Asymptotic Approximations of Apostol-Tangent Polynomials in Terms of Hyperbolic Functions

Cristina B Corcino,Wilson D Casta馿Da Jr,Roberto B Corcino

doi:10.32604/cmes.2022.019965

Cristina B Corcino, Wilson D Casta馿Da Jr + Show 1 more

Open Access

https://doi.org/10.32604/cmes.2022.019965

Copy DOI

Abstract

The tangent polynomials Tn (z) are generalization of tangent numbers or the Euler zigzag numbers Tn. In particular, Tn (0) = Tn. These polynomials are closely related to Bernoulli, Euler and Genocchi polynomials. One of the extensions and analogues of special polynomials that attract the attention of several mathematicians is the Apostoltype polynomials. One of these Apostol-type polynomials is the Apostol-tangent polynomials Tn(z, λ). When λ = 1, Tn (z, 1) = Tn(z). The use of hyperbolic functions to derive asymptotic approximations of polynomials together with saddle point method was applied to the Bernoulli and Euler polynomials by Lopez and Temme. The same method was applied to the Genocchi polynomials by Corcino et al. The essential steps in applying the method are (1) to obtain the integral representation of the polynomials under study using their exponential generating functions and the Cauchy integral formula, and (2) to apply the saddle point method. It is found out that the method is applicable to Apostol-tangent polynomials. As a result, asymptotic approximation of Apostol-tangent polynomials in terms of hyperbolic functions are derived for large values of the parameter n and uniform approximation with enlarged region of validity are also obtained. Moreover, higher-order Apostol-tangent polynomials are introduced. Using the same method, asymptotic approximation of higher-order Apostol-tangent polynomials in terms of hyperbolic functions are derived and uniform approximation with enlarged region of validity are also obtained. It is important to note that the consideration of Apostol-type polynomials and higher order Apostoltype polynomials were not done by Lopez and Temme. This part is first done in this paper. The accuracy of the approximations are illustrated by plotting the graphs of the exact values of the Apostol-tangent and higher-order Apostol-tangent polynomials and their corresponding approximate values for specific values of the parameters n, λ and m.

Highlights

The Apostol-tangent polynomials denoted by Tn (z; λ), λ = 0 are defined by generating function
The asymptotic approximations of the Apostol-tangent polynomials Tn (z; λ) for large n which are uniformly valid in some unbounded region of the complex variable z, are derived using saddle point method as used in [9] and [10]
Corresponding asymptotic formulas of the tangent polynomials are given as corollaries

Summary

Introduction

The Apostol-tangent polynomials denoted by Tn (z; λ) , λ = 0 are defined by generating function (see [1]). When λ = 1, the equation gives the generating function for the classical tangent polynomials Tn (z) given by (see [2,3]). The Apostol-tangent polynomials are extensions of the classical tangent polynomials. Asymptotic approximations of Apostol-tangent polynomials parallel to the results obtained in [9] and [10], are not mentioned and found in those studies and other related literature. The asymptotic approximations of the Apostol-tangent polynomials Tn (z; λ) for large n which are uniformly valid in some unbounded region of the complex variable z, are derived using saddle point method as used in [9] and [10]. Corresponding asymptotic formulas of the tangent polynomials are given as corollaries

Asymptotic Expansions of Apostol-Tangent Polynomials

Approximation of Higher-Order Apostol-Tangent Polynomials

Conclusion