Asymptotic Approximation of the Apostol-Tangent Polynomials Using Fourier Series

: Asymptotic approximations of the Apostol-tangent numbers and polynomials were estab-lished for non-zero complex values of the parameter λ . Fourier expansion of the Apostol-tangent polynomials was used to obtain the asymptotic approximations. The asymptotic formulas for the cases λ = 1 and λ = − 1 were explicitly considered to obtain asymptotic approximations of the corresponding tangent numbers and polynomials.


Introduction
The tangent polynomials T n (z) of a degree of n with a complex argument z are defined by the generating function (see [1,2]). ∞ ∑ n=0 T n (z) w n n! = 2 e 2w + 1 e zw , |w| < π 2 (1) These polynomials can be expressed in polynomial form as T n (z) = n ∑ k=0 n k T k z n−k where T k denotes the tangent numbers defined by It is worth mentioning that tangent numbers are the odd indices of the numbers A n of alternating permutations known as the Euler zigzag numbers. The first few values of these numbers are as follows: T 0 = 1, T 1 = −1, T 3 = 2, T 5 = −16, T 7 = 272, T 9 = −7936, T 11 = 353792.

Asymptotic Approximations
: k ∈ Z be the set of poles of the generating function Equation (6). The Fourier series expansion of the Apostol-tangent polynomials in terms of poles in T λ is given in the following theorem: where the logarithm is taken to be the principal branch.

Proof.
Consider the integral CN f n (z)dz where f n (z) = 2e xz (λe 2z + 1)z n+1 and the circle C N is a circle about the origin of radius 1 2 (2N − 1 + )π , N ∈ Z + with being a fixed real number such that πi ± log λ = 0 (mod πi).
The function f n (z) has poles at z = 0 of order n + 1 and at z k = 1 2 [(2k − 1)πi − log λ], k ∈ Z. The poles z k are simple poles. Using the Cauchy Residue Theorem, We observe that, using the basic property of integration, Thus, As N → ∞, the last expression goes to 0. Hence, as N → ∞, n ≥ 1, This implies that Now, the first residue Res ( f n (z), z = 0) is given as Note that the limit of each term of the expansion is 0 as z → 0 except the term when l = n. This gives On the other hand, the residue Res ( f n (z), z = z k ) is given by , Combining these residues gives, Hence,

Corollary 1.
Let λ ∈ C\{0}. For n ≥ 1, the Fourier series of the Apostol-tangent numbers is given by where the logarithm is taken to be the principal branch.
Proof. This follows from Theorem 1 by taking x = 0.
Proceeding as in [20], ordering of the poles of the generating function Equation (6) is carried out in the following lemma.
, where the logarithm is taken to be the principal branch.
To verify the chains Equations (9)- (12), let x = e γ and y = Im γ. Then for k ∈ Z, Now, we consider two cases: From this, one can see that the order of magnitude of u k , k ∈ Z given in Equation (9) holds.
The chain of values of u k can be derived similarly, in which the order of magnitude of u k , k ∈ Z given in Equation (10) holds. Case 3. Im λ = 0. This means that λ is a real number, which is either positive or negative but not zero. Hence, we have the following subcases: and so on. Hence, which is exactly (11).
from which it can easily be observed that which is exactly the chain in (12).
The asymptotic expansion of the Apostol-tangent numbers T n (0; λ) is given in the next theorem.

M+1
Let Consequently, We can see that C M,λ → 0 as n → ∞ for |M| > 2. Thus, the tail of the series is Moreover, for fixed M > 2 and n 0, C M,λ is bounded and independent of M. Hence, we can replace C M, λ with C λ . This completes the proof of the theorem.
When λ = 1, log λ = 0, and u k = 1 2 (2k − 1)πi, k ∈ Z. Take H = πi 2 , −πi 2 . Then v = 3π 2 , and the ordinary tangent numbers T n = T n (0; 1) satisfy T n n! = T n (0;1) An approximation of T n (0; 1) is given by For even n, n ≥ 2, it is known that T n = 0, which is also true when we use Equation (14). Then, we have T 2n (2n!) For odd indices, This value is very close to the exact value of T 5 which is −16. It is proved in the next theorem that an asymptotic approximation of the Apostoltangent polynomials can be obtained from its Fourier series (Theorem 1) by choosing an appropriate subset of T λ . Theorem 3. Given λ ∈ C\{0}, let H be a finite subset of T λ satisfying max{|u| : u ∈ H} < min{|u| : u ∈ T λ \H} := v.
For all integers n ≥ 2, we have uniformly for x in a compact subset K of C, where the constant implicit in the order term depends on λ, H and K. Moreover, for n 0, this constant can be made independent of K, equal to the constant for the Apostol-tangent numbers, corresponding to the case x = 0. (6) Hence,

Proof. From the generating function in Equation
For z ∈ C, writing z = 0 + z ( here y = z, x = 0), where the implicit constant c in the order term is that corresponding to z = 0 and only depends on H and λ. Note also that To complete the proof of the theorem, it remains to show that is bounded. Using the Mean Value Theorem (MVT) for Banach spaces (see also [20] where e + (w) = max{ e(w), 0}. Since |u| ≤ v, for all u ∈ H, we have where |H| denotes the number of elements in H. We give the argument that which certainly holds for n 0, uniformly for z in a compact subset K ⊂ C.

Corollary 2.
Let K be an arbitrary compact subset of C. The tangent polynomials satisfy uniformly on K the estimates where the implicit constant in the order term depends on the set K. Moreover, for n 0, this constant can be made independent of K, equal to the constant for the tangent numbers, corresponding to the case x = 0.
Proof. The tangent polynomials correspond to the case λ = 1 so that For even indices, For odd indices,

The Case When λ Is a Negative Real Number
When λ is a negative real number, writing λ = −|λ|, the generating function in Equation (6) can be written as The poles of the generating function (3.1) is given by The next theorem immediately follows from Theorem 3.
For all integers n ≥ 2, we have uniformly for x in a compact subset K o f C, where the constant implicit in the order term depends on λ , F, and K.
The Apostol-tangent numbers T n (0; −1) corresponding to the case λ = −1 have the generating function The set of poles is T −1 = {kπi : k ∈ Z\{0}}. An asymptotic formula for T n (0; −1) is given in the following theorem.

Conclusions and Recommendation
The method of Navas et al. [24] is a clever way to obtain an asymptotic approximation from the Fourier series. In this paper, the method was applied to obtain asymptotic approximations of the Apostol-tangent numbers and polynomials for nonzero complex values of the parameter λ. The case when λ is negative was explicitly considered because the poles are simply in terms of 1 2 ln|λ| plus odd multiples of π 2 i. Moreover, the cases λ = 1 and λ = −1 give beautiful approximations of the corresponding Tangent polynomials in terms of the sine and cosine functions depending on whether n is even or odd.
The author recommends finding Fourier expansion and asymptotic approximations of higher-order Apostol-Tangent numbers and polynomials using the method employed in this paper (see also [25]). Furthermore, one may also try to consider multiple generalized Tangent polynomials and their p-adic interpolation function [26].

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