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

The goal of this paper is to define the (p, q)-analogue of tangent numbers and polynomials by generalizing the tangent numbers and polynomials and Carlitz-type q-tangent numbers and polynomials. We get some explicit formulas and properties in conjunction with (p, q)-analogue of tangent numbers and polynomials. We give some new symmetric identities for (p, q)-analogue of tangent polynomials by using (p, q)-tangent zeta function. Finally, we investigate the distribution and symmetry of the zero of (p, q)-analogue of tangent polynomials with numerical methods.

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.

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N denotes the set of natural numbers.C denotes the set of complex numbers.
We remember that the classical tangent numbers T n and tangent polynomials T n (x) 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→1 [n] p,q = n with p = 1.Since [n] p,q = p n−1 [n] q p , we observe that (p, q)-numbers and p-numbers are different.In other words, by substituting q by 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]).Definition 1.For any complex x we define the Carlitz-type q-tangent polynomials, T n,q (x), by the equation The numbers T n,q (0) 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 (x).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 (x).In the next section we define the (p, q)-analogue of tangent numbers and polynomials.After that we will obtain some their properties.

(p, q)-Analogue of Tangent Numbers and Polynomials
Firstly, we construct (p, q)-analogue of tangent numbers and polynomials and derive some of their relevant properties.Definition 2. For 0 < q < p ≤ 1, the Carlitz-type (p, q)-tangent numbers T n,p,q and polynomials T n,p,q (x) are defined by means of the generating functions and 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 (x) respectively.Obviously, if we put p = 1, then we have T n,p,q (x) = T n,q (x), T n,p,q = T n,q .
Putting p = 1, we have Proof.By (4), we have Equating the coefficients of t n n! , we arrive at the desired result (6).
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 (x).So, we choose to omit the details involved.Proposition 1.For any positive integer n, one has [m] n p,q ∑ m−1 a=0 (−1) a q a T n,p m ,q m 2a+x m , (m = odd).

(p, q)-Analogue of Tangent Zeta Function
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 (x) at negative integers, respectively.From (4), we note that From the above equation, we construct new (p, q)-tangent zeta function as follows: Definition 3. We define the (p, q)-tangent zeta function for s ∈ C with Re(s) > 0 by Notice that ζ p,q (s) is a meromorphic function on C(cf.7).Remark that, if p = 1, q → 1, then ζ p,q (s) = ζ T (s) which is the tangent zeta function (see [19]).The relationship between the ζ p,q (s) and the T k,p,q is given explicitly by the following theorem.
Please note that ζ p,q (s) 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.
Definition 4. For s ∈ C with Re(s) > 0 and x / ∈ Z − 0 , we define Obverse that the function ζ p,q (s, x) is a meromorphic function on C. We note that, if p = 1 and q → 1, then ζ p,q (s, x) = ζ T (s, x) which is the Hurwitz tangent zeta function (see [19]).The function ζ p,q (−k, x) interpolates the numbers T k,p,q (x) at non-negative integers.Substituting s = −k with k ∈ N into (15), and using Theorem 2, we easily arrive at the following theorem.Theorem 6.Let k ∈ N. One has ζ p,q (−k, x) = T k,p,q (x).
4. Some Symmetric Properties About (P, Q)-Analogue of Tangent Zeta Function 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.Theorem 7. Let w 1 and w 2 be positive odd integers.Then we have [2] Proof.For any x, y ∈ C, we observe that [xy] p,q = [x] p y ,q y [y] p,q .By substituting w 1 x + 2w 1 i w 2 for x in Definition 4, replace p by p w 2 and replace q by q w 2 , respectively, we derive Since for any non-negative integer m and positive odd integer w 1 , there exist unique non-negative integer r such that m = w 1 r + j with 0 ≤ j ≤ w 1 − 1.Thus, this can be written as It follows from the above equation that [2] From the similar method, we can have that After some calculations in the above, we have Thus, from ( 16) and ( 17), we obtain the result.
Corollary 1.For s ∈ C with Re(s) > 0, we have Proof.Let w 2 = 1 in Theorem 7. Then we immediately get the result.
Next, we also derive some symmetric identities for Carlitz-type (p, q)-tangent polynomials by using (p, q)-tangent zeta function.
Theorem 8. Let w 1 and w 2 be any positive odd integers.The following multiplication formula holds true for the Carlitz-type (p, q)-tangent polynomials: (−1) j q w 2 j T n,p w 1 ,q w 1 w 2 x + 2w 2 j w 1 .
Proof.By substituting T n,p,q (x) for ζ p,q (s, x) in Theorem 7, and using Theorem 6, we can find that and Thus, by ( 18) and ( 19), this concludes our proof.
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 (h) n,p,q (x), we can obtain the following theorem.Theorem 9. Let w 1 and w 2 be any positive odd integers.Then one has [2] n−l,p w 2 ,q w 2 (w 1 x)T n,l,p w 1 ,q w 1 (w 2 ).
Tables 1 and 2 present the numerical results for approximate solutions of real zeros of T n,p,q (x).The numbers of zeros of T n,p,q (x) are tabulated in Table 1 for a fixed p = 1 2 and q = 1 10 .
Table 1.Numbers of real and complex zeros of T n,p,q (x), p = 1 2 , q = 1 10 .Table 2. Numerical solutions of T n,p,q (x) = 0, p = 1 2 , q = 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 (x).The zeros of the Carlitz-type (p, q)-tangent polynomials T n,p,q (x) for x ∈ C are plotted in Figure 1.

Degree
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 (x) 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 (x) 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 (x) = 0 for x ∈ R. The numerical solutions are tabulated in Table 2 for a fixed p = 1 2 and q = 1 10 and various value of n. .Zeros of T n,p,q (x).

Conclusions and Future Developments
This study constructed the Carlitz-type (p, q)-tangent numbers and polynomials.We have derived several formulas for the Carlitz-type (h, q)-tangent numbers and polynomials.Some interesting symmetric identities for Carlitz-type (p, q)-tangent polynomials are also obtained.Moreover, the results of [18] can be derived from ours as special cases when q = 1.By numerical experiments, we will make a series of the following conjectures: Conjecture 1. Prove or disprove that T n,p,q (x), x ∈ C, has Im(x) = 0 reflection symmetry analytic complex functions.Furthermore, T n,p,q (x) has Re(x) = a reflection symmetry for a ∈ R.
Many more values of n have been checked.It still remains unknown if the conjecture holds or fails for any value n (see Figure 1).Conjecture 2. Prove or disprove that T n,p,q (x) = 0 has n distinct solutions.
In the notations: R T n,p,q (x) denotes the number of real zeros of T n,p,q (x) lying on the real plane Im(x) = 0 and C T n,p,q (x) denotes the number of complex zeros of T n,p,q (x).Since n is the degree of the polynomial T n,p,q (x), we get R T n,p,q (x) = n − C T n,p,q (x) (see Tables 1 and 2).Conjecture 3. Prove or disprove that R T n,p,q (x) = 1, if n = odd, 2, if n = even.
We expect that investigations along these directions will lead to a new approach employing numerical method regarding the research of the Carlitz-type (p, q)-tangent polynomials T n,p,q (x) which appear in applied mathematics, and mathematical physics (see [11,[18][19][20]).