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ARTICLE

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

by Cristina B. Corcino^1,2, Wilson D. Castañeda³, Roberto B. Corcino^1,2,*

1 Research Institute for Computational Mathematics and Physics, Cebu Normal University, Cebu City, 6000, Philippines
2 Mathematics Department, Cebu Normal University, Cebu City, 6000, Philippines
3 Department of Mathematics, Cebu Technological University, Cebu City, 6000, Philippines

* Corresponding Author: Roberto B. Corcino. Email:

(This article belongs to the Special Issue: Trend Topics in Special Functions and Polynomials: Theory, Methods, Applications and Modeling)

Computer Modeling in Engineering & Sciences 2022, 132(1), 133-151. https://doi.org/10.32604/cmes.2022.019965

Received 27 October 2021; Accepted 30 December 2021; Issue published 02 June 2022

Abstract

The tangent polynomials T_n (z) are generalization of tangent numbers or the Euler zigzag numbers T_n. In particular, T_n (0) = T_n. 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 T_n(z, λ). When λ = 1, T_n (z, 1) = T_n(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.

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