Operational vs. Umbral Methods and Borel Transform

Umbral Calculus, a Different Mathematical Language

This thesis is intended to provide an account of the theory and applications of Operational Methods that allow the "translation" of the theory of special functions and polynomials into a "different" mathematical language. The language we are referring to is that of symbolic methods, largely based on a formalism of umbral type which provides a tremendous simplification of the derivation of the associated properties. The strategy we will follow is that of establishing the rules to replace higher trascendental functions in terms of elementary functions and to take advantage from such a recasting.

关于影算子的公式和分数指数

On formulas and fractional exponents for umbral operators

本文提出新的影算子公式，揭示其与迭代理论的联系，定义了影算子的分数次幂，并综合了操作微积分中的多项成果，扩展了拉格朗日多项式。

影子理论与形式幂级数的代数

Umbral theory and the algebra of formal power series

本文通过形式幂级数框架，构建了影子理论的严谨表述，引入影子算子并分析其收敛条件，从而为处理发散的影子恒等式提供理论基础，并提出新的影子图像用于高斯三角函数。

The spherical Bessel and Struve functions and operational methods

We review some aspects of the theory of spherical Bessel functions and Struve functions by means of an operational procedure essentially of umbral nature, capable of providing the straightforward evaluation of their definite integrals and of successive derivatives. The method we propose allows indeed the formal reduction of these family of functions to elementary ones of Gaussian type. We study the problem in general terms and present a formalism capable of providing a unifying point of view including Anger and Weber functions too. The link to the multi-index Bessel functions is also briefly d…

Operational Methods in the Study of Sobolev-Jacobi Polynomials

Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover…

勒雷、勒赫和 Legendre 赤函数与广义 Borel-Le Roy 变换

Le Roy, Lerch and Legendre chi functions and generalised Borel-Le Roy transform

本文基于最近的指示乌姆布拉理论（IUT）重新表述，统一研究勒雷、勒赫和 Legendre 赤函数的性质及推广，探讨通过适当求和技术扩展到发散级数的正式方法。