主講人：崔素平 青海師范大學教授
時間：2023年6月14日10：30
地點：三號樓332室
舉辦單位：數理學院
主講人介紹：崔素平，青海師范大學教授。青海省數學會副秘書長。曾獲南開大學優(yōu)秀畢業(yè)生、鐘家慶數學獎等榮譽稱號。一直從事組合及其應用等方向的研究,主要涉及同余式、仿theta函數、分拆的秩等。在《Advances in Mathematics》、《Advances in Applied Mathematics》、《The Ramanujan Journal》、《International Journal of Number Theory》、《Journal of the Australian Mathematical Society》等重要期刊發(fā)表或接受發(fā)表論文20多篇。
內容介紹：In his 1984 AMS memoir, Andrews introduced the family of $k$-colored generalized Frobenius partition functions. For any positive integer k, let $c\phi_k(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. Among many other things, Andrews proved that for any $n\geq0$, $c\phi_2(5n+3)\equiv0\pmod{5}$. Since then, many scholars considered subsequently congruence properties for various $k$-colored generalized Frobenius partition functions, typically with a small number of colors. In 2019, Chan, Wang and Yang studied systematically arithmetic properties of $\textrm{C}\Phi_k(q)$ with $2\leq k\leq17$ by utilizing the theory of modular forms, where $\textrm{C}\Phi_k(q)$ denotes the generating function of $c\phi_k(n)$. We notice that many coefficients in the expressions of $\textrm{C}\Phi_k(q)$ are not integers. In this paper, we first observe that $\textrm{C}\Phi_k(q)$ corresponds to the constant term of a family of bivariable functions, then establish a general symmetric and recurrence relation on the coefficients of these bivariable functions. Based on this relation, we next derive many bivariable identities. By extracting or computing the constant terms of these bivariable identities, we establish the representations of $\textrm{C}\Phi_k(q)$ with integral coefficients. Moreover, we prove some infinite families of congruences satisfied by $c\phi_k(n)$ where $k$ is allowed to grow arbitrary large.