Abstract
This paper provides asymptotic expansions for large values of of tangent and Apostol-tangent polynomials of complex order. The derivation is done using contour integration with the contour avoiding branch cuts.
1. Introduction
The tangent and Apostol-tangent polynomials of complex order and complex argument are defined by the relations where and is taken to be the principal branch.
When and , (1) and (2) reduce to the classical tangent polynomials, respectively (see [1]).
It is worth mentioning that results obtained in [2, 3] may have potential applications in mathematics and physics. More precisely, the numerical values of the zeros of the tangent polynomials may represent important values in engineering and physics while the twisted -analogue of tangent polynomials may be used in quantum physics, particularly in the study of quantum groups and their representation theory.
Ryoo [4] introduced a variation of tangent numbers and polynomials, known as twisted tangent numbers and polynomials, associated with the -adic integral on . Through his work, Ryoo presented intriguing findings and established connections related to these concepts. In addition, Ryoo [5] explored differential equations arising from the generating functions of generalized tangent polynomials and derived explicit identities for them. Furthermore, Ryoo [6] investigated the symmetry property of the deformed fermionic integral on , which is a mathematical structure defined over a prime field. Specifically, he focused on establishing recurrence identities for tangent polynomials and alternating sums of powers of consecutive even integers within this context. These discoveries expand our knowledge and understanding of this specialized area of mathematics, providing insights into its unique properties and potential applications across different domains. Moreover, a study by Corcino et al. [7] obtained the Fourier expansion of tangent polynomials of integer order.
In this paper, the same method described by Lόpez and Temme ([8], p. 4) has been followed in deriving the asymptotic expansion which only gives a first-order approximation. C. Corcino and R. Corcino ([9], p. 2) describe a similar method and provide a first-order and second-order approximations.
2. Asymptotic Expansions
In this section, the asymptotic expansions for large values of of tangent and Apostol-tangent polynomials of complex order are derived.
2.1. Tangent Polynomials of Complex Order
Applying Cauchy’s integral formula for derivatives to (1), we have where is a circle about 0 with radius .
We observe in (3) that the singularities at are the sources for the main asymptotic contributions. We integrate around a circle about 0 with radius avoiding the branch cuts running from to (see Figure 1). Denote the loops by and and the remaining part of the circle by . Then, we have where is the integrand on the right-hand side of (3). By the principle of deformation of paths,
The following lemma gives the contribution from the circular arc .
Lemma 1. The integral along is . That is,
Proof. Taking the modulus of the integral, we have for all . Since does not pass any singularity, is not zero. Thus, for some positive number . So that, This proves the lemma.
Remark 2. Lemma 1 shows that, for large values of (as ), the contribution from the circular arc is exponentially small with respect to the main contributions.
For the contributions from the loops, let and be the integrals along and , respectively. We first compute the integral :
Let . Then, and where is the image of under the transformation . is the contour that encircles the origin in the clockwise direction. Multiplying the last array by and since , where . Multiplying the last array by , where
To obtain an asymptotic expansion, we apply Watson’s lemma for loop integrals (see [10], p. 120). We expand
Substituting (16)–(14), becomes where with extended to . That is, the path of integration starts at with arg , encircles the origin in the clockwise direction, and returns to , now with arg .
Now, we evaluate . First, we turn the path by writing : where is the image of under the transformation . is the contour that starts at with arg , encircles the origin in clockwise direction, and returns to with arg .
We recall Hankel’s loop integral representation for the reciprocal gamma function (see [11, 12], p. 48 and p. 153, respectively): where is the Hankel contour (see Figure 2) that runs from with arg , encircles the origin in positive direction (that is, counterclockwise), and terminates at , now with arg .
Observe that, by deformation of paths, the contour is the Hankel contour traversed in the opposite direction. So that,
Let ; . Then,
Moreover, where , the rising factorial of of degree . Hence,
Applying (24)–(17) and noting that , we get where and .
Now, the integral along the loop , can be obtained similarly. After the substitution , we obtain where
We expand and interchange the summation and integration in (27) and get where are the integrals in (18). Applying (24) and noting that , we obtain where and .
We observe that is just the complex conjugate of (not considering and as complex numbers). So that, if we write (with real when and are real), then . Hence, by Remark 2 and applying (25) and (30), we obtain
Consequently, we have the following theorem.
Theorem 3. As , and are fixed complex numbers. where and .
Compute the first few values of and using Mathematica:
A first-order approximation is obtained by taking and for and , respectively, and taking the first term of the sum. This is given in the following theorem.
Theorem 4. As , and are fixed complex numbers. where and .
A second-order approximation is given as follows.
Theorem 5. As , and are fixed complex numbers. where and .
2.2. Apostol-Tangent Polynomials of Complex Order
We apply the same method as in the previous subsection.
For convenience, we take , where and . Then, (2) reduces to
Applying Cauchy’s integral formula for derivative to (36), we have where is a circle about 0 with radius .
We consider (37) and observe that the singularities at and are the source for the main asymptotic contribution. We integrate around a circle about 0 with radius avoiding the branch cuts running from to and to (see Figure 3). Denote the loops by and and the remaining part of the circle by . Then, we have where is the integrand on the right-hand side of (37). By the principle of deformation of paths,
Remark 6. It follows from Lemma 1 that the contribution from the circular arc is also , so that, for large values of (as ), it is exponentially small with respect to the main contributions.
We proceed to compute the contributions from the loops and . Let be the integral along the loop . Then,
Let . Then, and where is the image of under the transformation . is the contour that encircles the origin in the clockwise direction. Multiplying the last array by and since , where . Multiplying the last array by , where
We expand ; (49) becomes where are the integrals in (18).
Applying (24)–(51) and noting that , we get where and .
Next, let be the integral along loop . Then,
After the substitution , we obtain where
We expand and interchange the summation and integration in (49) and get where are the integrals in (18). Applying (24) to (51) and noting that , we get where and .
Then, by Remark 6 and applying (47) and (52), we obtain
Hence, we have the following theorem.
Theorem 7. As , and are fixed complex numbers. where and .
Remark 8. When , Theorem 7 reduces to Theorem 3.
Compute for the first few values of and using Mathematica:
A first-order approximation is obtained by taking and for and , respectively, and taking the first term of the sum. This is given in the following theorem.
Theorem 9. As , and are fixed complex numbers. where and .
Remark 10. When , Theorem 9 reduces to Theorem 4.
3. Summary
This paper derives asymptotic expansions for the tangent polynomials and Apostol-tangent with complex orders. The primary objective is to approximate these polynomials effectively when takes on large values. To accomplish this, the authors have employed a mathematical technique known as contour integration. This approach entails integrating the polynomials along specific paths in the complex plane, carefully avoiding branch cuts. By utilizing contour integration, the authors have derived expressions that offer valuable approximations for the tangent and Apostol-tangent polynomials as becomes increasingly large. More precisely, as , and are fixed complex numbers; the tangent polynomials have the following asymptotic expansion: where and . Consequently, the first-order approximation is obtained given as follows: and the second-order approximation is given by
On the other hand, the asymptotic expansion for the Apostol-tangent polynomials is given as follows: where and .
These findings contribute to our comprehension of the behaviors exhibited by these polynomials and can prove beneficial in various applications that necessitate approximations for significant values of .
Data Availability
The articles used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the Cebu Normal University (CNU) for funding this research project through its Research Institute for Computational Mathematics and Physics (RICMP).