Topics in special functions

复和调和分析中的一些问题，3

Some topics in complex and harmonic analysis, 3

这些笔记简要地考虑了施瓦茨类的一些方面迅速减少光滑函数，回火分布，和调和函数的多项式增长。

A Short Course on Orthogonal Polynomials and Special Functions

These notes contain part of the lectures of an introductory course on orthogonal polynomials and special functions that I gave in the joint PhD Program in Mathematics UC|UP in the academic years 2015-2016 (at University of Porto) and 2016-2017 (at University of Coimbra). The notes were written for students who have never contacted with the above topics. Most results presented here can be found in the available bibliography at the end of each text/chapter, although in general more detailed proofs have been included (a few of them different from the ones presented in the source references), hopi…

复和调和分析中的一些问题，4

Some topics in complex and harmonic analysis, 4

这些笔记简要考虑了调和分布与施瓦茨类函数的卷积。

On Trilinear Oscillatory Integral Inequalities and Related Topics

Inequalities are established for certain trilinear scalar-valued functionals. These functionals act on measurable functions of one real variable, are defined by integration over two- or three-dimensional spaces, and are controlled in terms of Lebesgue space norms of the functions, and of negative powers of large parameters describing a degree of oscillation. Related sublevel set inequalities are a central element of the analysis. The main results and the main lines of their proofs are largely unchanged in this draft, but some details have been corrected. The analysis has already been extended …

Special polynomials related to the supersymmetric eight-vertex model: A summary

We introduce and study symmetric polynomials, which as very special cases include polynomials related to the supersymmetric eight-vertex model, and other elliptic lattice models with $Δ=\pm 1/2$. There is also a close relation to affine Lie algebra characters. After a natural change of variables, our polynomials satisfy a non-stationary Schrödinger equation with elliptic potential, which is related to the Knizhnik-Zamolodchikov-Bernard equation and to the canonical quantization of Painlevé VI. Moreover, specializations of our polynomials can be identified with tau functions of Painlevé VI, obt…