TY - EJOU
AU - Corcino, Cristina B.
AU - Jr., Wilson D. Castañeda
AU - Corcino, Roberto B.
TI - Asymptotic Approximations of Apostol-Tangent Polynomials in Terms of Hyperbolic Functions
T2 - Computer Modeling in Engineering \& Sciences
PY - 2022
VL - 132
IS - 1
SN - 1526-1506
AB - 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*.
KW - Apostol-tangent polynomials; tangent polynomials; Genocchi polynomials; Hermite polynomials; asymptotic approximation
DO - 10.32604/cmes.2022.019965