嵌套导数：计算反函数级数展开式的一种简单方法

套导数的渐近分析

Asymptotic analysis of nested derivatives

我们分析嵌套衍生物的功能$\mathfrak{D}^{n}[f]\,(x)$渐近，作为$n\rightarrow\infty,$使用离散版本的射线方法。我们给出了一些例子来说明我们的公式的准确性。

On the higher derivatives of the inverse tangent function

In this paper, we find explicit formulas for higher order derivatives of the inverse tangent function. More precisely, we study polynomials which are induced from the higher-order derivatives of arctan(x). Successively, we give generating functions, recurrence relations and some particular properties for these polynomials. Connections to Chebyshev, Fibonacci, Lucas and Matching polynomials are established.

逆误差函数导数的渐近分析

Asymptotic analysis of the derivatives of the inverse error function

逆的误差函数，$\operatorname{inverf}(x),$扩散问题，化学势，超声成像等，我们分析的衍生物$\frac{d^{n}}{dz^{n}} \operatorname*{inverf}(z) |_{z=0}$，作为$n\to \infty$使用嵌套衍生物和离散射线方法的应用。通过对$x=0$的高阶Taylor展开，得到了$\operatorname{inverf}(x)$的一个很好的近似.我们给出的数值结果表明我们的公式的准确性。

基于广义拉普拉斯变换的时间尺度分数阶导数和积分

Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform

我们介绍了一个分数阶微积分的时间尺度使用理论的delta（或nabla）动力学方程。利用时间尺度上的拉普拉斯逆变换，提出了任意时间尺度上分数阶积分和分数阶导数的基本概念。证明了新的分数次算子的有用性质。

Higher derivatives of the inverse tangent function and a summation formula involving binomial coefficients

In 2017, O. Deiser and C. Lasser obtained an explicit formula for the $n$-th derivative of the inverse tangent function. We calculate this derivative by a different method based on Faà di Bruno's formula. Comparing the two results leads to the following identity for binomial coefficients: $$\sum_{i=m}^{[n/2]}\frac{(-1)^i}{4^i}\binom{i}{m}\binom{n-i}{i}=\frac{(-1)^m}{2^n}\binom{n+1}{2m+1},$$ where $n,m\in \mathbb{N}_0$ and $m\leq [n/2]$. As was pointed out to the author by C. Krattenthaler, this formula is a special case of Gauß's formula for the hypergeometric function $_2F_1$.

Analytic solution for one dimensional inverse heat conduction problem of semi-infinite bar

We present analytical formula along with its existence theorem for solution of inverse heat conduction problem of semi-infinite bar, equivalent to a Volterra integral equation of first kind, as an infinite series of fractional derivatives. The mathematical method is based on some properties of function space M[0, T] (proved here) with respect to fractional integration and derivatives.