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

Authors: Cristina B. Corcino; Roberto B. Corcino; Jay M. Ontolan
Journal: Journal of Applied Mathematics
 (2021)

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## Abstract

Asymptotic approximations of Tangent polynomials, Tangent-Bernoulli, and Tangent-Genocchi polynomials are derived using saddle point method and the approximations are expressed in terms of hyperbolic functions. For each polynomial there are two approximations derived with one having enlarged region of validity.


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## Body

## 1. Introduction

Several well-known special functions, numbers and polynomials (e.g. zeta functions, Bernoulli, Euler and Genocchi numbers and polynomials and derivative polynomials [1–6]) have been studied by many researchers in recent decade due to their wide-ranging applications from number theory and combinatorics to other fields of applied mathematics. With these, different variations and generalizations of these functions, numbers and polynomials have been constructed and investigated. For instance, papers in [7, 8] have introduced and investigated q-analogues of zeta functions and Euler polynomials. Some variations are constructed by mixing the concept of two special functions, numbers or polynomials. For example, poly-Bernoulli numbers and polynomials in [9–11] are constructed by mixing the concepts of polylogarithm and Bernoulli numbers and polynomials. Moreover, the Apostol-Genocchi polynomials, Frobenius-Euler polynomials, Frobenius-Genocchi polynomials and Apostol-Frobenius-type poly-Genocchi polynomials in [12–14] are constructed by mixing the concepts of Apostol, Frobenius, Genocchi and Euler polynomials.Another interesting mixture of special polynomials can be constructed by joining the concept of Tangent polynomials with Bernoulli and Genocchi polynomials. The Tangent polynomialsTnz, Tangent – Bernoulli TBnz and Tangent – Genocchi TGnz polynomials are defined by the generating functions(1)2ewze2w+1=∑n=0∞Tnzwnn!,w<π2(2)wewze2w−1=∑n=0∞TBnzwnn!,w<π(3)2wewze2w+1=∑n=0∞TGnzwnn!,w<π2.Whenz=0, 1.1 reduces to the generating function of the tangent numbers Tn given by(4)2e2w+1=∑n=0∞Tnwnn!,w<π2.In [15], the tangent polynomials can be determined explicitly using the tangent numbers Tn which is given by(5)Tnz=∑k=0nTknkzn−k.We are interested in finding asymptotic approximations of the Tangent polynomialsTnz, Tangent–Bernoulli TBnz and Tangent–Genocchi TGnz polynomials for large n which are uniformly valid in some unbounded region of the complex variable z. Equation Equation (1) yields a recurrence relation(6)Tnz=zn−∑k=1n2k−1nkTn−kz,with initial value T0z=1. The Tangent polynomials Tnz can be determined explicitly using (6). The first few values are(7)T0z=1,T1z=z−1,T2z=z2−2z,T3z=z3−3z2+2,T4z=z4−4z3+8z,T5z=z5−5z4+20z2−16.The Tangent-Bernoulli polynomials and Tangent-Genocchi polynomials satisfy the relations(8)TBnz=2n−1Bnz2(9)TGnz=2n−1Gnz2,where Bernoulli and Genocchi polynomials were defined in [7, 16], respectively. Some specific values are given below.(10)TB0z=12,TB1z=−12+z2,TB2z=162−6z+3z2,TB3z=12z2−3z+z2,TB4z=−415+2z2−2z3+z42,TB5z=16z−8+20z2−15z3+3z4,TG0z=0,TG1z=1,TG2z=−2+2z,TG3z=−6z+3z2TG4z=8−12z2+4z3,TG5z=40z−20z3+5z4.Applications of Bernoulli polynomials can be found in [16] while new formulas for Genocchi polynomials involving Bernoulli polynomials can be found in [17]. The Bernoulli polynomials and Genocchi polynomials were expressed in terms of hyperbolic function in [7, 16] as follows(11)Bnz+12=n!2πi∫Cwewz2sinhw/2dwwn+1Gnz+12=n!2πi∫Cwewzcoshw/2dwwn+1.where C is a circle about 0 with radius <2π (resp. <π). These integral representations were used to establish the asymptotic approximations of Bernoulli and Genocchi numbers.In this paper, the Tangent polynomials, Tangent-Bernoulli, and Tangent-Genocchi polynomials will be given asymptotic approximations using the method used in [19, 20].


## 2. Uniform Approximations

First, let us consider the approximation of Tangent polynomials. Using the saddle point method introduced we can establish the following theorem.Theorem 1.
Forz∈ℂ\0 such that Imz−1<π/2 and z±2i/π>2/π and n≥1,(12)Tnnz+1=nzncosh1/z1+12nz2sech21z−tanh21z+O1n2.Proof.
Applying the Cauchy-Integral Formula to (1)(13)Tnzn!=12πi∫C2ewze2w+1dwwn+1,where C is a circle about 0 with radius <π/2 . It follows from (13) that(14)Tnz+1=n!2πi∫C2ez+1we2w+1dwwn+1.
With2ewcoshw=e2w+1, (14) can be written as(15)Tnz+1=n!2πi∫Cewzcoshwdwwn+1.
Letfw=1/coshw . The singularities of fw are the zeros of coshw, which are wj=2j+1π/2i, j∈ℤ. Each of these singularities is a simple pole of fw while 0 is a pole of order n+1 of the integrand of (15).
Takez⟼nz and let nz⟼∞ by letting n⟶∞ with z fixed. Then(16)Tnnz+1=n!2πi∫Cfwewnzdwwn+1=n!2πi∫Cfwenwz−logwdww.
The main contribution of the integrand to the integral above originates at the saddle point of the argument of the exponential (see [18]). This saddle point is at w such that(17)ddwnwz−logw=0⟺w=1z=z−1.
Assume thatz−1 is not a pole of fw. Approximations of Tnnz+1 can be obtained by expanding fw around the saddle point [19]. With z−1 not a pole of fw, we can expand fw around z−1 . That is,(18)fw=∑k=0∞fkz−1k!w−z−1k,w−z−1<rwhere r is the distance from z−1 to the nearest singularity of fw. For w∈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/2 and z−1<z−1−wj for all j=0,±1,±2,⋯ . It follows from Lemma 1, Lemma 2 and Theorem 1 of [19] that(19)Tnnz+1=nzn∑k=0∞fkz−1k!pknnzk,where(20)p0n=1,p1n=0,p2n=−n,p3n=2n,(21)pkn=1−kpk−1n+npk−2n.
Writing the first few terms of (19), we have(22)Tnnz+1=nznf0z−1p0n+f1z−1p1nnz+f2z−12!p2nnz2+O1n2=nzn1cosh1/z−n2nz2−sech31z+sech1ztanh21z+O1n2=nzncosh1/z1+12nz21cosh21/z−tanh21z+O1n2=nzncosh1/z1+12nz2sech21z−tanh21z+O1n2.Figure1 below depicts the graphs of Tnnz+1 (in solid lines) generated using relation (6) and the graphs of the approximate values of Tnnz+1 (in dashed lines) generated using the expansion at the right-hand side of (12). The graphs below are generating using the software Mathematica.Figure 1
Solid lines representTnnz+1 for several values of n, whereas dashed lines represent the right-hand side of (12) with z≡x, both normalized by the factor 1+x/αn−1 where α=0.2.The graphs show the accuracy of approximation (12) for several values of n for real values of the uniform parameter z. For a real argument, the oscillatory region of Tnnz+1 is also contained in x≤2π−1, whereas the monotonic region contains x>2π−1. Therefore, the accuracy of approximation (12) is restricted to the monotonic region.The next theorem contains the approximation formula for Tangent-Bernoulli polynomials.Theorem 2.
Forz∈ℂ\0 such that Imz−1<π and z±1/π>1/π and n≥1,(23)TBnnz+1=nzn−1csch1/z8z24nz2−1+4zcoth1z−coth21z−3csch21z+O1n2Proof.
Applying the Cauchy-Integral Formula to (2)(24)TBnzn!=12πi∫Cwewze2w−1dwwn+1,where C is a circle about 0 with radius <π. It follows from (24) that(25)TBnz+1=n!2πi∫Cwez+1we2w−1dwwn+1.
With2ewsinhw=e2w−1, (25) can be written as(26)TBnz+1=n!2πi∫Cwewz2sinhwdwwn+1.
Letfw=w/2sinhw . The singularities of fw are the zeros of sinhw, which are.
wj=jπ, j∈ℤ. Each of these singularities is a simple pole of fw while 0 is a pole of order n+1 of the integrand of (26).
Takez⟼nz and let nz⟼∞ by letting n⟶∞ with z fixed. Then in view of Theorem 1, we can write(27)TBnnz+1=n!2πi∫Cfwenwz−logwdww.
Withz−1 not a pole of fw, we can expand fw around z−1. That is, if z−1 is in the strip Imz−1<π and z−1<z−1−wj for all j=0,±1,±2,⋯. It follows from Lemma 1, Lemma 2 and Theorem 1 of [19] that(28)TBnnz+1=nzn∑k=0∞fkz−1k!pknnzk,where pkn are given in (20). Writing the first few terms of (28), we have(29)TBnnz+1=nznf0z−1p0n+f1z−1p1nnz+f2z−12!p2nnz2+O1n2=nzn−1csch1/z8z24nz2−1+4zcoth1z−coth21z−3csch21z+O1n2.The following figure depicts the graphs ofTBnnz+1 (in solid lines) generated using relation (8) and the graphs of the approximate values of TBnnz+1 (in dashed lines) generated using the expansion at the right-hand side of (23). The graphs are generated using Mathematica.The graphs show the accuracy of approximation (23) for several values of n for real values of the uniformity parameter z. For a real argument, the oscillatory region of TBnnz+1 is also contained in x≤π−1, whereas the monotonic region contains x>π−1. Therefore, the accuracy of approximation (23) is restricted to the monotonic region.The following theorem contains the approximation formula for Tangent-Genocchi polynomials This theorem is proved similarly as the first two theorems so the proof is omitted.Theorem 3.
Forz∈ℂ\0 such that Imz−1<π/2 and z±2i/π>2/π and n≥1,(30)TGnnz+1=nznsech1/zz+sech31/z3−cosh2/z+2zsinh2/z4nz3+O1n2.Figure2 below depicts the graphs of TGnnz+1 (in solid lines) generated using relation (9) and the graphs of the approximate values of TGnnz+1 (in dashed lines) generated using the expansion at the right-hand side of (30).Figure 2
Solid lines representTGnnz+1 for several values of n, whereas dashed lines represent the right-hand side of (30) with z≡x, both normalized by the factor 1+x/αn−1 where α=0.2.


## 3. Expansion of Tangent, Tangent-Bernoulli, and Tangent-Genocchi Polynomials with Enlarged Region of Validity

The validity of the approximations obtained in Theorems1, 2, and 3, are restricted to the region z−1<z−1−wj for all j=1,2,⋯ . However, the region z−1<z−1−wj may be enlarged by isolating the contribution of the poles wj’s of fw. We will follow similar procedure done in [16, 19] to prove our next three results. A detailed discussion can be seen in Lemma 3.2 in [16] that allows us to write the polynomial Pnnz defined by(31)Pnnz=n!2πi∫Cfwewzdwwn+1with a meromorphic function fw analytic in the origin with simple poles w1,w2,⋯ represented for each integer m>0 as(32)Pnnz=−∑k=1mrkewknzwkn+1Γn+1,wknz+nzn∑k=0∞fkz−1+hmkz−1pknk!nzk,which is valid for all complex number z satisfying z−1<z−1−wj for all j=m+1,m+2,⋯, such that C is a circle whose center is at the origin and contains no poles of fw inside, Γn+1,wknz is the incomplete gamma function, pkn are polynomials defined by the relation (20) and hmkz−1 is the kth derivative at z−1of the function(33)hmw=−∑l=1mrlw−wl.The following theorem contains the asymptotic expansion of Tangent polynomials with enlarged region of validity.Theorem 4.
Forz∈ℂ\0 such that z−1<z−1±2k+1π/2i for k=0,1,2,⋯,m−1. Then, as n⟶∞,(34)Tnnz+1=i∑k=0m−1−1ke2k+1/2πinz2k+1/2πin+1Γn+1,2k+12πinz−e−2k+1/2πinz−2k+1/2πin+1Γn+1,−2k+12πinz+nznsech1z+∑k=0m−14−1k+12k+1πz24+2k+12π2z2+n2nz2sech31z−sech1ztanh21z+∑k=0m−132−12kπ+2kπ2z2π+2kπ2−12z44+z2π+2kπ23+O1n2.Proof.
To obtain the corresponding residues, we first let(35)fw=1coshw=pwqw.
Then(36)rk=pwkq′wk=1sinhwk,k=0,1,2⋯.,m−1.
Now, fork=0,1,2⋯.,m−1, sinhwk=−1ki. Hence, rk=−1k+1i.
On the other hand, fork=0,1,2⋯.,m−1, sinhw−k=−1k+1i. Thus, for k=0,1,2⋯.,m−1, r−k=−1ki. Hence, for k=0,1,2⋯.,m−1 we have(37)rk=−1k+1iforwk=2k+12πir−k=−1kiforw−k=−2k+12πi.
We then evaluate some derivatives of the functionhmw defined in (33) at the saddle point z−1. Then for k=0,1,2⋯.,m−1, we write(38)hmw=−∑k=0m−1rkw−wk−∑k=0m−1r−kw−w−k=∑k=0m−1−1kiw−2k+1/2πi−∑k=0m−1−1kiw+2k+1/2πi=∑k=0m−1−1k+12k+1πw2+2k+12/4π2.
Using (37) we can compute some derivatives of hmw and evaluate it at the saddle point z−1. We obtain the results below.(39)hm1w=∑k=0m−132−12k+1π+2kπ2w4w2+π+2kπ22,(40)hm2w=∑k=0m−132−12k+1π+2kπ2π+2kπ2−12w24w2+π+2kπ230
Equations (37) and (39) yield(41)hm0z−1=∑k=0m−14−1k+12k+1πz24+2k+12π2z2(42)hm2z−1=∑k=0m−132−12k+1π+2kπ2z2π+2kπ2−12z44+z2π+2kπ23.
Using (16) and (32) for k=0,1,2⋯.,m−1, we have(43)Tnnz+1=−∑k=0m−1−1k+1ie2k+1/2πinz2k+1/2πin+1Γn+1,2k+12πinz−∑k=0m−1−1kie−2k+1/2πinz−2k+1/2πin+1Γn+1,−2k+12πinz+nznfz−1+hmz−1p0n0!+f1z−1+hm1z−1p1n1!nz+f2z−1+hm2z−1p2n2!nz2+O1n2.
In Theorem1, we have p0n=1,p1n=0,p2n=−n,fz−1=1/cosh1/z, f2z−1=−sech31/z+sech1/ztanh21/z and using (40) and (41), we obtain (34).Figure3 below depicts the graphs of Tnnz+1 (in solid lines) generated using relation (6) and the graphs of the approximate values of Tnnz+1 (in dashed lines) generated using the expansion at the right-hand side of (34). We use Mathematica to graph the left-hand and right-hand side of (34).Figure 3
Solid lines representTnnz+1 for several values of n, whereas dashed lines represent the right-hand side of (34) with z≡x, both normalized by the factor 1+x/αn−1 where α=0.2.Figure 4
Solid lines representTBnnz+1 for several values of n, whereas dashed lines represent the right-hand side of (23) with z≡x, both normalized by the factor 1+x/αn−1 where α=0.2.The graphs show the accuracy of approximation (34) for several values of n for real values of the uniformity parameter z. For a real argument, the approximation of Tnnz+1 with enlarged validity in the monotonic region is better than the approximation in (12). It can also be observed that, in the oscillatory region, even if the accuracy is not that good the approximation is better compared to the approximation in (12).Now, let us consider the asymptotic expansion of Tangent-Bernoulli polynomials with enlarged region of validity.Theorem 5.
Forz∈ℂ\0 such that z−1<z−1±kπ for k=1,2,⋯,m. Then, as n⟶∞,(44)TBnnz+1=−π∑k=1mk1+ⅇ2kπⅇkπ1+nzkπn+1Γn+1,kπnz+ⅇkπ1−nz−kπn+1Γn+1,−kπnz+nzncsch1/z2z+∑k=1m2k2π2z2ⅇ−kπ+ⅇkπz2k2π2−1−n2nz2csch31/z3+cosh2/z−2zsinh2/z4z+∑k=1m4ⅇkπk2π2z4z2k2π2+31+ⅇ2kπz2k2π2−13+O1n2Proof.
To obtain the corresponding residues, we first let(45)fw=w2sinhw=pwqw.
Then,(46)rk=pwkq′wk=wk2coshwk,k=1,2⋯.,m.
Now, fork=1,2⋯.,m−1,(47)coshwk=coshkπ=ekπ+e−kπ2·
Hence,(48)rk=kπekπ+e−kπ·
On the other hand, fork=1,2⋯.,m,(49)coshw−k=cosh−kπ,k=1,2,⋯m=coshkπ=ekπ+e−kπ2·
Thus, fork=1,2⋯.,m,(50)r−k=−kπekπ+e−kπ·
Hence, fork=1,2⋯.,m we have(51)rk=kπekπ+e−kπforwk=kπ,r−k=−kπekπ+e−kπforw−k=−kπ·
We then evaluate some derivatives of the functionhmw defined in (33) at the saddle point z−1. Then for k=1,2⋯.,m, we write(52)hmw=−∑k=1mrkw−wk−∑k=1mr−kw−w−k=−∑k=1mkπ/ekπ+e−kπw−kπ−∑k=1m−kπ/ekπ+e−kπw+kπ=∑k=1mkπekπ+e−kπ1w+kπ−1w−kπ=∑k=1m2kπ2ekπ+e−kπk2π2−w2.
Using (51) we can compute some derivatives of hmw and evaluate it at the saddle point z−1. We obtain the results below.(53)hm1w=∑k=1m4k2π2wⅇ−kπ+ⅇkπk2π2−w22,(54)hm2w=∑k=1m4ⅇkπk2π2k2π2+3w21+ⅇ2kπk2π2−w23.
Equations (51) and (53) yield(55)hm0z−1=∑k=1m2k2π2z2ⅇ−kπ+ⅇkπz2k2π2−1(56)hm2z−1=∑k=1m4ⅇkπk2π2z4z2k2π2+31+ⅇ2kπz2k2π2−13.
Using equations (26) and (32) for k=1,2⋯.,m, we have(57)TBnnz+1=−∑k=1mkπ/ekπ+e−kπekπnzkπn+1Γn+1,kπnz−∑k=1m−kπ/ekπ+e−kπe−kπnz−kπn+1Γn+1,−kπnz+nznf0z−1+hm0z−1p0n0!+f1z−1+hm1z−1p1n1!nz+f2z−1+hm2z−1p2n2!nz2+O1n2.
In Theorem2, we have p0n=1,p1n=0,p2n=−n,f0z−1=csch1/z/2z, f2z−1=csch31/z3+cosh2/z−2zsinh2/z/4z and using (54) and (55), we obtain the desired result.The figure below depicts the graphs ofTBnnz+1 (in solid lines) generated using relation (8) and the graphs of the approximate values of TBnnz+1 (in dashed lines) generated using the expansion at the right-hand side of (43). Using Mathematica, we obtain the graph below.The graphs show the accuracy of approximation (43) for several values of n for real values of the uniform parameter z. For a real argument the approximation of TBnnz+1 with enlarged validity in the monotonic region is better than the approximation in (23). It can also be observed that in the oscillatory region the accuracy is better compare to the approximation in (23).The next theorem contains the asymptotic expansion of Tangent-Genocchi polynomials with enlarged region of validity.Theorem 6.
Forz∈ℂ\0 such that z−1<z−1±2k+1π/2i for k=0,1,⋯,m−1. Then, as n⟶∞,(58)TGnTGnnz+1=2nπnin+1∑k=0m−1−1k+12k+1ne2k+1/2πinzΓn+1,2k+12πinz+−1n+1e−2k+1/2πinzΓn+1,−2k+12πinz+nznsech1/zz+∑k=0m−14−1k+11+2kπz4+π2z+2kz2−n2nz2sech31/z−3+cosh2/z−2zsinh2/z2z+∑k=0m−132−1k1+2kπz3−4+3π2z+2kz24+π2z+2kz23+O1n2.Proof.
We let(59)fw=wcoshw=pwqw.
Then(60)rk=pwkq′wk=wksinhwk,k=0,1,2⋯.,m−1.
Now, forwk=2k+1π/2i,k=0,1,2⋯.,m−1, we see from Theorem 4 that(61)sinhwk=−1ki,so that(62)rk=2k+1π/2i−1ki=−1k2k+1π2.
On the other hand, forw−k=−2k+1π/2i,k=0,1,2⋯.,m−1, we see from Theorem 4 that(63)sinhw−k=−1k+1i.
This gives(64)r−k=−2k+1π/2i−1k+1i=−1k2k+1π2.
Hence, fork=0,1,2⋯.,m−1 we have(65)rk=r−k=−1k2k+1π2,for both(66)wk=2k+12πiandw−k=−2k+12πi·
Evaluating some derivatives of the functionhmw at the saddle point z−1 for k=0,1⋯.,m−1, we have(67)hmw=−∑k=0m−1−1k2k+1π/2w−2k+1/2πi−∑k=0m−1−1k2k+1π/2w+2k+1/2πi=∑k=0m−14−1k+11+2kπwπ+2kπ2+4w2,which yields,(68)hm1w=∑k=0m−14−1k+11+2kππ+2kπ2−4w2π+2kπ2+4w22,(69)hm2w=∑k=0m−132−1k1+2kπw3π+2kπ2−4w2π+2kπ2+4w23′(70)hm0z−1=∑k=0m−14−1k+11+2kπz4+π2z+2kz2(71)hm2z−1=∑k=0m−132−1k1+2kπz3−4+3π2z+2kz24+π2z+2kz23.
Using equations (2.18) and (32) for k=0,1,2⋯.,m−1, we have(72)TGnnz+1=−∑k=0m−1−1k2k+1π/2e2k+1/2πinz2k+1/2πin+1Γn+1,2k+12πinz−∑k=0m−1−1k2k+1π/2e−2k+1/2πinz−2k+1/2πin+1Γn+1,−2k+12πinz+nznf0z−1+hm0z−1p0n0!+f1z−1+hm1z−1p1n1!nz+f2z−1+hm2z−1p2n2!nz2+O1n2.
In Theorem3, we have p0n=1,p1n=0,p2n=−n,f0z−1=sech1/z/z,(73)f2z−1=sech31/z−3+cosh2/z−2zsinh2/z2z,and using (58) and (73) for k=0,1,2⋯.,m−1, we obtain (59).The following figure depicts the graphs ofTBnnz+1 (in solid lines) generated using relation (9) and the graphs of the approximate values of TBnnz+1 (in dashed lines) generated using the expansion at the right-hand side of (58).The graphs show the accuracy of approximation (58) for several values of n for real values of the uniform parameter z. In both monotonic and oscillatory regions, the approximation in (58) shows better accuracy than in (30).


## 4. Conclusion

Two sets of approximation formulas for Tangent polynomials, Tangent-Bernoulli polynomials and Tangent-Genocchi polynomials are obtained using the saddle point method and the integral representation of these polynomials in terms of hyperbolic functions. The last set of formulas which have enlarged validity give more accurate approximations compared to the first set of formulas as shown in Figures1–6. It is interesting to establish the asymptotic approximation of other variations of Tangent polynomials like Apostol-Tangent polynomials, Apostol-Tangent-Bernoulli polynomials and Apostol-Tangent-Genocchi polynomials as well as the higher- order versions of these polynomials.Figure 5
Solid lines representTBnnz+1 for several values of n, whereas dashed lines represent the right-hand side of (43) with z≡x, both normalized by the factor 1+x/αn−1 where α=0.2.Figure 6
Solid lines representTGnnz+1 for several values of n, whereas dashed lines represent the right-hand side of (58) with z≡x, both normalized by the factor 1+x/αn−1 where α=0.2.



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