Constant terms and k-colored generalized Frobenius partitions


主讲人:崔素平 青海师范大学教授




主讲人介绍:崔素平,青海师范大学教授。青海省数学会副秘书长。曾获南开大学优秀毕业生、钟家庆数学奖等荣誉称号。一直从事组合及其应用等方向的研究,主要涉及同余式、仿theta函数、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》等重要期刊发表或接受发表论文20多篇。

内容介绍: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.