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

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.

Asymptotic Approximations of Higher-Order Apostol–Frobenius–Genocchi Polynomials with Enlarged Region of Validity

“…In this section, approximation with an enlarged region of validity is derived by isolating the contribution of the poles. Motivated by the study of Corcino et al [10], the approximation uses the method of contour integration, which introduces the incomplete gamma function in the formula. The following theorem describes the said approximation.…”

“…Corcino et al produced related studies on asymptotic approximations of some special polynomials in terms of hyperbolic functions (see [10][11][12]). It is observed that there is a resemblance in the generating function of the Apostol-tangent polynomials in [10] and the Apostol-Frobenius-Euler polynomials.…”

“…Corcino et al produced related studies on asymptotic approximations of some special polynomials in terms of hyperbolic functions (see [10][11][12]). It is observed that there is a resemblance in the generating function of the Apostol-tangent polynomials in [10] and the Apostol-Frobenius-Euler polynomials. However, the approximation of the Apostol-Frobenius-Genocchi polynomials parallel to the results of Corcino et al remains to be unexplored in other related studies.…”

“…In this study, the uniform approximation of the Apostol-Frobenius-Genocchi polynomials of order α for large n valid in some unbounded region of the complex variable z are derived using the saddle-point method. Using the technique of the contour integration in [10], an approximation with an enlarged region of validity is also obtained. Moreover, the same methods are applied to obtain the approximations for the case of the generalized Apostol-type Frobenius-Genocchi polynomials of order α with parameters a, b, and c denoted by G…”

Fourier Series for Bernoulli-Type Polynomials, Euler-Type Polynomials and Genocchi-Type Polynomials of Integer Order

“…The Apostol-type of these polynomials were mentioned in [9] in the introduction of the paper. Fourier series for the tangent type of these polynomials were obtained in [7] while the Fourier series for the Apostol-Tangent polynomials were obtained in [6]. Integral representation and explicit formula at rational arguments of tangent polynomials of higher order were derived in [8].…”