报告题目：Algebraic Properties of q-Difference Operators and Mock-Theta Functions

报告人：张长贵教授

报告时间：2022/03/23 14:00-15:30

报告地点：腾讯会议603-175-641

报告摘要：One calls q-difference operator any expression of the form $$\displaystyle a_0+a_1\sigma_q+...+a_n\sigma_q^n\in\mathcal{K}[\sigma_q],$$ where q is some given constant different from zero and one, $a_0$,..., $a_n$ are known functions belonging to a given fields $\mathcal{K}$ and where $\sigma_q$ denotes the q-shift operator defined by the relation $\sigma_qf(x) =f(qx)$. Such operators, appearing often in combinatorics or number theory,

may be viewed as q-analog of ordinary differential operators.

In our talk, we will consider the q-difference operators whose coefficients are polynomials or analytic functions at zero in the complex plane $\mathbf{C}$, this is to say, $\mathcal{K}=\mathbf{C}(x)$ or $\mathbf{C}[x^{-1}]\{x\}$. We will see how to write everyone of such operators as a product of one finite number of first order operators in some appropriate analytic setting. This factorization permits to define a q-summation procedure for solving the corresponding q-difference equation when zero becomes a non-Fuchsian singular point. By using this q-summation method, we will also give some remarks about Ramanujan's third order mock-theta functions.

报告人简介：张长贵，法国著名数学家，师从国际数学大师、法国科学院院士Ramis教授，2000-2001年任法国国家科学研究中心研究员，2001年起任法国里尔大学数学系教授。 主要从事特殊函数、常微分方程、偏微分方程以及差分方程的解析理论研究工作，并且通过采用复分析和方程的观点，来对解析数论与模函数，尤其是与q-级数相关的特殊函数进行研究。目前在《Advances in Mathematics》《Asterisque》等著名期刊上发表论文30多篇。