Approximations of Tangent Polynomials, Tangent –Bernoulli and Tangent – Genocchi Polynomials in terms of Hyperbolic Functions

Corcino, Cristina B.; Corcino, Roberto B.; Ontolan, Jay M.

doi:10.1155/2021/8244000

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…Remark 2.4. Taking λ = 1, Theorem 2.1 and Theorem 2.3, respectively, will give uniform approximationformula and an asypmtotic expansion with enlarged region of validity which are same formulas as those obtained in [17] for the tangent polynomials.…”

Section: Asymptotic Expansions Of Apostol-tangent Polynomialsmentioning

confidence: 63%

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

Corcino¹,

Casta馿da²,

Corcino³

2022

Computer Modeling in Engineering &Amp; Sciences

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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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Section: Asymptotic Expansions Of Apostol-tangent Polynomialsmentioning

confidence: 63%

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

Corcino¹,

Casta馿da²,

Corcino³

2022

Computer Modeling in Engineering &Amp; Sciences

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“…This validity can be enlarged by isolating the contribution of the poles. This method was done in [4], [5]. The authors recommend to obtain approximation formulas with enlarged region of validity for the polynomials studied here.…”

Section: Comparing Coefficients Yieldsmentioning

confidence: 99%

“…Asymptotic approximations for higher order Genocchi polynomials using residues were done in [2] and [3]. Approximations for the Bernoulli and Euler polynomials using hyperbolic functions were obtained in [4] and approximations for Genocchi polynomials in terms of hyperbolic functions were obtained in [5]. At the time of the search, there were no approximations for the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials found in the literature.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic approximations for Generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials in terms of Hyperbolic Functions

Corcino

2023

Eur. J. Pure Appl. Math.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

2023

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In this paper, the uniform approximations of the Apostol–Frobenius–Genocchi polynomials of order α in terms of the hyperbolic functions are derived through the saddle-point method. Moreover, motivated by the works of Corcino et al., an approximation with enlarged region of validity for these polynomials is also obtained. It is found out that the methods are also applicable for the case of the higher order generalized Apostol-type Frobenius–Genocchi polynomials and Apostol–Frobenius-type poly-Genocchi polynomials with parameters a, b, and c. These methods demonstrate the techniques of computing the symmetries of the defining equation of these polynomials. Graphs are illustrated to show the accuracy of the exact values and corresponding approximations of these polynomials with respect to some specific values of its parameters.

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Approximations of Tangent Polynomials, Tangent –Bernoulli and Tangent – Genocchi Polynomials in terms of Hyperbolic Functions

Cited by 4 publications

References 13 publications

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

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

Asymptotic approximations for Generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials in terms of Hyperbolic Functions

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

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