Rademacher符号的双曲类比

立方过分割的 Rademacher 型精确公式与高阶 Turán 不等式

Rademacher-type exact formula and higher order Turán inequalities for cubic overpartitions

本文为立方过分割建立 Rademacher 型精确公式，利用显式误差项证明对数凹性，并进一步推出高阶 Turán 不等式、对数次可加性和广义对数凹性。

Function Field Analogue of Shimura's Conjecture on Period Symbols

In this paper we introduce the notion of Shimura's period symbols over function fields in positive characteristic and establish their fundamental properties. We further formulate and prove a function field analogue of Shimura's conjecture on the algebraic independence of period symbols. Our results enable us to verify the algebraic independence of the coordinates of any nonzero period vector of an abelian t-module with complex multiplication whose CM type is non-degenerate and defined over an algebraic function field. This is an extension of Yu's work on Hilbert-Blumenthal t-modules.

Rademacher-Carlitz多项式

Rademacher-Carlitz Polynomials

我们介绍和研究\emph{Rademacher-Carlitz polynomial} \[ \RC(u, v, s, t, a, b) := \sum_{k = \lceil s \rceil}^{\lceil s \rceil + b - 1} u^{\fl{\frac{ka + t}{b}}} v^k \]，其中$a, b \in \Z_{>0}$，$s, t \in \R$，和$u$和$v$是变量。这些多项式推广和统一了各种戴德金式的和多项式;最自然的是，人们可以将$\RC(u, v, s, t, a, b)$视为\emph{Dedekind-Rademacher sum} \[ \r_t(a,b) := \sum_{k=0}^{b-1}\left(\left(\frac{ka+t}{b} \right)\right) \left(\left(\frac{k}{b} \right)\right), \]的多项式模拟（在Carlitz的意义上），\emph{Dedekind-Rademacher sum}出现在各种数论、组合、几何和计算环境中。我们的结果有三种口味：我们证明了Rademacher-Carlitz多项式的互易定理，我们展示了它们是如何成为任何有理多面体$\mathcal{P}$的整数点变换\[ σ(x,y):=\sum_{(j,k) \in \mathcal{P…

k秩泰勒族：典范调和提升、主部与拉德马赫级数

The k-Rank Taylor Family: Canonical Harmonic Lifts, Principal Parts, and Rademacher Series

该研究针对k≥2的Garvan修正k秩矩，构造k秩泰勒族的典范调和提升，推导其主部、拉德马赫展开及精确递归，k=2时得到2标记Durfee符号数的精确公式。

Pochhammer符号的连续模拟

Continuous Analogues for the Pochhammer Symbol

本研究沿Díaz与Cano的组合对象连续化思路，提出Pochhammer符号的连续模拟，拓展了离散组合对象的连续对应理论。

An analogue of $k$-marked Durfee symbols for strongly unimodal sequences

In a seminal 2007 paper, Andrews introduced a class of combinatorial objects that generalize partitions called $k$-marked Durfee symbols. Multivariate rank generating functions for these objects have been shown by many to have interesting modularity properties at certain vectors of roots of unity. Motivated by recent studies of rank generating functions for strongly unimodal sequences, we apply methods of Andrews to define an analogous class of combinatorial objects called $k$-marked strongly unimodal symbols that generalize strongly unimodal sequences. We establish a multivariate rank generat…

On the q-binomial identities involving the Legendre symbol modulo 3

I use polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 . to prove six identities involving the $q$-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.

特征$p$中第$p^n$个希尔伯特符号的类似物（更新）

Analogues of the $p^n$th Hilbert symbol in characteristic $p$ (updated)

来自特征$\neq p$的$p$阶希尔伯特符号$(\cdot,\cdot )_p:K^\times/K^{\times p}\times K^\times/K^{\times p}\to{}_p{\rm Br}(K)$在特征$p$中有两个类似物，$$[\cdot,\cdot )_p:K/\wp (K)\times K^\times/K^{\times p}\to{}_p{\rm Br}(K),$$，其中$\wp$是Artin-Schreier映射$x\mapsto x^p-x$，和$$((\cdot,\cdot ))_p:K/K^p\times K/K^p\to{}_p{\rm Br}(K).$$符号$[\cdot,\cdot )_p$通过维特向量推广到$(\cdot,\cdot )_{p^n}$的类似物，$$[\cdot,\cdot )_{p^n}:W_n(K)/\wp (W_n(K))\times K^\times/K^{\times p^n}\to{}_{p^n}{\rm Br}(K).$$这里$W_n(K)$是$p$ -典型维特向量环的长度$n$的截断，即$W_{\{1,p,\ldots,p^{n-1}\}}(K)$。在本文中，我们构造类似的$((\cdot,\cdot ))_p$的推广。我们的构造涉及到Witt向量和Weyl代数。在这个过程中，我们得到了一类新的Weyl代…

算术中的三元符号

Triple symbols in arithmetic

本文受算术Chern–Simons理论启发，提出含本原$n$次单位根的数域上特征的模$n$三元符号上同调表述，验证其在$n=2$、$n=3$时分别与已有定义一致。

与富克斯群和高阶Eisenstein级数相关的模Dedekind符号

Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series

设$E(z,s)$是模群$SL(2,{\mathbb Z})$的非全纯Eisenstein级数。经典的Kronecker极限公式表明，$E(z,s)$在$s=1$上的Laurent展开式的第二项本质上是Dedekind eta函数的对数.这个eta函数是一个加权$1/2$模形式，而戴德金则用戴德金和表示其乘子系统。Goldstein的工作的基础上，我们将这些结果从模群推广到更一般的富克斯群$Γ$。eta函数的模拟有一个乘数系统，可以用一个映射$S:Γ\to {\mathbb R}$来表示，我们称之为模戴德金符号。我们通过极限公式得到了这些符号的详细性质。将通常的具有加性同态的幂的爱森斯坦级数从$Γ$扭曲到${\mathbb C}$，产生高阶爱森斯坦级数。这些级数共享$E(z,s)$的许多性质，尽管它们具有更复杂的自同构条件。它们满足Kronecker极限公式，并产生高阶Dedekind符号$S^*:Γ\to {\mathbb R}$。作为一般结果的应用，我们证明了亏格为1的同余群$Γ_0(N)$的高阶Dedekind符号是有理数的.

Some variations of a "divergent" Ramanujan-type $q$-supercongruence

Using the $q$-Wilf--Zeilberger method and a $q$-analogue of a "divergent" Ramanujan-type supercongruence, we give several $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. One of them is a $q$-analogue of a supercongruence recently proved by Wang: for any prime $p>3$, $$ \sum_{k=0}^{p-1} (3k-1)\frac{(\frac{1}{2})_k (-\frac{1}{2})_k^2 }{k!^3}4^k\equiv p-2p^3 \pmod{p^4}, $$ where $(a)_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer symbol.