q-特殊函数概览
q-Special functions, an overview
AI总结：
本文概述q-特殊函数理论，涵盖q-超几何级数、Askey–Wilson多项式、Macdonald多项式及椭圆超几何级数，并讨论其在数论、组合、量子群和统计力学中的应用。
摘要：
本文简要介绍q-特殊函数，即经典特殊函数的q-模拟。这里q是一个形变参数，通常0<q<1，而q=1对应经典情形。将要处理的主要主题是q-超几何级数，包括一些选定的求值与变换公式，以及q-超几何正交多项式，尤其是Askey–Wilson多项式。一些较新的主题，如非对称模拟和q=-1极限，也将被讨论。在多变量情形下，我们讨论与根系相关的Macdonald多项式，特别是A_n和BC_n情形。椭圆超几何级数理论也得到一定关注。还将讨论q-级数在数论和组合学中的出现。最后，我们指出其在量子群、Chevalley群、仿射Lie代数和统计力学中的应用与解释。
英文摘要：
This article gives a brief introduction to $q$-special functions, i.e., $q$-analogues of the classical special functions. Here $q$ is a deformation parameter, usually $0<q<1$, where $q=1$ is the classical case. The main topics to be treated are $q$-hypergeometric series, with some selected evaluation and transformation formulas, and the $q$-hypergeometric orthogonal polynomials, most notably the Askey--Wilson polynomials. Some newer topics as nonsymmetric analogues and $q=-1$ limits will also be addressed. In several variables we discuss Macdonald polynomials associated with root systems, in particular the $A_n$ and the $BC_n$ case. The theory of elliptic hypergeometric series also gets some attention. The occurrence of $q$-series in number theory and combinatorics will be discussed. Finally we indicate applications and interpretations in quantum groups, Chevalley groups, affine Lie algebras and statistical mechanics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
