that |𝑞|<1. For 𝑓∈𝑈𝐷(ℤ_𝑝,ℂ_𝑝)={𝑓∣𝑓∶ℤ_𝑝→ℂ_𝑝isuniformlydifferentiablefunction}, the 𝑝-adic 𝑞-integral (or 𝑞-Volkenborn integration) was defined as 𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁_𝑞𝑝^𝑁−1_𝑥=0𝑓(𝑥)𝑞^𝑥,(1.1) where [𝑥]_𝑞=(1−𝑞^𝑥)/(1−𝑞), compare with [1–8]. Thus, we note that 𝐼₁(𝑓)=lim_𝑞→1𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇₁(𝑥)=lim_𝑁→∞1𝑝^𝑁_{0≤𝑥<𝑝𝑁}𝑓(𝑥),comparewith[1,2,3,4,5,6].(1.2) By (1.2), we easily see that 𝐼₁𝑓₁=𝐼₁(𝑓)+𝑓^(0),comparewith[1,2,3,4,5,6],(1.3) where 𝑓₁(𝑥)=𝑓(𝑥+1),𝑓^(0)=(𝑑/𝑑𝑥)𝑓(𝑥)|_𝑥=0.

In (1.3), if we take 𝑓(𝑥)=𝑞^ℎ𝑥𝑒^𝑥𝑡, then we have _ℤ𝑝𝑞^ℎ𝑥𝑒^𝑥𝑡𝑑𝜇₁(𝑥)=ℎlog𝑞+𝑡𝑞^ℎ𝑒^𝑡−1,comparewith[6],(1.4) 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 [6]). The (ℎ,𝑞)-extension of Bernoulli numbers 𝐵^{(ℎ)𝑛,𝑞} and polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥) is defined by means of the generating functions as follows: 𝐹^𝑞(ℎ)(𝑡)=ℎlog𝑞+𝑡𝑞^ℎ𝑒^𝑡−1=^∞_𝑛=0𝐵^{(ℎ)𝑛,𝑞}𝑡^𝑛𝐹𝑛!_𝑞(ℎ),(𝑡,𝑥)=ℎlog𝑞+𝑡𝑞^ℎ𝑒^𝑡−1𝑒^𝑥𝑡=^∞_𝑛=0𝐵^{(ℎ)𝑛,𝑞}(𝑥)𝑡^𝑛𝑛!.(1.5)
Note that 𝐵^{(ℎ)𝑛,𝑞}(0)=𝐵^{(ℎ)𝑛,𝑞},lim_𝑞→1𝐵^{(ℎ)𝑛,𝑞}(𝑥)=𝐵_𝑛(𝑥), and 𝐵^(0)𝑛,𝑞(𝑥)=𝐵_𝑛(𝑥), where 𝐵_𝑛 are the 𝑛th Bernoulli numbers.

By (1.4) and (1.5), we have the following Witt formula. For ℎ∈ℤ,𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝≤𝑝^{−1/(𝑝−1)}, we have _ℤ𝑝_ℤ𝑝𝑞^ℎ𝑥𝑥^𝑛𝑑𝜇₁(𝑥)=𝐵^{(ℎ)𝑛,𝑞},𝑞^ℎ𝑦(𝑥+𝑦)^𝑛𝑑𝜇₁(𝑦)=𝐵^{(ℎ)𝑛,𝑞}(𝑥).(1.6) 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 𝐵^{(ℎ)𝑛,𝑞}(𝑥). The outline of this paper is as follows. In Section 2, we study the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥). In Section 3, we describe the beautiful zeros of the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥) using a numerical investigation. Also we display distribution and structure of the zeros of the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(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 𝐵^{(ℎ)𝑛,𝑞}(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 𝐵^{(ℎ)𝑛,𝑞}(𝑥).

2. Basic Properties for the (ℎ,𝑞)-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 𝐵^{(ℎ)𝑛,𝑞} and Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥) is defined by means of the following generating function: 𝐹^𝑞(ℎ)(𝑡)=ℎlog𝑞+𝑡𝑞^ℎ𝑒^𝑡−1=^∞_𝑛=0𝐵^{(ℎ)𝑛,𝑞}𝑡^𝑛𝑛!,(2.1)𝐹^𝑞(ℎ)(𝑥,𝑡)=ℎlog𝑞+𝑡𝑞^ℎ𝑒^𝑡−1𝑒^𝑥𝑡=^∞_𝑛=0𝐵^{(ℎ)𝑛,𝑞}(𝑥)𝑡^𝑛𝑛!,(2.2) respectively.

Here is the list of the first (ℎ,𝑞)-extension of Bernoulli numbers 𝐵^{(ℎ)𝑛,𝑞}. 𝐵^(ℎ)0,𝑞=ℎlog𝑞−1+𝑞^ℎ,𝐵_1,𝑞=1−1+𝑞^ℎ−ℎ𝑞^ℎlog𝑞−1+𝑞^ℎ²𝐵_{(ℎ)2,𝑞ℎ},=−2𝑞^ℎ−1+𝑞^ℎ²−ℎ𝑞^ℎlog𝑞−1+𝑞^ℎ²+2ℎ𝑞^2ℎlog𝑞−1+𝑞^ℎ³𝐵_{(ℎ)3,𝑞ℎ},=−3𝑞^ℎ−1+𝑞^ℎ²+6𝑞^2ℎ−1+𝑞^ℎ³−ℎ𝑞^ℎlog𝑞−1+𝑞^ℎ²+6ℎ𝑞^2ℎlog𝑞−1+𝑞^ℎ³−6ℎ𝑞^3ℎlog𝑞−1+𝑞^ℎ⁴,…,(2.3)

because 𝜕𝜕𝑥𝐹^𝑞(ℎ)(𝑥,𝑡)=𝑡𝐹^𝑞(ℎ)(𝑥,𝑡)=^∞_𝑛=0𝑑𝑑𝑥𝐵^{(ℎ)𝑛,𝑞}(𝑥)𝑡^𝑛𝑛!,(2.4) it follows the important relation 𝑑𝑑𝑥𝐵^{(ℎ)𝑛,𝑞}(𝑥)=𝑛𝐵^{(ℎ)𝑛−1,𝑞}(𝑥).(2.5) We have the integral formula as follows: ^𝑏𝑎𝐵^{(ℎ)𝑛−1,𝑞}(𝑥)𝑑𝑥=1𝑛𝐵^{(ℎ)𝑛,𝑞}(𝑏)−𝐵^{(ℎ)𝑛,𝑞}(𝑎).(2.6)

Here is the list of the first (ℎ,𝑞)-extension of Bernoulli Polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥). 𝐵^(ℎ)0,𝑞=ℎlog𝑞−1+𝑞^ℎ𝐵_(ℎ)1,𝑞,=1−1+𝑞^ℎ−ℎ𝑞^ℎlog𝑞−1+𝑞^ℎ²+ℎ𝑥log𝑞𝐵_(ℎ)2,𝑞−1+𝑞^ℎ,=2𝑞^ℎ−1+𝑞^ℎ²+2𝑥−1+𝑞^ℎ+2ℎ𝑞^2ℎlog𝑞−1+𝑞^ℎ³−ℎ𝑞^ℎlog𝑞−1+𝑞^ℎ²−2ℎ𝑞^ℎ𝑥log𝑞−1+𝑞^ℎ²+ℎ𝑥²log𝑞−1+𝑞^ℎ,….(2.7)

Since ^∞_𝑙=0𝐵^{(ℎ)𝑙,𝑞}(𝑥+𝑦)𝑡^𝑙𝑙!=ℎ𝑞l^ℎo𝑒g^𝑡𝑞−+1𝑡𝑒^{(𝑥+𝑦)𝑡}=^∞_𝑛=0𝐵^{(ℎ)𝑛,𝑞}(𝑥)𝑡^𝑛𝑛!^∞_𝑚=0𝑦^𝑚𝑡^𝑚=𝑚!_∞_𝑙=0^𝑙_𝑛=0𝐵^{(ℎ)𝑛,𝑞}(𝑥)𝑡^𝑛𝑛!𝑦^𝑙−𝑛𝑡^𝑙−𝑛=(𝑙−𝑛)!_∞_𝑙=0^𝑙_𝑛=0𝑛𝑙𝐵^{(ℎ)𝑛,𝑞}(𝑥)𝑦^𝑙−𝑛𝑙𝑡!^𝑙,(2.8) we have the following theorem.

Theorem 2.1. (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥) satisfies the following relation: 𝐵^{(ℎ)𝑙,𝑞}(𝑥+𝑦)=^𝑙_𝑛=0𝑛𝑙𝐵^{(ℎ)𝑛,𝑞}(𝑥)𝑦^𝑙−𝑛.(2.9)
From (2.2), we can derive the following equality: ^∞_𝑛=0𝑞^ℎ𝐵^{(ℎ)𝑛,𝑞}(𝑥+1)−𝐵^{(ℎ)𝑛,𝑞}(𝑥)𝑡^𝑛𝑛!=^∞_𝑛=0𝑥^𝑛ℎlog𝑞+𝑛𝑥^𝑛−1𝑡^𝑛𝑛!.(2.10)
Hence, we obtain the following difference equation.

Theorem 2.2. For any positive integer 𝑛, we obtain 𝑞^ℎ𝐵^{(ℎ)𝑛,𝑞}(𝑥+1)−𝐵^{(ℎ)𝑛,𝑞}(𝑥)=𝑥^𝑛ℎlog𝑞+𝑛𝑥^𝑛−1.(2.11)

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 𝐵^{(ℎ)𝑛,𝑞}(𝑥). We display the shapes of the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵_𝑛,𝑞(𝑥) and we investigate the zeros of the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥). For 𝑛=1,…,10, we can draw a plot of the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥), respectively. This shows the ten plots combined into one. We display the shape of 𝐵^{(ℎ)𝑛,𝑞}(𝑥),−1≤𝑥≤1,𝑞=1/2 (Figure 1). We investigate the beautiful zeros of the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥) by using a computer. We plot the zeros of the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^(3)𝑛,𝑞(𝑥) for 𝑛=15,20,25,30 and 𝑥∈ℂ (Figure 2).

Figure 1 

Curveof𝐵(3)𝑛,1/2(𝑥).

(a)

(b)

(c)

(d)

Figure 2 

Zerosof𝐵(3)𝑛,1/2(𝑥)for𝑛=15,20,25,30.

Our numerical results for approximate solutions of real zeros of 𝐵^{(ℎ)𝑛,1/2}(𝑥) are displayed (Tables 1 and 2).

Table 1 

Numbers of real and complex zeros of 𝐵^{(ℎ)𝑛,𝑞}(𝑥).

Table 2 

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

We plot the zeros of (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(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 𝐵^{(ℎ)𝑛,𝑞}(𝑥) for 𝑛=30,𝑞=9/10,99/100, and 𝑥∈ℂ (Figure 4).

(a)

(b)

(c)

(d)

Figure 3 

Zerosof𝐵30,1/2(𝑥)forℎ=5,7,9,11.

(a)

(b)

Figure 4 

Zerosof𝐵(3)𝑛,30(𝑥)for𝑞=9/10,99/100.

We observe a remarkably regular structure of the complex roots of the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥). We hope to verify a remarkably regular structure of the complex roots of the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(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 𝐵^{(ℎ)𝑛,𝑞}(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 𝐵^{(ℎ)𝑛,𝑞}(𝑥). The results are given in Table 2.

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

Figure 5 

Stacksofzerosof𝐵(ℎ)𝑛,𝑞(𝑥),1≤𝑛≤30.

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

Figure 6 

Realofzerosof𝐵(3)𝑛,𝑞(𝑥),𝑞=1/2,1≤𝑛≤30.

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

Figure 7 

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

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

Figure 8 

Realofzerosof𝐵𝑛(𝑥),1≤𝑛≤30.

4. Direction for Further Research

In [7], we observed the behavior of complex roots of the Bernoulli polynomials 𝐵_𝑛(𝑥), using numerical investigation. Prove that 𝐵_𝑛(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 𝐵_𝑛(𝑥) will also inherit these symmetries. If𝐵_𝑛𝑥₀=0,then𝐵_𝑛1−𝑥₀=0=𝐵_𝑛𝑥^∗0=𝐵_𝑛1−𝑥^∗0,(4.1) where ∗ denotes complex conjugation (see [7]). Finally, we shall consider the more general problems. Prove that 𝐵_𝑛(𝑥)=0 has 𝑛 distinct solutions. If 𝐵_2𝑛+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, 𝐵_𝑛12=0∀odd𝑛.(4.2) Prove that 𝐵^{(ℎ)𝑛,𝑞}(𝑥)=0 has 𝑛 distinct solutions, that is, all the zeros are nondegenerate. Find the numbers of complex zeros 𝐶_𝐵^{(ℎ)𝑛,𝑞}_(𝑥) of 𝐵^{(ℎ)𝑛,𝑞}(𝑥),Im(𝑥)≠0. Since 𝑛 is the degree of the polynomial 𝐵^{(ℎ)𝑛,𝑞}(𝑥), the number of real zeros 𝑅_𝐵^{(ℎ)𝑛,𝑞}_(𝑥) lying on the real plane Im(𝑥)=0 is then 𝑅_𝐵^{(ℎ)𝑛,𝑞}_(𝑥)=𝑛−𝐶_𝐵^{(ℎ)𝑛,𝑞}_(𝑥), where 𝐶_𝐵^{(ℎ)𝑛,𝑞}_(𝑥) denotes complex zeros. See Table 1 for tabulated values of 𝑅_𝐵^{(ℎ)𝑛,𝑞}_(𝑥) and 𝐶_𝐵^{(ℎ)𝑛,𝑞}_(𝑥). Find the equation of envelope curves bounding the real zeros lying on the plane. We prove that 𝐵^{(ℎ)𝑛,𝑞}(𝑥),𝑥∈ℂ, has Im(𝑥)=0 reflection symmetry analytic complex functions. If 𝐵^{(ℎ)𝑛,𝑞}(𝑥)=0, then 𝐵^{(ℎ)𝑛,𝑞}(𝑥^∗)=0, where ∗ denotes complex conjugate (see Figures 2, 3, and 4). Observe that the structure of the zeros of the Bernoulli polynomials 𝐵_𝑛(𝑥) resembles the structure of the zeros of the 𝑞-Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥) as 𝑞→1 (see Figures 3, 4, and 5). In order to study the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥), we must understand the structure of the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(𝑥). Therefore, using computer, a realistic study for the (ℎ,𝑞)-extension of Bernoulli polynomials 𝐵^{(ℎ)𝑛,𝑞}(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 𝐵^{(ℎ)𝑛,𝑞}(𝑥) to appear in mathematics and physics. For related topics, the interested reader is referred to [3–8].