Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions

Abstract

The study of special functions has become an enthralling area in mathematics because of its properties and wide range of applications that are relevant into other fields of knowledge. Developing topics in special functions involves the investigation of Apostol-type polynomials encompassing the combinations, extensions, and generalizations of some classical polynomials such as Bernoulli, Euler, Genocchi, and tangent polynomials. One particular type of these polynomials is the Apostol-Frobenius-Euler polynomials of order aa denoted by Hnα(z;u;λ){H}_{n}^{\alpha }\left(z;\hspace{0.33em}u;\hspace{0.33em}\lambda ). Using the saddle point method, Corcino et al. obtained approximations for the higher-order tangent polynomials. They also established a new method to derive its approximations with enlarged region of validity. In this article, it is found that these methods are applicable to the higher-order Apostol-Frobenius-Euler polynomials. Consequently, approximations of higher-order Apostol-Frobenius-Euler polynomials in terms of the hyperbolic functions are obtained for large values of the parameter nn, and its uniform approximations with enlarged region of validity are also derived. Moreover, approximations of the generalized Apostol-type Frobenius-Euler polynomials of order α\alpha with parameters a,b,a,b, and cc are obtained by applying the same methods. Graphs are provided to show the accuracy of the exact values of these polynomials and their corresponding approximations for some specific values of the parameters.

Keywords

• apostol-frobenius-euler polynomials
• generalized apostol-type froebnius-euler polynomials
• euler polynomials
• hermite polynomials
• asymptotic approximation
• 11b68
• 11b83
• 41a60