Elsevier

Applied Mathematics and Computation

Volume 270, 1 November 2015, Pages 136-141
Applied Mathematics and Computation

Closed formulas for computing higher-order derivatives of functions involving exponential functions

https://doi.org/10.1016/j.amc.2015.08.051Get rights and content

Abstract

For integers k ≥ 1 and n ≥ 0, the functions 1/(1λeαt)k and the derivatives (1/(1λeαt))(n) can be expressed each other by linear combinations. Based on this viewpoint, we find several new closed formulas for higher-order derivatives of trigonometric and hyperbolic functions, derive a higher-order convolution formula for the tangent numbers, and generalize a recurrence relation for the tangent numbers.

Introduction

Motivated by two identities in [10] the following problem was posed in the recently published paper [6]. For t ≠ 0 and kN, determine the numbers ak,i1 for 1 ≤ ik such that 1(1et)k=1+i=1kak,i1(1et1)(i1).Stimulated by this problem, eight identities involving an exponential function were established in [6]. The authors [15] answered this question alternatively with combinatorial technique and unified the eight identities due to Guo and Qi to two identities involving two parameters. We state them as the following theorems:

Theorem 1.1

Let n ≥ 0 be an integer and let α and λ be two real parameters. Then (11λeαt)(n)=k=1n+1αn(1)nk+1(k1)!S(n+1,k)(1λeαt)k,where S(n, k) are the Stirling numbers of the second kind.

Theorem 1.2

Let n ≥ 1 be an integer and let α and λ be two real parameters. Then 1(1λeαt)n=k=1n(1)nks(n,k)(n1)!αk1(11λeαt)(k1),where s(n, k) are the Stirling numbers of the first kind.

The above two theorems imply that the functions 1/(1λeαt)k and the derivatives (1/(1λeαt))(n) can be expressed each other by linear combinations. Guo and Qi [7] gave elementary proofs for these two theorems.

Finding closed expressions for higher-order derivatives of trigonometric functions is a subject of recurrent interest. In a recent paper, Adamchik [1] solved a long-standing problem on finding a closed-form expression for the higher-order derivatives of the cotangent function by showing that dndxncotx=(2i)n(cotxi)k=1nk!2kS(n,k)(icotx1)k,where i=1. Independently, Boyadahiev [2] derived derivative polynomials for the hyperbolic and trigonometric tangent, cotangent and secant in explicit form.

Using basic operations on the Zeon algebra, a simple and short proof of Formula (1.4) was given in [11]. Recently, by using the derivative polynomials introduced by Hoffman [9], Cvijović [8] derived closed-form higher derivative formulas for eight trigonometric and hyperbolic functions, which involve the Carlitz-Scoville higher-order tangent and secant numbers [3], [4]. More recently, Qi [12] found explicit formulas for higher order derivatives of the tangent and cotangent functions as well as powers of the sine and cosine functions, obtained explicit formulas for two Bell polynomials of the second kind for successive derivatives of sine and cosine functions, presented curious identities for the sine function and discovered explicit formulas and recurrence relations for the tangent numbers, the Bernoulli numbers, the Genocchi numbers, special values of the Euler polynomials at zero, and special values of the Riemann zeta function at even numbers. Moreover, in [12] some comments on five different forms of higher order derivatives for the tangent function and on derivative polynomials of the tangent, cotangent, secant, cosecant, hyperbolic tangent, and hyperbolic cotangent functions were given.

This paper is a sequel to the work of Xu and Cen [15]. By directly applying Theorems 1.1 and 1.2, we will find several new explicit expressions for higher-order derivatives of trigonometric and hyperbolic functions which are different from those of Adamchik and Boyadahiev. By means of a key equality in [15], we present a proof to show that they are equivalent. Furthermore, we derive a higher-order convolution formula for the tangent numbers, which generalizes the recurrence relation for the tangent numbers given in [12].

Section snippets

Main results

First of all, recall briefly that Stirling numbers of two kinds play very important roles in combinatorial analysis and number theory. Let S(n, k) be the Stirling number of the second kind and let s(n, k) be the Stirling number of the first kind. The Stirling number of the second kind S(n, k) counts the number of partitions of a set of n elements into k indistinguishable boxes in which no box is empty, and the Stirling number of the first kind s(n, k) counts the number of arrangements of n

Acknowledgments

We thank the anonymous referee for his/her careful reading of our manuscript and very helpful comments. Ai-Min Xu’s work was supported by the National Natural Science Foundation of China under grants 11201430, 61303144 and the Ningbo Natural Science Foundation under grant 2014A610021. Zhong-Di Cen’s work was supported by the Youth Fund of Humanities and Social Sciences of the Ministry of Education under grant 14YJC900006, the Project of Philosophy and Social Science Research in Zhejiang

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