DERIVATIVE POLYNOMIALS OF A FUNCTION ...

DERIVATIVE POLYNOMIALS OF A FUNCTION RELATED TO THE APOSTOL–EULER AND FROBENIUS–EULER NUMBERS JIAO-LIAN ZHAO, JING-LIN WANG, AND FENG QI

Abstract. In the paper, the authors find a simple and significant expression in terms of the Stirling numbers for derivative polynomials of a function with a parameter related to the higher order Apostol–Euler numbers and to the higher order Frobenius–Euler numbers. Moreover, the authors also present a common solution to a sequence of nonlinear ordinary differential equations.

1. Introduction In [4], Theorem 2.1 states that the differential euqation n dn F (t, λ) X (−1)n−k bk (n)λk F k+1 (t, λ) = d tn

(1)

k=0

for n ∈ N has a solution F (t, λ) =

1 , et + λ

(2)

where b0 (n) = 1,

bn (n) = n!,

(3)

and bk (n) = k!

n−k k −k X n−i X

···

ik =0 ik−1 =0

n−ikX −···i3 −k n−ikX −···i2 −k i2 =0

2i1 3i2 · · · (k + 1)ik

(4)

i1 =0

for 1 ≤ k ≤ n. Theorems 2 and 3 in [4] read that m X 1 (k+1) En+m,1/λ = (−1)m−k k En,1/λ bk (m) + (−1)m En,1/λ 2

(5)

k=1

and Hn+m (−λ) =

m X

(−1)

k=1

m−k



λ 1+λ

k

Hn(k+1) (−λ)

bk (m) + (−1)m Hn (−λ) k!

(6) (k)

for n ≥ 0, m ∈ N, and λ ∈ C with λ 6= 0, −1, where the quantities En,λ , En,λ , (k)

Hn (u), and Hn (u) for k ∈ N, n ≥ 0, λ ∈ C with λ 6= 0, and u ∈ C with u 6= 1 are generated by  k X  k X ∞ ∞ n 2 1−u tn (k) t (k) = E , = H (u) , n n,λ λet + 1 n! et − u n! n=0 n=0 2010 Mathematics Subject Classification. Primary 11B68; Secondary 11B73, 34A34. Key words and phrases. derivative polynomial; Stirling number; nonlinear ordinary differential equation; solution. This paper was typeset using AMS-LATEX. 1

2

J.-L. ZHAO, J.-L. WANG, AND F. QI

and (1)

En,λ = En,λ ,

Hn (u) = Hn(1) (u).

In the literature, the quantities En,λ and Hn (u) are respectively called the Apostol– (k) Euler numbers and the Frobenius–Euler numbers, and the quantities En,λ and (k)

Hn (u) are respectively called the higher order Apostol–Euler numbers and the higher order Frobenius–Euler numbers. The quantities in (3) and (4) were derived in [4] by elementary, inductive, and recurrent method. The method and idea in [4] have been employed in some papers, such as [5, 6, 7, 8, 9] and closely related reference therein, for many years. This means that the method used in [4, 5, 6, 7, 8, 9] and other similar papers is effectual and applicable. On the other hand, a defect of the method is also apparent: those results obtained in the above mentioned papers are too complicated to be computed and understood in mathematics. In recent years, the defect has been discussed, avoided, and corrected in some papers such as [11, 12, 13, 14, 15, 16, 17, 18, 19] and closely related references therein. Alternatively, we can regard the differential equation (1) as derivative polynomials n X Pn (x) = (−1)n−k bk (n)λk xk+1 (7) k=0

with respect to the variable t of the function F (t, λ) defined by (2). As usual, a sequence of polynomials Pn are called the derivative polynomials of a function f (x) if and only if f (n) (x) = Pn (f (x)) for n ≥ 0. For more detailed information on the concept of derivative polynomials, please refer to [3, 10, 16, 17] and closely related references therein. In our eyes, the expression (4) is also not simple, not significant, not meaningful, and not easy to be computed by hands and computer softwares. In this paper, we will simply find derivative polynomials Pn (x) for the function F (t, λ), which is simpler, more significant, and more meaningful than the one in (7). In other words, we will simply derive a simpler form for the quantities bk (n). Our main results can be stated as the following theorems. Theorem 1. For λ 6= 0 and n ∈ N, the derivative polynomials of the function F (t, λ) = et 1+λ can be computed by Pn (x) =

n X

(−1)n−k k!S(n + 1, k + 1)λk xk+1 ,

(8)

k=0

where S(n, k), which can be generated by ∞ X (ex − 1)k xn = S(n, k) , k! n!

k ∈ N,

n=k

stands for the Stirling numbers of the second kind. Remark 1. Comparing (1) with (8) reveals that bk (n) = k!S(n + 1, k + 1) for n ≥ k ≥ 0 and that S(n + 1, k + 1) =

n−k k −k X n−i X ik =0 ik−1 =0

···

n−ikX −···i3 −k n−ikX −···i2 −k i2 =0

i1 =0

2i1 3i2 · · · (k + 1)ik

DERIVATIVE POLYNOMIALS RELATED TO APOSTOL–FROBENIUS–EULER NUMBERS 3

for n ≥ k ≥ 1. Hence, the identities (5) and (6) can be significantly simplified as En+m,1/λ =

m X

(−1)m−k S(m + 1, k + 1)

k=1

k! (k+1) E + (−1)m En,1/λ 2k n,1/λ

and Hn+m (λ) = (−1)m

m X

 S(m + 1, k + 1)

k=1

λ 1−λ

k

Hn(k+1) (λ) + (−1)m Hn (λ)

for n ≥ 0, m ∈ N, and λ ∈ C with λ 6= 0, 1. These imply that the derivative polynomials Pn (x) in (8) is simpler, more significant, more meaningful, and easier to be computed by hands and computer softwares than the result in [4, Theorem 2.1] mentioned above. Theorem 2. For λ 6= 0 and n ∈ N, the nonlinear differential equations n X

(−1)k s(n + 1, k + 1)

k=0

have a common solution F (t, λ) =

dk F (t, λ) = (−1)n n!λn F n+1 (t, λ) d tk

1 et +λ ,

where s(n, k), which can be generated by

∞ X xn [ln(1 + x)]k s(n, k) , = k! n!

|x| < 1

n=k

for k ∈ N, stands for the Stirling numbers of the first kind. 2. A lemma In order to prove Theorems 1 and 2, we need the following lemma. Lemma 1 ([1, Theorem 2.1] and [22, Theorems 3.1 and 3.2]). Let α, β 6= 0 be real constants and k ∈ N. When β > 0 and t 6= − lnαβ or when β < 0 and t ∈ R, we have   m  k+1 X 1 1 dk k k = (−1) α (m − 1)!S(k + 1, m) (9) d tk βeαt − 1 βeαt − 1 m=1 and 

1 βeαt − 1

k

  k X 1 (−1)m−1 dm−1 1 = s(k, m) m−1 . (k − 1)! m=1 αm−1 dt βeαt − 1

(10)

Remark 2. For motivations and further developments about the papers [1, 22], please refer to [2, 20, 21] and closely related references therein. 3. Proofs of Theorems 1 and 2 Now we are in a position to prove our Theorems 1 and 2. Proof of Theorem 1. Taking α = 1 and β = − λ1 for λ 6= 0 in (9) gives    m k+1 X dk 1 1 k = (−1) (m − 1)!S(k + 1, m) d tk −et /λ − 1 −et /λ − 1 m=1

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J.-L. ZHAO, J.-L. WANG, AND F. QI

which can be rearranged as    m k+1 X dk 1 1 k+1 m m−1 = (−1) (−1) (m − 1)!S(k + 1, m)λ d tk et + λ et + λ m=1  m+1 k X 1 = (−1)k+1 (−1)m+1 m!S(k + 1, m + 1)λm t . e +λ m=0 The proof of Theorem 1 is thus complete.



Proof of Theorem 2. Letting α = 1 and β = − λ1 for λ 6= 0 in (10) arrives at   k  k X 1 1 1 dm−1 m−1 = (−1) s(k, m) m−1 −et /λ − 1 (k − 1)! m=1 dt −et /λ − 1 which can be rewritten as  k   k X dm−1 1 1 1 (−1)m−1 s(k, m) m−1 t (−1)k−1 λk−1 t = . e +λ (k − 1)! m=1 dt e +λ The proof of Theorem 2 is thus complete.



Acknowledgements. This work was partially supported by the China Postdoctoral Science Foundation with Grant No. 2015M582619. References [1] B.-N. Guo and F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math. 272 (2014), 251–257; Available online at http://dx.doi.org/10.1016/j.cam.2014.05.018. [2] B.-N. Guo and F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math. 255 (2014), 568–579; Available online at http://dx.doi. org/10.1016/j.cam.2013.06.020. [3] M. E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly 102 (1995), no. 1, 23–30; Available online at http://dx.doi.org/10.2307/2974853. [4] T. Kim, G.-W. Jang, and J. J. Seo, Revisit of identities for Apostol-Euler and FrobeniusEuler numbers arising from differential equation, J. Nonlinear Sci. Appl. 10 (2017), no. 1, 186–191; Available online at http://dx.doi.org/10.22436/jnsa.010.01.18. [5] T. Kim and D. S. Kim, Differential equations associated with Catalan–Daehee numbers and their applications, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM 111 (2017), in press; Available online at http://dx.doi.org/10.1007/s13398-016-0349-4. [6] T. Kim and D. S. Kim, Some identities of Eulerian polynomials arising from nonlinear differential equations, Iran. J. Sci. Technol. Trans. Sci. (2016), in press; Available online at http://dx.doi.org/10.1007/s40995-016-0073-0. [7] T. Kim, D. S. Kim, L.-C. Jang, and H. I. Kwon, On differential equations associated with squared Hermite polynomials, J. Comput. Anal. Appl. 23 (2017), no. 5, 1252–1264. [8] T. Kim, D. S. Kim, J.-J. Seo, and D. V. Dolgy, Some identities of Chebyshev polynomials arising from non-linear differential equations, J. Comput. Anal. Appl. 23 (2017), no. 5, 820–832. [9] D. Lim, Differential equations for Daehee polynomials and their applications, J. Nonlinear Sci. Appl. (2017), in press. [10] F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput. 268 (2015), 844–858; Available online at http://dx.doi.org/10.1016/j.amc.2015.06.123. [11] F. Qi, Explicit formulas for the convolved Fibonacci numbers, ResearchGate Working Paper (2016), available online at http://dx.doi.org/10.13140/RG.2.2.36768.17927. [12] F. Qi and B.-N. Guo, Explicit formulas and nonlinear ODEs of generating functions for Eulerian polynomials, ResearchGate Working Paper (2016), available online at http://dx. doi.org/10.13140/RG.2.2.23503.69288.

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[13] F. Qi and B.-N. Guo, Some properties of a solution to a family of inhomogeneous linear ordinary differential equations, Preprints 2016, 2016110146, 11 pages; Available online at http://dx.doi.org/10.20944/preprints201611.0146.v1. [14] F. Qi and B.-N. Guo, Some properties of the Hermite polynomials and their squares and generating functions, Preprints 2016, 2016110145, 14 pages; Available online at http://dx. doi.org/10.20944/preprints201611.0145.v1. [15] F. Qi and B.-N. Guo, Viewing some nonlinear ODEs and their solutions from the angle of derivative polynomials, ResearchGate Working Paper (2016), available online at http: //dx.doi.org/10.13140/RG.2.1.4593.1285. [16] F. Qi and B.-N. Guo, Viewing some nonlinear ODEs and their solutions from the angle of derivative polynomials, ResearchGate Working Paper (2016), available online at http: //dx.doi.org/10.13140/RG.2.1.4593.1285. [17] F. Qi and B.-N. Guo, Viewing some ordinary differential equations from the angle of derivative polynomials, Preprints 2016, 2016100043, 12 pages; Available online at http: //dx.doi.org/10.20944/preprints201610.0043.v1. [18] F. Qi and J.-L. Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, J. Differ. Equ. Appl. (2017), in press. [19] F. Qi and J.-L. Zhao, The Bell polynomials and a sequence of polynomials applied to differential equations, Preprints 2016, 2016110147, 8 pages; Available online at http: //dx.doi.org/10.20944/preprints201611.0147.v1. [20] C.-F. Wei and B.-N. Guo, Complete monotonicity of functions connected with the exponential function and derivatives, Abstr. Appl. Anal. 2014 (2014), Article ID 851213, 5 pages; Available online at http://dx.doi.org/10.1155/2014/851213. [21] A.-M. Xu and G.-D. Cen, Closed formulas for computing higher-order derivatives of functions involving exponential functions, Appl. Math. Comput. 270 (2015), 136–141; Available online at http://dx.doi.org/10.1016/j.amc.2015.08.051. [22] A.-M. Xu and Z.-D. Cen, Some identities involving exponential functions and Stirling numbers and applications, J. Comput. Appl. Math. 260 (2014), 201–207; Available online at http://dx.doi.org/10.1016/j.cam.2013.09.077. (Zhao) Department of Mathematics and Informatics, Weinan Teachers University, Weinan City, Shaanxi Province, 714000, China; State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an City, Shaanxi Province, 710071, China E-mail address: [email protected], [email protected] (Wang) Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China E-mail address: [email protected] URL: http://orcid.org/0000-0001-6725-533X (Qi) Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China E-mail address: [email protected], [email protected] URL: https://qifeng618.wordpress.com