Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2022.019965

ARTICLE

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

1Research Institute for Computational Mathematics and Physics, Cebu Normal University, Cebu City, 6000, Philippines

2Mathematics Department, Cebu Normal University, Cebu City, 6000, Philippines

3Department of Mathematics, Cebu Technological University, Cebu City, 6000, Philippines

*Corresponding Author: Roberto B. Corcino. Email: rcorcino@yahoo.com

Received: 27 October 2021; Accepted: 30 December 2021

Abstract: The tangent polynomials

Keywords: Apostol-tangent polynomials; tangent polynomials; Genocchi polynomials; Hermite polynomials; asymptotic approximation

The Apostol-tangent polynomials denoted by

where

when

Setting

where

First few values of the Apostol-tangent polynomials are given below:

The Apostol-tangent polynomials are extensions of the classical tangent polynomials. The latter have become an interesting area for many mathematicians for their extensions and analogues possess properties that are relevant in analytic number theory and physics (see [5–8]). In [1], the 2-variable q generalized tangent-Apostol type polynomials were introduced and investigated as a new class of q-hybrid special polynomials.

Asymptotic approximations for Bernoulli polynomials

where the contour C encircles the origin in the counterclockwise direction and contains no poles of

In this study, the asymptotic approximations of the Apostol-tangent polynomials

2 Asymptotic Expansions of Apostol-Tangent Polynomials

Theorem 2.1.

where

Proof. Applying the Cauchy Integral Formula [11] to Eq. (1), we have

where C is a circle about the origin with radius

where

Now take

The main contribution of the integrand above to the integral occurs at the saddle point of the argument of the exponential [12]. This saddle point is at the point

where

where

Computing the derivatives

Expanding the sum in Eq. (10) and keeping only the first three terms yield

The accuracy of the asymptotic formula obtained in Eq. (5) is shown in Fig. 1.

To enlarge the region of validity of Eq. (5) and obtain an asymptotic expansion valid in a larger region, the following theorem will be utilized.

Theorem 2.2. [9] The polynomials

where

that is valid for

where the residues

The second asymptotic formula for

Theorem 2.3. Let

Proof. We start by computing the residues

where

On the other hand, for

Thus, the residues

Next, the derivatives of

Computing the derivatives yields

At the saddle point

From Theorem

Keeping only the first three terms of the infinite sum in (28) and using

The accuracy of the asymptotic formula obtained in Eq. (18) is shown in Fig. 2. The accuracy of the approximation in the oscillatory region is better that that the of the formula in Eq. (5).

Remark 2.4. Taking

3 Approximation of Higher-Order Apostol-Tangent Polynomials

Higher-order Apostol-tangent polynomials are defined by the generating function

In this section, it is shown that the method in Section 2 can be extended to obtain asymptotic expansion of the Apostol-tangent polynomials of order

Theorem 3.1 For

when

Proof. Applying the Cauchy Integral Formula to Eq. (29),

where

where

Likewise, the approximations of

where

Expanding the sum in (34) and keeping only the first three terms give

The accuracy of the asymptotic formula obtained in Eq. (30) is shown in Fig. 3.

Corollary 3.2. For

Proof. This follows from Theorem 3.1 by taking

Theorem 3.3. For

where the polynomisals

Are the given principal parts of the Laurent series corresponding to the poles

Proof. With

where

Recall from Eq. (33),

where

Let

Repeating the process to prove Theorem

Assume that

where

where

It follows from Lemma 1 [9] that

where

valid for

This gives

where

The expansion of

valid for

On the other hand, to obtain an expansion for

where

giving

Similarly,

so that

It follows that

Then Eq. (51) becomes

First, we compute

Note that when

This gives

It can be computed that

where

Thus, Eq. (53) can be written

while

Note also that

Now the incomplete gamma function

gives

Let

Consequently,

or

Take note

Substituting Eqs. (55) and (56) to Eq. (52) yields

Using Eq. (57) into Eq. (58) we have

Since

we can write Eq. (59) as follows:

Substituting Eqs. (52) and (61) into Eq. (44) we have

The comparison of the accuracy of the asymptotic formula obtained in Eq. (30) and Eq. (39) is shown in Fig. 4.

valid for

Note that if

since

Corollary 3.4. For

where

are the given principal parts of the Laurent series corresponding to the poles

Proof. This follows from Theorem 3.3 by taking

The saddle-point method and the use of hyperbolic functions are shown to give good approximations to the Apostol-tangent polynomials. Uniform approximations of the Apostol-tangent polynomials and of higher-order Apostol-tangent polynomials are derived. Moreover, approximation formulas with larger region of validity are obtained. The computations to derive the approximation formulas with larger region of validity for the case of Apostol-tangent polynomials of order m are quite tedious and the formulas obtained are original. Corollaries are being stated to explicitly give the corresponding formulas for the special case

For future research work, one may try to investigate more properties of Apostol-tangent and higher order Apostol-tangent polynomials and establish

Funding Statement: This research is funded by Cebu Normal University through its Research Institute for Computational Mathematics and Physics (RICMP).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

- Yasmin, G., & Muhyi, A. (2020). Certain results of 2-variable q-generalized tangent-Apostol type polynomials.
*Journal of Mathematics and Computer Science*,*22(3)*, 238-251. [Google Scholar] [CrossRef] - Ryoo, C. S. (2013). A note on the tangent numbers and polynomials.
*Advance Studies in Theoretical Physics*,*7(9)*, 447-454. [Google Scholar] [CrossRef] - Ryoo, C. S. (2014). A numerical investigation on the zeros of the tangent polynomials.
*Journal of Applied Mathematics and Informatics*,*32(3–4)*, 315-322. [Google Scholar] [CrossRef] - Ryoo, C. S. (2016). Differential equations associated with tangent numbers.
*Journal of Applied Mathematics and Informatics*,*34(5–6)*, 487-494. [Google Scholar] [CrossRef] - Bildirici, C., Acikgoz, M., & Araci, S. (2014). A note on analogues of tangent polynomials.
*Journal for Algebra and Number Theory Academica*,*4(1)*, 21. [Google Scholar] - Ryoo, C. S. (2013). Generalized tangent numbers and polynomials associated with p-adic integral on Zp.
*Applied Mathematical Sciences*,*7*, 4929-4934. [Google Scholar] [CrossRef] - Ryoo, C. S. (2013). A note on the symmetric properties for the tangent polynomials.
*International Journal of Mathematical Analysis*,*7(52)*, 2575-2581. [Google Scholar] [CrossRef] - Yasmin, G., Ryoo, C. S., & Islahi, H. (2020). A numerical computation of zeros of q-generalized tangent-Appell polynomials.
*Mathematics*,*8(3)*, 383. [Google Scholar] [CrossRef] - López, J. L., & Temme, N. M. (1999). Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions.
*Studies in Applied Mathematics*,*103(3)*, 241-258. [Google Scholar] [CrossRef] - Corcino, C. B., Corcino, R. B., Ontolan, J. M., & Castañeda, W. D. (2019). Approximations of Genocchi polynomials in terms of hyperbolic functions.
*Journal of Mathematical Analysis*,*10(3)*, 76-88. [Google Scholar] - Wong, M. W. (2008). Complex analysis: Series on analysis, applications and computation. Singapore: World Scientific Publishing Co. Pvt, Ltd.
- Wong, R. (1989). Asymptotic approximations of integrals. New York: Academic Press.
- Temme, N. M. (1983). Uniform asymptotic expansions of Laplace integrals.
*Analysis*,*3(1–4)*, 221-250. [Google Scholar] [CrossRef] - Temme, N. M. (1985). Laplace type integrals: transformation to standard form and uniform asymptotic expansions.
*Quarterly of Applied Mathematics*,*43(1)*, 103-123. [Google Scholar] [CrossRef] - Temme, N. M. (1990). Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions.
*SIAM Journal on Mathematical Analysis*,*21(1)*, 241-261. [Google Scholar] [CrossRef] - Temme, N. M. (1995). Uniform asymptotic expansions of integrals: A selection of problems.
*Journal of Computational and Applied Mathematics*,*65(1–3)*, 395-417. [Google Scholar] [CrossRef] - Corcino, C. B., Corcino, R. B., & Ontolan, J. M. (2021). Approximations of tangent polynomials, tangent-bernoulli and tangent-genocchi polynomials in terms of hyperbolic functions.
*Journal of Applied Mathematics*,*2021(2)*, 10. [Google Scholar] [CrossRef] - Korn, G. A., Korn, T. M. (2013). Mathematical handbook for scientists and engineers. New York: Dover Publications, Inc.
- Solomentsev, E. D. (2017). Encyclopedia of mathematics: Mittag-Leffler theorem. http://encyclopediaofmath.org/index.php?title=Mittag-Leffler_theorem&oldid=41565.
- Ryoo, C. S. (2013). On the analogues of Tangent numbers and polynomials associated with p-adic integral on Zp.
*Applied Mathematical Sciences*,*7(64)*, 3177-3183. [Google Scholar] [CrossRef] - Ryoo, C. S. (2013). On the twisted q-tangent numbers and polynomials.
*Applied Mathematical Sciences*,*7(99)*, 4935-4941. [Google Scholar] [CrossRef] - Ryoo, C. S. (2018). Explicit identities for the generalized tangent polynomials.
*Nonlinear Analysis and Differential Equations*,*6(1)*, 43-51. [Google Scholar] [CrossRef]

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