特殊函数论中一些问题的研究

Special functions can be defined as useful mathematical functions appearing frequently in science and engineering. They are roughly classfied into two categories, the classic special functions and q-special functions. Noticeable examples for the classical special functions are Hermite polynomials, Laguerre polynomials, Airy functions and Bessel functions etc. q-Special functions are the natural generalizations to classical special functions, they include Stieltjes-Wigert polynomials, q-Laguerre polynomials, q^(-1)-Hermite polynomials, Ramanujan's entire function(a.k.a q-Airy function) and q-Bessel functions etc. Their formulas and asymptotics have been crucial to the success of many projects in science and engineering. In the concluding project (2017), by applying Fourier analysis and other mathematical techniques we have made significant progress in deriving a wide range of formulas and asymptotics for many interesting special functions. Based on the success of the previous project, starting with 8 concrete problems, we propose to study special function problems arising from mathematics and physics, especially for those related to combinatorics and functions of a complex variable.

特殊函数是那些在科学工程中经常出现而且非常有用的数学函数，它包括经典特殊函数和它们的q模拟, 后者是对前者的自然推广。正交多项式如Hermite、Laguerre、Stieltjes-Wigert、q-Laguerre、q^(-1)-Hermite、 Airy、Bessel、Ramanujan 函数和q-Bessel 都是特殊函数的范例, 它们的公式和渐近性质一直是科学家和工程师所关注的重要问题。在2017年即将结题的项目中应用Fourier 分析方法与其他数学技巧我们研究了q-正交多项和其他q-级数所满足的公式和渐近性质并取得了不小进展。本项目将以前面项目的工作积累为基础，从八个具体问题入手，进一步研究在数学和物理学中出现的一些特殊函数相关的问题，特别是与组合数学和函数论相关的特殊函数问题。

特殊函数是那些在科学工程中经常出现而且非常有用的数学函数，它包括经典特殊函数和 它们的q模拟, 后者是对前者的自然推广。正交多项式如Hermite、Laguerre、Stieltjes-Wigert、q-Laguerre、q^(-1)-Hermite、 Airy、Bessel、Ramanujan、$\zeta$ 函数和q-Bessel 都是特 殊函数的范例, 它们的公式、零点分布和渐近性质一直是科学家和工程师所关注的重要问题。在即将结题的项目中继续应用Fourier 分析方法与其他数学技巧（如算子理论）我们研究了q-特殊函数所满足的公式与单调性并取得了不小进展. 遗憾的是，将近一半工作没有来得及发表。