Symmetric Identities for (P,Q)-Analogue of Tangent Zeta Function

The field of the special polynomials such as tangent polynomials, Bernoulli polynomials, Euler polynomials, and Genocchi polynomials is an expanding area in mathematics (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]). Many generalizations of these polynomials have been studied (see [1,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18]). Srivastava [14] developed some properties and q-extensions of the Euler polynomials, Bernoulli polynomials, and Genocchi polynomials. Choi, Anderson and Srivastava have discussed q-extension of the Riemann zeta function and related functions (see [5,17]). Dattoli, Migliorati and Srivastava derived a generalization of the classical polynomials (see [6]).

It is the purpose of this paper to introduce and investigate a new some generalizations of the Carlitz-type q-tangent numbers and polynomials, q-tangent zeta function, Hurwiz q-tangent zeta function. We call them Carlitz-type $(p,q)$-tangent numbers and polynomials, $(p,q)$-tangent zeta function, and Hurwitz $(p,q)$-tangent zeta function. The structure of the paper is as follows: In Section 2 we define Carlitz-type $(p,q)$-tangent numbers and polynomials and derive some of their properties involving elementary properties, distribution relation, property of complement, and so on. In Section 3, by using the Carlitz-type $(p,q)$-tangent numbers and polynomials, $(p,q)$-tangent zeta function and Hurwitz $(p,q)$-tangent zeta function are defined. We also contains some connection formulae between the Carlitz-type $(p,q)$-tangent numbers and polynomials and the $(p,q)$-tangent zeta function, Hurwitz $(p,q)$-tangent zeta function. In Section 4 we give several symmetric identities about $(p,q)$-tangent zeta function and Carlitz-type $(p,q)$-tangent polynomials and numbers. In the following Section, we investigate the distribution and symmetry of the zero of Carlitz-type $(p,q)$-tangent polynomials using a computer. Our paper ends with Section 6, where the conclusions and future developments of this work are presented. The following notations will be used throughout this paper.

We remember that the classical tangent numbers $T_n$ and tangent polynomials $T_n\left(x\right)$ are defined by the following generating functions (see [19]) and respectively. Some interesting properties of basic extensions and generalizations of the tangent numbers and polynomials have been worked out in [11,12,18,19,20]. The $(p,q)$-number is defined as

It is clear that $(p,q)$-number contains symmetric property, and this number is q-number when $p=1$. In particular, we can see ${\lim }_{q\to 1}{\left[n\right]}_{p,q}=n$ with $p=1$. Since ${\left[n\right]}_{p,q}=p^{n-1}{\left[n\right]}_{\frac{q}{p}}$, we observe that $(p,q)$-numbers and p-numbers are different. In other words, by substituting q by $\frac{q}{p}$ in the definition q-number, we cannot have $(p,q)$-number. Duran, Acikgoz and Araci [7] introduced the $(p,q)$-analogues of Euler polynomials, Bernoulli polynomials, and Genocchi polynomials. Araci, Duran, Acikgoz and Srivastava developed some properties and relations between the divided differences and $(p,q)$-derivative operator (see [1]). The $(p,q)$-analogues of tangent polynomials were described in [20]. By using $(p,q)$-number, we construct the Carlitz-type $(p,q)$-tangent polynomials and numbers, which generalized the previously known tangent polynomials and numbers, including the Carlitz-type q-tangent polynomials and numbers. We begin by recalling here the Carlitz-type q-tangent numbers and polynomials (see [18]).

The numbers $T_{n,q}\left(0\right)$ are called the Carlitz-type q-tangent numbers and are denoted by $T_{n,q}$. Based on this idea, we generalize the Carlitz-type q-tangent number $T_{n,q}$ and q-tangent polynomials $T_{n,q}\left(x\right)$. It follows that we define the following $(p,q)$-analogues of the the Carlitz-type q-tangent number $T_{n,q}$ and q-tangent polynomials $T_{n,q}\left(x\right)$. In the next section we define the $(p,q)$-analogue of tangent numbers and polynomials. After that we will obtain some their properties.

Firstly, we construct $(p,q)$-analogue of tangent numbers and polynomials and derive some of their relevant properties.

Setting $p=1$ in (4) and (5), we can obtain the corresponding definitions for the Carlitz-type q-tangent numbers $T_{n,q}$ and q-tangent polynomials $T_{n,q}\left(x\right)$ respectively. Obviously, if we put $p=1$, then we have

Putting $p=1$, we have

If we put $p=1$ in Theorem 1, we obtain (cf. [18])

Next, we construct the Carlitz-type $(h,p,q)$-tangent polynomials $T_{n,p,q}^{\left(h\right)}\left(x\right)$. Define the Carlitz-type $(h,p,q)$-tangent polynomials $T_{n,p,q}^{\left(h\right)}\left(x\right)$ by

Furthermore, by (7) and Theorem 2, we have

From (4) and (5), we can derive the following properties of the Carlitz-type tangent numbers $T_{n,p,q}$ and polynomials $T_{n,p,q}\left(x\right)$. So, we choose to omit the details involved.

Using Carlitz-type $(p,q)$-tangent numbers and polynomials, we define the $(p,q)$-tangent zeta function and Hurwitz $(p,q)$-tangent zeta function. These functions have the values of the Carlitz-type $(p,q)$-tangent numbers $T_{n,p,q}$, and polynomials $T_{n,p,q}\left(x\right)$ at negative integers, respectively. From (4), we note that

From the above equation, we construct new $(p,q)$-tangent zeta function as follows:

Notice that ${\zeta }_{p,q}\left(s\right)$ is a meromorphic function on $\mathbb{C}$(cf.7). Remark that, if $p=1,q\to 1$, then ${\zeta }_{p,q}\left(s\right)={\zeta }_T\left(s\right)$ which is the tangent zeta function (see [19]). The relationship between the ${\zeta }_{p,q}\left(s\right)$ and the $T_{k,p,q}$ is given explicitly by the following theorem.

Please note that ${\zeta }_{p,q}\left(s\right)$ function interpolates $T_{k,p,q}$ numbers at non-negative integers. Similarly, by using Equation (5), we get and Furthermore, by (13) and (14), we are ready to construct the Hurwitz $(p,q)$-tangent zeta function.

Obverse that the function ${\zeta }_{p,q}(s,x)$ is a meromorphic function on $\mathbb{C}$. We note that, if $p=1$ and $q\to 1$, then ${\zeta }_{p,q}(s,x)={\zeta }_T(s,x)$ which is the Hurwitz tangent zeta function (see [19]). The function ${\zeta }_{p,q}(-k,x)$ interpolates the numbers $T_{k,p,q}\left(x\right)$ at non-negative integers. Substituting $s=-k$ with $k\in \mathbb{N}$ into (15), and using Theorem 2, we easily arrive at the following theorem.

Our main objective in this section is to obtain some symmetric properties about $(p,q)$-tangent zeta function. In particular, some of these symmetric identities are also related to the Carlitz-type $(p,q)$-tangent polynomials and the alternate power sums. To end this section, we focus on some symmetric identities containing the Carlitz-type $(p,q)$-tangent zeta function and the alternate power sums.

Next, we also derive some symmetric identities for Carlitz-type $(p,q)$-tangent polynomials by using $(p,q)$-tangent zeta function.

Considering $w_1=1$ in the Theorem 8, we obtain as below equation.

Furthermore, by applying the addition theorem for the Carlitz-type $(h,p,q)$-tangent polynomials $T_{n,p,q}^{\left(h\right)}\left(x\right)$, we can obtain the following theorem.

The purpose of this section is to support theoretical predictions using numerical experiments and to discover new exciting patterns for zeros of the Carlitz-type $(p,q)$-tangent polynomials $T_{n,p,q}\left(x\right)$. We propose some conjectures by numerical experiments. The first values of the $T_{n,p,q}\left(x\right)$ are given by

Table 1 and Table 2 present the numerical results for approximate solutions of real zeros of $T_{n,p,q}\left(x\right)$. The numbers of zeros of $T_{n,p,q}\left(x\right)$ are tabulated in Table 1 for a fixed $p=\frac{1}{2}$ and $q=\frac{1}{10}$.

The use of computer has made it possible to identify the zeros of the Carlitz-type $(p,q)$-tangent polynomials $T_{n,p,q}\left(x\right)$. The zeros of the Carlitz-type $(p,q)$-tangent polynomials $T_{n,p,q}\left(x\right)$ for $x\in \mathbb{C}$ are plotted in Figure 1.

In Figure 1(top-left), we choose $n=10,p=1/2$ and $q=1/10$. In Figure 1(top-right), we choose $n=20,p=1/2$ and $q=1/10$. In Figure 1(bottom-left), we choose $n=30,p=1/2$ and $q=1/10$. In Figure 1(bottom-right), we choose $n=40,p=1/2$ and $q=1/10$. It is amazing that the structure of the real roots of the Carlitz-type $(p,q)$-tangent polynomials $T_{n,p,q}\left(x\right)$ is regular. Thus, theoretical prediction on the regular structure of the real roots of the Carlitz-type $(p,q)$-tangent polynomials $T_{n,p,q}\left(x\right)$ is await for further study (Table 1). Next, we have obtained the numerical solution satisfying Carlitz-type $(p,q)$-tangent polynomials $T_{n,p,q}\left(x\right)=0$ for $x\in \mathbb{R}$. The numerical solutions are tabulated in Table 2 for a fixed $p=\frac{1}{2}$ and $q=\frac{1}{10}$ and various value of n.