Approximations of Tangent Polynomials, Tangent –Bernoulli and Tangent – Genocchi Polynomials in terms of Hyperbolic Functions

Several well-known special functions, numbers and polynomials (e.g. zeta functions, Bernoulli, Euler and Genocchi numbers and polynomials and derivative polynomials [1–6]) have been studied by many researchers in recent decade due to their wide-ranging applications from number theory and combinatorics to other fields of applied mathematics. With these, different variations and generalizations of these functions, numbers and polynomials have been constructed and investigated. For instance, papers in [7, 8] have introduced and investigated q-analogues of zeta functions and Euler polynomials. Some variations are constructed by mixing the concept of two special functions, numbers or polynomials. For example, poly-Bernoulli numbers and polynomials in [9–11] are constructed by mixing the concepts of polylogarithm and Bernoulli numbers and polynomials. Moreover, the Apostol-Genocchi polynomials, Frobenius-Euler polynomials, Frobenius-Genocchi polynomials and Apostol-Frobenius-type poly-Genocchi polynomials in [12–14] are constructed by mixing the concepts of Apostol, Frobenius, Genocchi and Euler polynomials. Another interesting mixture of special polynomials can be constructed by joining the concept of Tangent polynomials with Bernoulli and Genocchi polynomials. The Tangent polynomials TnðzÞ, Tangent – Bernoulli ðTBÞnðzÞ and Tangent – Genocchi ðTGÞnðzÞ polynomials are defined by the generating functions


Introduction
Several well-known special functions, numbers and polynomials (e.g. zeta functions, Bernoulli, Euler and Genocchi numbers and polynomials and derivative polynomials [1][2][3][4][5][6]) have been studied by many researchers in recent decade due to their wide-ranging applications from number theory and combinatorics to other fields of applied mathematics. With these, different variations and generalizations of these functions, numbers and polynomials have been constructed and investigated. For instance, papers in [7,8] have introduced and investigated q-analogues of zeta functions and Euler polynomials. Some variations are constructed by mixing the concept of two special functions, numbers or polynomials. For example, poly-Bernoulli numbers and polynomials in [9][10][11] are constructed by mixing the concepts of polylogarithm and Bernoulli numbers and polynomials. Moreover, the Apostol-Genocchi polynomials, Frobenius-Euler polynomials, Frobenius-Genocchi polynomials and Apostol-Frobenius-type poly-Genocchi polynomials in [12][13][14] are constructed by mixing the concepts of Apostol, Frobenius, Genocchi and Euler polynomials.
Another interesting mixture of special polynomials can be constructed by joining the concept of Tangent polynomials with Bernoulli and Genocchi polynomials. The Tangent polynomials T n ðzÞ, Tangent -Bernoulli ðTBÞ n ðzÞ and Tangent -Genocchi ðTGÞ n ðzÞ polynomials are defined by the generating functions we wz 2we wz e 2w + 1 = 〠 ∞ n=0 TG ð Þ n z ð Þ w n n! , w j j < π 2 : When z = 0, ð1:1Þ reduces to the generating function of the tangent numbers T n given by In [15], the tangent polynomials can be determined explicitly using the tangent numbers T n which is given by We are interested in finding asymptotic approximations of the Tangent polynomials T n ðzÞ, Tangent-Bernoulli ðTBÞ n ðzÞ and Tangent-Genocchi ðTGÞ n ðzÞ polynomials for large n which are uniformly valid in some unbounded region of the complex variable z. Equation Equation (1) yields a recurrence relation with initial value T 0 ðzÞ = 1. The Tangent polynomials T n ðzÞ can be determined explicitly using (6). The first few values are The Tangent-Bernoulli polynomials and Tangent-Genocchi polynomials satisfy the relations where Bernoulli and Genocchi polynomials were defined in [7,16], respectively. Some specific values are given below.
Applications of Bernoulli polynomials can be found in [16] while new formulas for Genocchi polynomials involving Bernoulli polynomials can be found in [17]. The Bernoulli polynomials and Genocchi polynomials were expressed in terms of hyperbolic function in [7,16] as follows where C is a circle about 0 with radius <2π (resp. <π). These integral representations were used to establish the asymptotic approximations of Bernoulli and Genocchi numbers.
In this paper, the Tangent polynomials, Tangent-Bernoulli, and Tangent-Genocchi polynomials will be given asymptotic approximations using the method used in [19,20].

Uniform Approximations
First, let us consider the approximation of Tangent polynomials. Using the saddle point method introduced we can establish the following theorem. Theorem 1. For z ∈ ℂ \ f0g such that jI m z −1 j < π/2 and jz ± ð2i/πÞj > 2/π and n ≥ 1, Proof. Applying the Cauchy-Integral Formula to (1) where C is a circle about 0 with radius <π/2 . It follows from (13) that With 2e w cosh w = e 2w + 1, (14) can be written as Let f ðwÞ = 1/cosh w . The singularities of f ðwÞ are the zeros of cosh w, which are w j = ð2j + 1Þðπ/2Þi, j ∈ ℤ. Each of these singularities is a simple pole of f ðwÞ while 0 is a pole of order n + 1 of the integrand of (15).
Take z ⟼ nz and let nz ⟼ ∞ by letting n ⟶ ∞ with z fixed. Then The main contribution of the integrand to the integral above originates at the saddle point of the argument of the exponential (see [18]). This saddle point is at w such that Assume that z −1 is not a pole of f ðwÞ. Approximations of T n ðnz + 1Þ can be obtained by expanding f ðwÞ around the saddle point [19]. With z −1 not a pole of f ðwÞ, we can expand f ðwÞ around z −1 . That is, 2 Journal of Applied Mathematics where r is the distance from z −1 to the nearest singularity of f ðwÞ. For w ∈ ℂ, the above series is absolutely convergent if the saddle point z −1 is closer to the origin than to any of the singularities w j : That is, if z −1 is in the strip jIm z −1 j < π/2 and jz −1 j < jz −1 − w j j for all j = 0, ±1,±2, ⋯ . It follows from Lemma 1, Lemma 2 and Theorem 1 of [19] that where Writing the first few terms of (19), we have Figure 1 below depicts the graphs of T n ðnz + 1Þ (in solid lines) generated using relation (6) and the graphs of the approximate values of T n ðnz + 1Þ (in dashed lines) generated using the expansion at the right-hand side of (12). The graphs below are generating using the software Mathematica.
The graphs show the accuracy of approximation (12) for several values of n for real values of the uniform parameter z. For a real argument, the oscillatory region of T n ðnz + 1Þ is also contained in jxj ≤ 2π −1 , whereas the monotonic region contains jxj > 2π −1 . Therefore, the accuracy of approximation (12) is restricted to the monotonic region.
The next theorem contains the approximation formula for Tangent-Bernoulli polynomials.

Theorem 2.
For z ∈ ℂ \ f0g such that jI m z −1 j < π and jz ± ð1/πÞj > 1/π and n ≥ 1, Proof. Applying the Cauchy-Integral Formula to (2) where C is a circle about 0 with radius <π. It follows from (24) that With 2e w sinh w = e 2w − 1, (25) can be written as Let f ðwÞ = w/2 sinh w . The singularities of f ðwÞ are the zeros of sinh w, which are. w j = jπ, j ∈ ℤ. Each of these singularities is a simple pole of f ðwÞ while 0 is a pole of order n + 1 of the integrand of (26).
Take z ⟼ nz and let nz ⟼ ∞ by letting n ⟶ ∞ with z fixed. Then in view of Theorem 1, we can write 3 Journal of Applied Mathematics It follows from Lemma 1, Lemma 2 and Theorem 1 of [19] that where p k ðnÞ are given in (20). Writing the first few terms of (28), we have The following figure depicts the graphs of ðTBÞ n ðnz + 1Þ (in solid lines) generated using relation (8) and the graphs of the approximate values of ðTBÞ n ðnz + 1Þ (in dashed lines) generated using the expansion at the right-hand side of (23). The graphs are generated using Mathematica.
The graphs show the accuracy of approximation (23) for several values of n for real values of the uniformity parameter z. For a real argument, the oscillatory region of ðTBÞ n ðnz + 1Þ is also contained in jxj ≤ π −1 , whereas the monotonic region contains jxj > π −1 . Therefore, the accuracy of approximation (23) is restricted to the monotonic region.
The following theorem contains the approximation formula for Tangent-Genocchi polynomials This theorem is proved similarly as the first two theorems so the proof is omitted.

Theorem 3.
For z ∈ ℂ \ f0g such that jI m z −1 j < π/2 and jz ± ð2i/πÞj > 2/π and n ≥ 1, Figure 2 below depicts the graphs of ðTGÞ n ðnz + 1Þ (in solid lines) generated using relation (9) and the graphs of the approximate values of ðTGÞ n ðnz + 1Þ (in dashed lines) generated using the expansion at the right-hand side of (30).

Expansion of Tangent, Tangent-Bernoulli, and Tangent-Genocchi Polynomials with Enlarged Region of Validity
The validity of the approximations obtained in Theorems 1, 2, and 3, are restricted to the region jz −1 j < jz −1 − w j j for all j = 1, 2, ⋯ . However, the region jz −1 j < jz −1 − w j j may be enlarged by isolating the contribution of the poles w j 's of f ðwÞ. We will follow similar procedure done in [16,19] to prove our next three results. A detailed discussion can be seen in Lemma 3.2 in [16] that allows us to write the polynomial P n ðnzÞ defined by P n nz ð Þ = n! 2πi with a meromorphic function f ðwÞ analytic in the origin  r k e w k nz w n+1 which is valid for all complex number z satisfying jz −1 j < jz −1 − w j j for all j = m + 1, m + 2, ⋯, such that C is a circle whose center is at the origin and contains no poles of f ðwÞ inside, Γðn + 1, w k nzÞ is the incomplete gamma function, p k ðnÞ are polynomials defined by the relation (20) and h ðkÞ m ðz −1 Þ is the kth derivative at z −1 of the function The following theorem contains the asymptotic expansion of Tangent polynomials with enlarged region of validity.

Theorem 4.
For z ∈ ℂ \ f0g such that jz −1 j < jz −1 ± ð2k + 1Þ ðπ/2Þij for k = 0, 1, 2, ⋯, m − 1. Then, as n ⟶ ∞, Proof. To obtain the corresponding residues, we first let Then We then evaluate some derivatives of the function h m ðwÞ defined in (33) at the saddle point z −1 . Then for k = 0, 1, 2 ⋯ :, m − 1, we write Using (37) we can compute some derivatives of h m ðwÞ and evaluate it at the saddle point z −1 . We obtain the results below.
The graphs show the accuracy of approximation (34) for several values of n for real values of the uniformity parameter z. For a real argument, the approximation of T n ðnz + 1Þ with enlarged validity in the monotonic region is better than the approximation in (12). It can also be observed that, in the oscillatory region, even if the accuracy is not that good the approximation is better compared to the approximation in (12). Now, let us consider the asymptotic expansion of Tangent-Bernoulli polynomials with enlarged region of validity.
The figure below depicts the graphs of ðTBÞ n ðnz + 1Þ (in solid lines) generated using relation (8) and the graphs of the approximate values of ðTBÞ n ðnz + 1Þ (in dashed lines) generated using the expansion at the right-hand side of (43). Using Mathematica, we obtain the graph below.
The graphs show the accuracy of approximation (43) for several values of n for real values of the uniform parameter z. For a real argument the approximation of ðTBÞ n ðnz + 1Þ with enlarged validity in the monotonic region is better than the approximation in (23). It can also be observed that in the oscillatory region the accuracy is better compare to the approximation in (23).
The graphs show the accuracy of approximation (58) for several values of n for real values of the uniform parameter z. In both monotonic and oscillatory regions, the approximation in (58) shows better accuracy than in (30).

Conclusion
Two sets of approximation formulas for Tangent polynomials, Tangent-Bernoulli polynomials and Tangent-Genocchi polynomials are obtained using the saddle point method and the integral representation of these polynomials in terms of hyperbolic functions. The last set of formulas which have enlarged validity give more accurate approximations compared to the first set of formulas as shown in Figures 1-6. It is interesting to establish the asymptotic approximation of other variations of Tangent polynomials like Apostol-Tangent polynomials, Apostol-Tangent-Bernoulli polynomials and Apostol-Tangent-Genocchi polynomials as well as the higherorder versions of these polynomials.

Data Availability
The articles used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The author's declare that they have no conflicts of interest.