报告题目： On the Biconfluent Heun and its connection with Painleve IV equation
报告人： 虞国富教授 （上海交通大学0029cc金沙贵宾会）
时间：4月23日（星期二）下午1:50-2:50
地点：第五教学楼5205教室
摘要： In this talk, we first apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions of certain Hill equation are Liouvillian solutions. That are solutions obtainable by suitable differential field extensions construction. We establish a full correspondence between solutions of non-oscillatory type equations and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained for this Hill equation whose potential owns four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of biconfluent Heun equation. We further show that these Liouvillian solutions exhibit novel single and double orthogonality and a Fredholm integral equation over suitable integration regions in complex plane that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lame and Whittaker-Hill equations, discovered by Whittaker and Ince almost a century ago. In the second part, we discuss special solutions of Painleve IV equation, that comes from the pioneering works of Okamoto and Noumi. We report that the linear equation that gives raise to $P_{IV}$ via isomonodromy deformation in the classical works of Garnier and
Jimbo-Miwa also possesses special properties with the same parameter space as the $P_{IV}$. This is a joint work with Yik-Man Chiang and Chun-Kong Law.
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