A remarkable congruence involving Appell polynomials

Proofs of Some Conjectures of Chan on Appell-Lerch Sums

On page 3 of his lost notebook, Ramanujan defines the Appell-Lerch sum $$ϕ(q):=\sum_{n=0}^\infty \dfrac{(-q;q)_{2n}q^{n+1}}{(q;q^2)_{n+1}^2},$$ which is connected to some of his sixth order mock theta functions. Let $\sum_{n=1}^\infty a(n)q^n:=ϕ(q)$. In this paper, we find a representation of the generating function of $a(10n+9)$ in terms of $q$-products. As corollaries, we deduce the congruences $a(50n+19)\equiv a(50n+39)\equiv a(50n+49)\equiv0~(\textup{mod}~25)$ as well as $a(1250n+250r+219)\equiv 0~(\textup{mod}~125)$, where $r=1$, $3$, and $4$. The first three congruences were conjectured …

由截断Appell级数产生的超同余式

Supercongruences Arising from Truncated Appell Series

研究截断Appell级数和截断q - Appell级数的同余性质，利用组合恒等式和‘创造性显微镜’方法，给出截断Appell级数\(F_1\)和\(F_2\)一些同余式的q - 类似，建立与\(F_4\)相关新q - 超同余式并猜想。

算术函数的贝尔变换：欧拉乘积、同余关系与多项式序列

Bell Transforms of Arithmetic Functions: Euler Products, Congruences, and Polynomial Sequences

本文提出一个统一的代数框架，利用形式贝尔变换将算术函数的狄利克雷卷积与无限欧拉型乘积的组合结构联系起来，通过分析指数生成函数的对数导数，建立了贝尔指数与莫比乌斯逆运算之间的显式映射，并应用于推导经典序列的精确消失性质和同余继承性质，包括拉马努金的tau函数和素数着色分区。此外，证明了逆贝尔变换能够无缝恢复经典分区递推关系，并提供了一个离散的组合引擎来生成特殊多项式族，包括经典的Appell和Sheffer序列。

On some remarkable congruences between two elliptic curves

We exhibit two non-isogenous rational elliptic curves with $17$-torsion subgroups isomorphic as Galois modules.

Andrews-Beck $NT$ 函数的变换性质与广义 Appell-Lerch 级数

Transformation properties of Andrews-Beck $NT$ functions and generalized Appell-Lerch series

本文加强并扩展了Mao关于 $NT$ 函数变换性质的工作，提供了Hickerson和Mortenson关于秩函数工作的类比，并推导出包含 $NT(r,m,n)$ 及 $M_ω(r,m,n)$ 的恒等式，同时给出了广义 Appell-Lerch 级数的新性质。