$p$-adic超几何函数$\widehat{\mathscr{F}}_{a,...,a}^{(\sigma)}(t)$的同余关系及其变换公式

关于$p$-adic超几何函数的变换公式

On transformation formulas of $p$-adic hypergeometric functions

本文回顾了三种$p$-adic超几何函数及其猜想变换公式，并证明了一个变换公式蕴含另一个。

Representations of the Dynamical Affine Quantum Group $U_{q,x}(\widehat{sl_2})=U_{q,λ}(\widehat{sl_2})$ and Hypergeometric Functions

The representation theory of the Hopf algebroid $U_{q,x}(\widehat{sl_2})=U_{q,λ}(\widehat{sl_2})$ is initiated and it is established that the intertwiner between the tensor products of dynamical evaluation modules is a well-poised balanced $_{10}W_9$ symbol, confirming a conjecture of Konno, that the degeneration of the elliptic $_{12}V_{11}$ series to $_{10}W_9$ can be proven based on the representation theory of $U_{q,x}(\widehat{sl_2})$, the degeneration of the elliptic algebra $U_{q,p}(\widehat{sl_2})$ as $p \to 0$.

On poly-Euler numbers of the second kind

For an integer $k$, define poly-Euler numbers of the second kind $\widehat E_n^{(k)}$ ($n=0,1,\dots$) by $$ \frac{{\rm Li}_k(1-e^{-4 t})}{4\sinh t}=\sum_{n=0}^\infty\widehat E_n^{(k)}\frac{t^n}{n!}\,. $$ When $k=1$, $\widehat E_n=\widehat E_n^{(1)}$ are {\it Euler numbers of the second kind} or {\it complimentary Euler numbers} defined by $$ \frac{t}{\sinh t}=\sum_{n=0}^\infty\widehat E_n\frac{t^n}{n!}\,. $$ Euler numbers of the second kind were introduced as special cases of hypergeometric Euler numbers of the second kind in \cite{KZ}, so that they would supplement hypergeometric Euler number…

p-adic Generalized Hypergeometric Equations from the Viewpoint of Arithmetic D-modules

We study the $p$-adic (generalized) hypergeometric equations by using the theory of multiplicative convolution of arithmetic $\mathscr{D}$-modules. As a result, we prove that the hypergeometric isocrystals with suitable rational parameters have a structure of overconvergent $F$-isocrystals.

高斯超几何函数与循环矩阵

Gaussian hypergeometric functions and cyclotomic matrices

基于高斯超几何函数与循环矩阵，研究非零平方元素的算术性质，推导出循环矩阵行列式的表达式，涉及雅可比和与高斯和的乘积形式。