Umbral-Algebraic Methods and Asymptotic Properties of Special Polynomials

利用umbral（本影）技术揭示超几何函数的新视角

Unveiling new perspectives of hypergeometric functions using umbral techniques

本文引入Gauss超几何函数的umbral表述并推广至其广义形式，证明该方法能有效简化相关计算，尤其适合求解含超几何函数及其他特殊函数的积分。

Symbolic calculus and integrals of Laguerre polynomials

An umbral type formalism is used to derive integrals involving products of Laguerre polynomials and other special functions.

二元退化 Hermite 多项式及其部分正交性的运算本影方法

Operational-umbral approach to bivariate degenerate Hermite polynomials and their partial orthogonality

本文运用运算微积与本影形式体系研究二元退化 Hermite 多项式，重新推导并扩展其微分方程、单项性、运算恒等式及部分正交性，获得负阶情形的新结果。

Symbolic methods for the evaluation of sum rules of Bessel functions

The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermore, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics.

Hermite Calculus

We develop a new method of umbral nature to treat blocks of Hermite and of Hermite like polynomials as independent algebraic quantities. The Calculus we propose allows the formulation of a number of practical rules allowing significant simplifications in computational problems.

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature $(0,n)$ the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function ({\bf EGF}) carrying the {\it continuum} Dirac operator $D=\sum_{j=1}^n\e_j\partial_{x_j}$ together with the Lie-algebraic representation of raising and lowering operators acting on the lattice $h\BZ^n$ is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership o…

A note on some identities involving special functions from the hypergeometric solution of algebraic equations

From the algebraic solution of $x^{n}-x+t=0$ for $n=2,3,4$ and the corresponding solution in terms of hypergeometric functions, we obtain a set of reduction formulas for hypergeometric functions. By differentiation and integration of these results, and applying other known reduction formulas of hypergeometric functions, we derive new reduction formulas of special functions as well as the calculation of some infinite integrals in terms of elementary functions.

特殊仿射康托集的代数差特征化

Characterization of the algebraic difference of special affine Cantor sets

本文研究了由参数l,r,p∈N生成的自相似康托集S-康托集，分析了其代数差的结构特征，揭示了其可能为区间、康托集、L-康托瓦尔、R-康托瓦尔或M-康托瓦尔等形式。