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Computer Modeling in Engineering & Sciences
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DOI: 10.32604/cmes.2022.019965

ARTICLE

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

Cristina B. Corcino1,2, CastañedaWilson D. Jr.3 and Roberto B. Corcino1,2,*

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 Tn(z) are generalization of tangent numbers or the Euler zigzag numbers Tn. In particular, Tn(0)=Tn. 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 Apostol-type polynomials. One of these Apostol-type polynomials is the Apostol-tangent polynomials Tn(z,λ). When λ=1, Tn(z,1)=Tn(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 Apostol-type 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.

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

1  Introduction

The Apostol-tangent polynomials denoted by Tn(z;λ),λ≠0 are defined by generating function (see [1])

2ezwλe2w+1=∑n=0∞Tn(z;λ)wnn!, (1)

where λϵC and the validity of the series in Eq. (1) is given as follows:

|w|<{π2whenλ=1πwhenλ≠1.

when λ=1, the equation gives the generating function for the classical tangent polynomials Tn(z) given by (see [2,3])

2ezwe2w+1=∑n=0∞Tn(z)wnn!,|w|<π2. (2)

Setting z=0 in Eqs. (1) and (2), we obtain

Tn(0,λ):=Tn(λ) and Tn(0):=Tn, (3)

where Tn(λ) and Tn are called the Apostol-tangent numbers and classical tangent numbers, respectively (see [1,4]).

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

T0(z;λ)=21+λ,T1(z;λ)=2[z+(−2+z)λ](1+λ)2,T2(z;λ)=2[−4λ+(z+(−2+z)λ)2](1+λ)3,T3(z;λ)=2(1+λ)4[−6z2λ(1+λ)2]+z3(1+λ)3−8λ(−4+λ)λ)+12λ(−1+λ2)],T4(z;λ)=2(1+λ)5[24z2(−1+λ)λ(1+λ)2−8z3λ(1+λ)3+z4(1+λ)4+16(−1+λ)λ(1+(−10+λ)λ)−32λ(1+λ)(1+(−4+λ)λ)].

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 Bn(nz+12) and Euler polynomials En(nz+12) in terms of hyperbolic functions are established in [9]. In the study of Corcino et al. [10], the Genichi polynomials are expressed as

Gn(z+12)=n!2πi∫Cwewzcosh⁡(w/2)dwwn+1 (4)

where the contour C encircles the origin in the counterclockwise direction and contains no poles of 1/cosh⁡(w/2). With this, they have derived the asymptotic formulas for Gn(z+12) in terms of hyperbolic functions. However, asymptotic approximations of Apostol-tangent polynomials parallel to the results obtained in [9] and [10], are not mentioned and found in those studies and other related literature.

In this study, the asymptotic approximations of the Apostol-tangent polynomials Tn(z;λ) for large n which are uniformly valid in some unbounded region of the complex variable z, are derived using saddle point method as used in [9] and [10]. Moreover, asymptotic expansion of higher-order Apostol-tangent polynomials Tnm(z;λ) is obtained. Corresponding asymptotic formulas of the tangent polynomials are given as corollaries.

2  Asymptotic Expansions of Apostol-Tangent Polynomials

Theorem 2.1. For λϵC−{0}, and zϵC such that |Imz−1|<π−Argλ2 and |z−1|<|z−1−(πi2−δ)| and n≥1,

Tn(nz+1;λ)=nnznsech (z−1+δ)λ{1−1−2sech2(z−1+δ)2nz2+O(1n2)}, (5)

where δ=(log⁡λ)/2 and the logarithim is taken to be the principal branch.

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

Tn(z;λ)=n!2πi∫C2ewze2δ+2w+1dwwn+1, (6)

where C is a circle about the origin with radius <|πi2−δ|. With 2e(w+δ)cosh⁡(w+δ=e(2w+2δ))+1, it follows from Eq. (6) that

Tn(z+1;λ)=n!2πiλ∫Cf(w)ezwwwn+1, (7)

where λ=e(log⁡λ)/2=eδ and f(w)=1/cosh⁡(w+δ). The function f(w) is meromorphic function with simple poles at the zeros of cosh⁡(w+δ) which are given by wj=(2j+1)πi2−δ,j=0,±1,±2,…

Now take z⟼nz and let nz⟼∞ with z fixed. It follows from Eq. (7) that

Tn(nz+1;λ)=n!2πiλ∫Cf(w)en(zw−log⁡w)dww. (8)

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 w=1/z=z−1,z≠0. Assume that z−1 is not a pole of f(w). Then approximations of Tn(nz+1;λ) can be obtained by expanding f(w) around the saddle point [13–16]. Let

f(w)=∑k=0∞f(k)(z−1)k!(w−z−1)k,|w−z−1|<r (9)

where r is the distance from z−1 to the nearest singularity of f(w). For w is the circle C, the above series is absolutely convergent if the saddle point z−1 is closer to the origin than to any of the singularities wj. That is, if z−1 is in the strip |Imz−1|<π−Argλ2 and |z−1|<|z−1−wj| for j=0,±1,±2,... It follows from Lemma 1, Lemma 2 and Theorem 1 of [16] that

Tn(nz+1;λ)=(nz)nλ∑k=0∞f(k)(z−1)k!pk(n)(nz)k, (10)

where

p0(n)=1,p1(n)=0,p2(n)=−n,p3(n)=2n (11)

pk(n)=(1−k)pk−1(n)+npk−2(n),k≥3. (12)

Computing the derivatives f(k)(z−1) for k=0,1,2 give

f(z−1)=sech(z−1+δ), (13)

f(1)(z−1)=−tanh⁡(z−1+δ)sech(z−1+δ), (14)

f(2)(z−1)=sech(z−1+δ)(1−2sech2(z−1+δ)). (15)

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

Tn(nz+1;λ)=(nz)nλ[u(z−1)0!+u(1)(z−1)1!p1(n)nz+u(2)(z−1)2!p2(n)(nz)2+O(1n2)]=nnzzλ{sech(1z+δ)−sech(1z+δ)(1−2sech2(1z+δ))2nz2+O(1n2)}=nnzn(sech(1z+δ))λ{1−1−2sech2(1z+δ)2nz2+O(1n2)}.

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

Pn(z)=n!2πi∫Cf(w)ewzdwwn+1, (16)

where f(w) is analytic at the origin with simple poles w1,w2,⋯ (and respective residues r1,r2,⋯), can be represented, for each integer m>0, as

Pn(nz)=−∑k=1mrkewknzwkn+1Γ(n+1,wknz)+(nz)n∑k=0∞f(k)(z−1)+hm(k)(z−1)k!pk(n)(nz)k, (17)

that is valid for z∈C,|z−1|<|z−1−wj|, for all j=m+1,m+2,⋯, where the polynomials pk(n) are given in Eq. (12) and hm(k) is the kth derivative of the function

hm(w)=−∑l=1mrlw−wl,

where the residues wl are ordered by increasing modulus |wl|≤|wl+1|. Each term of the finite sum in the above equation equals n!rk/wkn+1 multiplied by the Taylor polynomial of degree n in z=0 of ewknz.

The second asymptotic formula for Tn(nz+1;λ) with enlarged region of validity is given in the following theorem.

Figure 1: Solid lines represent Tn(nx+1;λ) for several values of n, whereas dashed lines represent the right-hand side of (5) with z=x, both normalized by the factor (1+|xα|n)−1 where we choose α=0.2 (a) n=7 and λ=4 (b) n=14 and λ=9

Theorem 2.3. Let z∈C\{0} such that |z−2±(2k+1)πi−2δ|>2(2k+1)π+2δ for k=0,1,2,⋯,m−1 and λ∈C{0}. Then, as n→∞,

Tn(nz+1;λ)=−2n+1i(λ)nz+1∑k=0m−1(−1)k[−e(2k+1)πinz2[(2k+1)πi−2δ]n+1Γ(n+1,[(2k+1)πi2−δ]nz)+e−(2k+1)πinz2[−(2k+1)πi−2δ]n+1Γ(n+1,[−(2k+1)πi2−δ]nz)]+(nz)nλ{sech(1z+δ)+∑k=0m−1(−1)k+14(2k+1)π4(1z+δ)2+(2k+1)2π2−sech(1z+δ)(1−2sech2(1z+δ))2nz2−∑k=0m−1(−1)k16(2k+1)π[(2k+1)2π2−12(1z+δ)2]nz2{4(1z+δ)2+(2k+1)2π2}3+O(1n2)}. (18)

Proof. We start by computing the residues rl for the Apostol-tangent polynomials. Observe that the case is the function f(w)=sec⁡h(w+δ)=1cos⁡h(w+δ)=p(w)q(w) which has simple poles at wl=±(2l+1)πi2−δ,l=0,1,2,…,m−1. Thus, the corresponding residues are

rl=p(wl)q′(wl)=1sin⁡h(wl+δ), (19)

where

sin⁡h(wl+δ)=sin⁡h((l+12)πi)=isin⁡h(lπ+π2)=(−1)li. (20)

On the other hand, for w−l=−(2l+1)πi2−δ,l=0,1,2,…,m−1,

sin⁡h(w−l+δ)=−sin⁡h((l+12)πi)=−isin⁡h(lπ+π2)=(−1)l+1. (21)

Thus, the residues rl,l=0,1,2,…m−1 of the function f(w) are

rl=1(−1)liand rl=1(−1)l+1i=(−1)l. (22)

Next, the derivatives of hm(w) at the saddle point z−1 will be computed. With the simple poles wl=(2l+1)πi2−δ and w−1=−(2l+1)πi2−δ of the function f(w), an expression for hm(w) is obtained as follows:

hm(w)=−∑l=0m−1rlw−wl−∑l=0m−1rlw−w−l=−∑l=0m−1(−1)l+1iw−[(2l+1)πi2−δ]−∑l=0m−1(−1)liw−[−(2l+1)πi2−δ]=−∑l=0m−1(−1)li(−1[w+δ]−(2l+1)πi2)+1[w+δ]+(2l+1)πi2))=−∑l=0m−1(−1)li(−(2l+1)πi[w+δ]2+(2l+1)2π24)=∑l=0m−1(−1)l+14(2l+1)π4(w+δ)2+(2l+1)2π2.

Computing the derivatives yields

hm(1)(w)=∑l=0m−1(−1)l32(2l+1)π[w+δ]{4(w+δ)2+(2l+1)2π2}2, (23)

hm(2)(w)=∑l=0m−1(−1)l32(2l+1)π[(2l+1)2π2−12(w+δ)2]{4(w+δ)2+(2l+1)2π2}3. (24)

At the saddle point z−1,

hm(0)(z−1)=∑l=0m−1(−1)l+14(2l+1)π4(1z+δ)2+(2l+1)2π2, (25)

hm(1)(z−1)=∑l=0m−1(−1)l32(2l+1)π(1z+δ){4(1z+δ)2+(2l+1)2π2}2, (26)

hm(2)(z−1)=∑l=0m−1(−1)l32(2l+1)π((2l+1)2π2−12(1z+δ)2){4(1z+δ)2+(2l+1)2π2}3. (27)

From Theorem 2.2,

Tn(nz+1;λ)=−(λ)−12∑rkewknzwkn+1Γ(n+1,wknz)+(nz)nλ∑k=0∞f(k)(z−1)+hm(k)(z−1)k!Pk(n)(nz)k. (28)

Keeping only the first three terms of the infinite sum in (28) and using Pk(n) in Eq. (11), f(k)(z−1) given in Eqs. (13)–(15) and hm(k)(z−1) given Eqs. (25)–(27) with wk=±(2k+1)πi2−δ,rk=(−1)k+1iλ and (−1)kiλ,k=0,1,…,m−1, the desired asymptotic formula is obtained.

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).

Figure 2: Solid lines represent Tn(nx+1;λ) for several values of n, whereas dashed lines represent the right-hand side of Eq. (18) with z≡x, both normalized by the factor (1+|xα|n)−1 where we choose α=0.2. (a) n=7 and λ=4 (b) n=14 and λ=9

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.

3  Approximation of Higher-Order Apostol-Tangent Polynomials

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

(2λe2w+1)mezw=∑n=0∞Tnm(z;λ)wnn!,|w|<π2when λ=1and |w|<π when λ≠1:λϵC\{0} (29)

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 m.

Theorem 3.1 For λϵC\{0}, and zϵC\{0} such that |Imz−1|<π−Argλ2 and |z−1|<|z−1−(πi2−δ)| and n,m≥1, the Apostol-tangent polynomials of order m satisfy

Tnm(nz+m;λ)=nnznsechm(z−1+δ)(λ)m{1−m(m−(m+1)sech2(z−1+δ))2nz2+O(1n2)}, (30)

when δ=(log⁡λ)/2 and the logarithm is taken to be the principal branch.

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

Tnm(z;λ)=n!2πi∫C2mezw(eIn(λ)+2w+1)mdwwn+1, (31)

where C is a circle about 0 with radius less than |π−In(λ)|2. With (2e(δ+w))m(cosh(w+δ))m=(e2δ+2w+1)m, it follows from Eq. (31) that

Tnm(z;λ)=n!2πi(λ)m∫Cf(w)ezwewmdwwn+1 (32)

where λm=(elog⁡(λ)/2)m=eδm and f(w)=1coshm(w+δ). The function f(w) is a meromorphic function with poles of order m at the zeros of coshm(w+δ) which are given by wj=(2j+1)πi2−δ,j=0,±1,±2,⋯. It follows that by taking z⟼nz an letting nz→∞ with fixed z,

Tnm(nz+m;λ)=n!2πi(λ)m∫Cf(w)en(zw−log⁡w)dwwn. (33)

Likewise, the approximations of Tnm(nz+m;λ) can be obtained by expanding f(w) around the saddle point w=z−1. Using Lemma 1, Lemma 2, and Theorem 1 of [9],

Tnm(nz+m;λ)=(nz)n(λ)m∑k=0∞f(k)(z−1)k!Pk(n)(nz)k, (34)

where Pk(n) are the polynomials given in Eqs. (11) and (12). The derivative of f(k)(z−1) for k=0,1,2 are given by

f(0)(z−1)=f(z−1)=sechm(z−1+δ), (35)

f(1)(z−1)=−mtanh⁡(z−1+δ)sechm(z−1+δ), (36)

f(2)(z−1)=m sechm(z−1+δ)(m−(m+1)sech2(z−1+δ)). (37)

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

Tnm(nz+m;λ)=(nz)n(λ)m[v(z−1)0!+v(1)(z−1)1!p1(n)nz+v(2)(z−1)2!p2(n)(nz)2+O(1n2)]=nnzn(λ)m{sechm(1z+δ)−msechm(1z+δ)(m−(m+1)sech2(1z+δ))2nz2+O(1n2)}=nnzn(sechm(1z+δ))(λ)m{1−m(m−(m+1)sech2(1z+δ))2nz2+O(1n2)}.

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

Figure 3: Solid lines represent Tnm(nx+m;λ) for several values of n and m, whereas dashed lines represent the right-hand side of Eq. (30) with z≡x, both normalized by the factor (1+|xα|n)−1 where we choose α=0.2 (a) m=7,n=10 and λ=5 (b) m=8,n=7 and λ=6

Corollary 3.2. For z∈C\{0} such that |Imz−1|<π2,|z−1|<|z−1−πi2| and n,m≥1,

Tnm(nz+m)=nnznsechm(z−1){1−m(m−(m+1)sech2(z−1+δ))2nz2+O(1/n2)}. (38)

Proof. This follows from Theorem 3.1 by taking λ=1. To enlarge the region of validity of Eq. (30) and obtain an asymptotic expansion valid in a larger region the following theorem will be used.

Theorem 3.3. For λ∈C\{0},m∈Z+ and z∈C such that |z−1|<|z−1−wk| for all k=l+1,l+2,⋯, the Apostol-tangent polynomials of order m satisfy

Tnm(nz+m;λ)=λ−m2{∑k=1l∑j−1mewknzrkj[∑s=0n(ns)(−1)(j−1)⟨j−1⟩s(wk)--(j−1+s)((n−s)!wkn−s+1−Γ(n−s+1,wknz)wkn−s+1)+(−1)j⟨j⟩nwkj+n]+(nz)n∑k=0∞f(k)(z−1)−hl(k)(z−1)k!Pk(n)(nz)k}, (39)

where the polynomisals pk(n) are given in Eq. (12) hl(k) is the kth derivative of the function hl(w) given by Eq. (49) and

∑j=1mrkj(w−wk)j

Are the given principal parts of the Laurent series corresponding to the poles wk, where the entire function h(z) is determined by f(z).

Proof. With f(w)=cosh−m⁡(w+δ), it follows from Mittag-Leffler’s Theorem (see [18,19]) that

f(w)=∑k=1l[∑j=1mrkj(w−wk)j+qk(w)]+g(w)=∑k=1l∑j=1mrkj(w−wk)j+∑k=1lqk(w)+g(w)=∑k=1l∑j=1mrkj(w−wk)j+fl(w), (40)

where

fl(w)=∑k=1lqk(w)+g(w),

qk(w) is a polynomial of w,rkj are residues of f(w) at wk,k=1,2,…,l. Note that inside the disk |w|<|wm+1|,fl(w) has no poles.

Recall from Eq. (33),

Tnm(nz+m;λ)=1λm2n!2πi∫Cf(w)ewnzdwwn+1, (41)

where f(w)=1/coshm(w+δ)=sechm(w+δ). Substituting Eq. (40) to Eq. (41) gives

Tnm(nz+m;λ)=1λm2n!2πi∫C(∑k=1l∑j=1mrkj(w−wk)j+fl(w))ewnzdwwn+1=λ−m2n!2πi∫C∑k=1l∑j=1mrkj(w−wk)jewnzdwwn+1+λ−m2n!2πi∫Cfl(w)ewnzdwwn+1.| (42)

Let

Xln,m(z)=λ−m2n!2πi∫Cfl(w)ewnzdwwn+1, (43)

Yln,m(z)=λ−m2n!2πi∫C∑k=1l∑j=1mrkj(w−wk)jewnzdwwn+1=λ−m2∑k=1l∑j=1mn!2πi∫Crkj(w−wk)jewnzdwwn+1. (44)

Repeating the process to prove Theorem 3.1 where f(w) there is replaced by fl(w), we have

Xln,m(z)=λ−m2n!2πi∫Cfl(w)en(wz−log⁡w)dww. (45)

Assume that z−1 is not a pole of fl(w). We can expand fl(w) around the saddle point. That is

fl(w)=∑k=0∞fl(k)(z−1)k!(w−z−1)k,|w−z−1|<r (46)

where r is the distance from z−1 to the nearest singularity of fl(w). Substitute Eq. (46) to Eq. (43)

Xln,m(z)=λ−m2n!2πi∫C∑k=0∞fl(k)(z−1)k!(w−z−1)kewnzdwwn+1=λ−m2(nz)n∑k=0∞fl(k)(z−1)k!1(nz)nn!2πi∫C(w−z−1)kewnzdwwn+1=λ−m2(nz)n∑k=0∞fl(k)(z−1)k!uk(n,z),

where

uk(n,z)=1(nz)nn!2πi∫C(w−z−1)kewnzdwwn+1.

It follows from Lemma 1 [9] that

uk(n,z)=pk(n)(nz)k, (47)

where pk(n) are the polynomials in Eqs. (11) and (12). Thus,

Xln,m(z)=λ−m2(nz)n∑k=0∞fl(k)(z−1)k!pk(n)(nz)k, (48)

valid for m∈Z+,z∈C\{0} such that |z−1|<|z−1−wj| for j=l+1,l+2,… given the first 2l poles of f(w). From Eq. (40),

fl(w)=f(w)−∑k=1l∑j=1mrkj(w−wk)j.

This gives

fl(k)(w)=fk(w)−hl(k)(w),

where

hl(w)=−∑k=1l∑j=1mrkj(w−wk)j. (49)

The expansion of Xln,m(z) in Eq. (48) becomes

Xln,m(z)=λ−m2(nz)n∑k=0∞fl(k)(z−1)−hl(k)(z−1)k!pk(n)(nz)k,| (50)

valid for |z−1|<|z−1−wj|,j=l+1,l+2,… and z≠0. This range of validity is larger than that in Theorem 2.1 and Theorem 3.1.

On the other hand, to obtain an expansion for Yln,m(z), shift the integration contour in Eq. (44) by w=wk+t. Then dw=dt and

Yln,m(z)=λ−m2∑k=1l∑j=1mn!2πi∫C′rkjtje(wk+t)nzdt(wk+t)n+1=λ−m2∑k=1l∑j=1mewknzrkjn!2πi∫C′etnztjdt(wk+t)n+1, (51)

where C′:t=−wk+Reiθ,−π<θ≤π is a circle with radius R and center at −wk. Note that 0 is not on the wk′s. This C′ is the image of C:w=Reiθ through the shift w=wk+t. Note that

∫0zetxdx=etxt|z0=etzt−1t,

giving

etzt=∫0zetxdx+1t.

Similarly,

∫02etxtj−1dx=etxtj|z0=etztj−1t.

so that

etztj=∫0zetxtj−1+1tj.

It follows that

etnztj=∫0nzetxtj−1+1tj.

Then Eq. (51) becomes

Yn,ml(z)=λ−m2∑k=1l∑j=1mewknzrkjn!2πi∫C′(∫0nzetxtj−1dx+1tj)dt(wk+t)n+1. (52)

First, we compute

n!2πi∫C′etxtj−1dt(wk+t)n+1=n!2πi∫C′etxt−(j−1)dt(wk+t)n+1=dndtn(etxt−(j−1))|t=−wk. (53)

Note that when j=1, the RHS of Eq. (53) is xn. For j≥1, we use the Leibniz rule for differentiation.

This gives

dndtn(etxt−(j−1))=∑s=0n(ns)xn−setxdsdtst−(j−1)|t=−wk. (54)

It can be computed that

dsdtst−(j−1)=(−1)s(j−1)j(j+1)…(j−1+(s−1))t−(j−1+s)=(−1)s⟨j−1⟩st−(j−1+s),

where ⟨j−1⟩s denotes the rising factorial of j−1 with increment s. Then Eq. (54) becomes

dndtn(etxt−(j−1))=∑s=0n(ns)xn−se−wkx(−1)s⟨j−1⟩s(−wk)−(j−1+s)=∑s=0n(ns)xn−se−wkx(−1)(j−1)⟨j−1⟩s(wk)−(j−1+s).

Thus, Eq. (53) can be written

n!2πi∫C′etxtj−1dt(wk+t)n+1=∑s=0n(ns)xn−se−wkx(−1)(j−1)⟨j−1⟩s(wk)−(j−1+s), (55)

while

n!2πi∫C′t−jdt(wk+t)n+1=dndtn(t−j)|t=−wk=(−1)n⟨j⟩n(−wk)−j−n=(−1)j⟨j⟩n(wk)−(j+n)=(−1)j⟨j⟩nwkj+n. (56)

Note also that

∫0nzxn−se−wkxdx=∫0nztn−se−wkxdt.

Now the incomplete gamma function

Γ(a,z)=∫z∞e−tta−1dt,

gives

Γ(n−s+1,wkz)=∫wkz∞e−ttn−sdt.

Let η=twk. Then t=ηwk and wkdη=dt. Moreover, t=∞⟺η=∞;t=wkz⟺η=z.

Consequently,

Γ(n−s+1,wkz)=∫z∞e−wkη(wkη)η−swkdηΓ(n−s+1,wkz)wkn−s+1=∫0∞e−wkηηn−sdη=∫0∞e−wkηηn−sdn−∫0ze−wkηηn−sdη

or

∫0ze−wkηηn−sdη=∫0∞e−wkηηn−sdη−Γ(n−s+1.wkz)wkn−s+1.

Take note z⟼nz. Then 

∫0nze−wkηηn−sdη=∫0∞e−wkηηn−sdη−Γ(n−s+1,wknz)wkn−s+1. (57)

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

Yln,m(z)=λ−m2∑k=1l∑j=1mewknzrkj[(∫0nz∑s=0n(ns)xn−se−wkx(−1)(j−1)⟨j−1⟩s(wk)−(j−1+s))dx+(−1)j⟨j⟩nwkj+n]=λ−m2∑k=1l∑j=1mewknzrkj[∑s=0n(ns)(−1)(j−1)⟨j−1⟩s(wk)−(j−1+s)(∫0nzxn−se−wkx)dx]+(−1)j⟨j⟩nwkj+n]. (58)

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

Yln,m(z)=λ−m2∑k=1l∑j=1mewknzrkj[∑s=0n(ns)(−1)(j−1)⟨j−1⟩s(wk)−(j−1+s)(∫0∞e−wkttn−sdt−Γ(n−s+1,wknz)wkn−s+1)+(−1)j⟨j⟩nwkj+n]. (59)

Since

∫0∞tn−se−wktdt=(n−s)!wkn−s+1,n≥s, (60)

we can write Eq. (59) as follows:

Yln,m(z)=λ−m2∑k=1l∑j=1mewknzrkj[∑s=0n(ns)(−1)(j−1)⟨j−1⟩s(wk)−(j−1+s)((n−s)!wkn−s+1−Γ(n−s+1,wknz)wkn−s+1)+(−1)j⟨j⟩nwkj+n]. (61)

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

Tnm(nz+m;λ)=λ−m2{∑k=1l∑j=1mewknzrkj[∑s=0n(ns)(−1)(j−1)⟨j−1⟩s(wk)−(j−1+s)((n−s)!wkn−s+1−Γ(n−s+1,wknz)wkn−s+1)+(−1)j⟨j⟩nwkj+n]+(nz)n∑k=0∞f(k)(z−1)−hl(k)(z−1)k!pk(n)(nz)k}

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

Figure 4: Solid lines in (a) and (b) represent Tmn(nx+m;A) for n=3,m=3, whereas dashed lines in (a) and (b) represent the right hand side of Eqs. (30) and (39), respectively, with z≡x, both normalized by the factor (1+|xa|n)−1 where we choose a=0.5 (a) n=3,m=3 and λ=3 (b) n=3,m=3 and λ=3

valid for m∈Z+,z∈C\{0} such that |z−1|<|z−1--wk| for all k=l+1,l+2,…, where the polynomials pk(n) are given in Eq. (12) and hl(k) is the kth derivative of hl(w) given by Eq. (49).

Note that if m=1, Eq. (61) reduces to

Yln(z)=λ−12(−∑k=1lewknzrkjwkn+1Γ(n+1,wknz)),

since ⟨1⟩0=1 and ⟨0⟩1=0. This is exactly the first term Eq. (28).

Corollary 3.4. For z∈C\{0} such that |z−1|<|z−1--wk| for all k=l+1,l+2,…,m,n∈Z+, the tangent polynomials of order m satisfy,

Tnm(nz+m)=∑k=1l∑j=1mewknzrkj[∑s=0n(ns)(−1)(j−1)⟨j−1⟩s(wk)−(j−1+s)((n−s)!wkn−s+1−Γ(n−s+1,wknz)wkn−s+1)+(−1)j⟨j⟩nwkj+n]+(nz)n∑k=0∞f(k)(z−1)−hl(k)(z−1)k!pk(n)(nz)k, (62)

where wk=(2k+1)πi2, the polynomials pk(n) are given in Eq. (12), hl(k) is the kth derivative of the function hl(w) given by Eq. (49) and

∑j=1mrkj(w−wk)j

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

Proof. This follows from Theorem 3.3 by taking λ=1.

4  Conclusion

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 λ=1 and can be used as check formulas of the general case. It will be interesting also to investigate if the methods used in the paper will be applicable to the Apostol-tangent-Bernoulli polynomials and Apostol-tangent-Genocchi polynomials of higher order.

For future research work, one may try to investigate more properties of Apostol-tangent and higher order Apostol-tangent polynomials and establish q-analogues of these polynomials (see [20–22]).

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.

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