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

(1) 2ezwλe2w+1=∑n=0∞Tn(z;λ)wnn!, 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]) (2) 2ezwe2w+1=∑n=0∞Tn(z)wnn!,|w|<π2.

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

(3) Tn(0,λ):=Tn(λ) and Tn(0):=Tn,

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

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.

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

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

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

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

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

(7) Tn(z+1;λ)=n!2πiλ∫Cf(w)ezwwwn+1,

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

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

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

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

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

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

where

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

(12) pk(n)=(1−k)pk−1(n)+npk−2(n),k≥3.

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

(13) f(z−1)=sech(z−1+δ),

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

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

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

(16) Pn(z)=n!2πi∫Cf(w)ewzdwwn+1,

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 (17) 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, 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.

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→∞, (18) 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)}.

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

(19) rl=p(wl)q′(wl)=1sin⁡h(wl+δ),

where

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

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

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

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

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

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

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

(24) 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.

At the saddle point z−1,

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

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

(27) 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.

From Theorem 2.2,

(28) 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.

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).

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.

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

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

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

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

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

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

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

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

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

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,

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

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],

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

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

(35) f(0)(z−1)=f(z−1)=sechm(z−1+δ),

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

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

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.

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

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

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

(39) 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}, 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

(40) 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), 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),

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

where f(w)=1/coshm(w+δ)=sechm(w+δ). Substituting Eq. (40) to Eq. (41) gives (42) 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.|

Let

(43) Xln,m(z)=λ−m2n!2πi∫Cfl(w)ewnzdwwn+1, (44) 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.

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

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

Assume that z−1 is not a pole of fl(w). We can expand fl(w) around the saddle point. That is (46) fl(w)=∑k=0∞fl(k)(z−1)k!(w−z−1)k,|w−z−1|<r 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),