发表机构
University of California, San Diego（加利福尼亚大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明对每个模4余3的素数p，两个受限分拆函数在模3下满足由二次剩余决定的二分性同余，并给出无穷自相似同余族。
AI 中文摘要
设$f_{0,1,4}(n)$表示将$n$分拆为模5余0、1或4的部数且每个部数至多使用两次的分拆数，并类似地定义$f_{0,2,3}(n)$为模5余0、2或3的部数且每个部数至多使用两次的分拆数。我们证明，对于每个素数$p \equiv 3 \pmod 4$，存在由$20a(p) \equiv -9$和$20b(p) \equiv -1 \pmod{p^2}$确定的显式非负整数$a(p)$和$b(p)$，使得当$p \equiv \pm 1 \pmod 5$时，$f_{0,1,4}(p^2m + a(p))$和$f_{0,2,3}(p^2m + b(p))$分别模3同余于$f_{0,1,4}(m)$和$f_{0,2,3}(m)$；当$p \equiv \pm 2 \pmod 5$时，分别模3同余于$f_{0,2,3}(m)$和$f_{0,1,4}(m)$。由二次互反律，当5是模$p$的二次剩余时，这两个函数被保持；否则它们互换。最小情形为$f_{0,1,4}(9m) \equiv f_{0,2,3}(m)$和$f_{0,2,3}(9m+4) \equiv f_{0,1,4}(m)$。证明利用$(1-x)^{\ell} \equiv 1-x^{\ell} \pmod{\ell}$和Jacobi三重积，将每个生成函数模3化为Rogers-Ramanujan型theta函数的平方，然后利用$-1$是模$p$的二次非剩余这一事实来剖分所得的二元二次型。作为推论，我们得到：对于每个这样的$p$，在模$p$的等差数列上存在带例外的同余式，以及在模$p^{dn}$（其中$d=2$或$d=4$）的等差数列上，常数分别为$9(p^{dn}-1)/20$和$(p^{dn}-1)/20$的无穷自相似同余族。
英文摘要
Let $f_{0,1,4}(n)$ denote the number of partitions of $n$ into parts congruent to 0, 1 or 4 modulo 5 with each part used at most twice, and let $f_{0,2,3}(n)$ be defined analogously for parts congruent to 0, 2 or 3 modulo 5. We prove that for every prime $p \equiv 3 \pmod 4$ there are explicit non-negative integers $a(p)$ and $b(p)$, determined by $20a(p) \equiv -9$ and $20b(p) \equiv -1 \pmod{p^2}$, such that $f_{0,1,4}(p^2m + a(p))$ and $f_{0,2,3}(p^2m + b(p))$ are congruent modulo 3 to $f_{0,1,4}(m)$ and $f_{0,2,3}(m)$ when $p \equiv \pm 1 \pmod 5$, and to $f_{0,2,3}(m)$ and $f_{0,1,4}(m)$ when $p \equiv \pm 2 \pmod 5$. By quadratic reciprocity the two functions are preserved exactly when 5 is a quadratic residue modulo $p$, and interchanged otherwise. The smallest cases are $f_{0,1,4}(9m) \equiv f_{0,2,3}(m)$ and $f_{0,2,3}(9m+4) \equiv f_{0,1,4}(m)$. The proof reduces each generating function modulo 3 to the square of a Rogers-Ramanujan-type theta function by means of $(1-x)^{\ell} \equiv 1-x^{\ell} \pmod{\ell}$ and the Jacobi triple product, and then dissects the resulting binary quadratic form using the fact that $-1$ is a quadratic non-residue modulo $p$. As corollaries we obtain, for each such $p$, a congruence with exceptions on a progression of modulus $p$, and an infinite family of self-similarity congruences on progressions of modulus $p^{dn}$ with $d = 2$ or $d = 4$ and constants $9(p^{dn}-1)/20$ and $(p^{dn}-1)/20$.
