In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the characteristic function (CF). The CF can be further expressed by a Taylor series involving the moments of the random variable. However, in some circumstances, the moments do not exist and the Taylor expansion of the CF is useless. This happens for example in the case of $α$--stable random variables. Here, the problem of representing the CF or the PDF of random variables (r.vs) is examined by introducing fractional calculus. Two very remarkable results are obtained. Firstly, it is shown that the fractional derivatives of the CF in zero coincide with fractional moments. This is true also in case of CF not derivable in zero (like the CF of $α$--stable r.vs). Moreover, it is shown that the CF may be represented by a generalized Taylor expansion involving fractional moments. The generalized Taylor series proposed is also able to represent the PDF in a perfect dual representation to that in terms of CF. The PDF representation in terms of fractional moments is especially accurate in the tails and this is very important in engineering problems, like estimating structural safety.
Applications of local fractional calculus to engineering in fractal time-space: Local fractional differential equations with local fractional derivative
Series expansion in fractional calculus and fractional differential equations
分数阶微积分是非整数阶的微分和积分的微积分。在最近的一篇论文（Annals of Physics 323（2008）2756-2778）中，分数阶微积分的基本定理被强调。基于这一定理，本文将分数阶级数展开法引入分数阶微积分中。定义了一类无穷分数阶可微函数的分数阶泰勒级数。进一步，基于我们的定义，我们推广了超几何函数，并导出了相应的微分方程。对于非无穷分数可微函数，我们发现其非无穷分数可微性是由多个分数阶指数引起的。我们展开函数的两个分数指数，并显示如何这种级数展开可以帮助解决分数阶微分方程。
Introduction to the techniques of the fractional calculus to investigate some models of the mathematical physics (in Portuguese)
在本文中，我们诉诸拉普拉斯变换方法，以显示其效率时，接近某些类型的分数阶微分方程。特别是，我们提出了一些这样的方法的应用时，适用于某些物理问题的线性粘弹性和谐振子的可能的概括，证明分数微积分是非常适合的建模和解决问题通常由普通整数微积分处理，具有前途的优势，能够提供更准确的理论预测，以适应与实验数据。OBS：葡萄牙文文章，在RBEF（Revista Brasileira de Ensino de Física）上发表。
