论文标题
$ p $ -ADIC设置中的超几何功能的零
Zeros of hypergeometric functions in the $p$-adic setting
论文作者
论文摘要
令$ p $为奇数，$ \ mathbb {f} _p $是带有$ p $元素的有限字段。 McCarthy \ Cite {McCarthy-Pacific}在$ P $ ADIC设置中启动了一项对超几何功能的研究。该功能可以理解为高斯高几何函数的$ p $ - adiC类似物，也可以将Greene超小几幅功能的某种扩展超过$ \ Mathbb {f} _p $。在本文中，我们调查了麦卡锡的两个通用家族的超几何函数的值，由$ {_ ng_n}（t）$表示，以及$ {_ n \ widetilde {g} _n} _n}（t）$ for $ n \ geq3 $，以及$ n \ geq3 $，以及$ t \ in \ in \ in \ mathbbbbbb {f f} _p $。函数$ {_ ng_n}（t）$的值当然取决于$ t $是$ n $ - th $ th power down down doptue modulo $ p $。同样，函数的值$ {_ n \ widetilde {g} _n}（t）$依赖于$ y^n-y^n-y^{n-y^{n-y^{n-y^{n-1}+\ frac {（n-1）{（n-1）^{n-1} t} t} t} n^n^n} {n^n} {n^n} {n^n} {n^n} {n^n} {n^n} {n} {这些结果概括了Whipple定理和Dixon的经典超小几阶系列定理的特殊情况。我们检查函数的零零$ {_ ng_n}（t）$，以及$ {_ n \ widetilde {g} _n}（t）$ over $ \ mathbb {f} _p $。此外，我们研究了$ {_ ng_n}（t）= 0 $的$ t $的值。例如，我们表明有很多数量$ {_ {_ {2k} g_ {2k}}}（ - 1）= 0 $。相比之下，对于$ t \ neq0 $，$ {_ {_ {2k} \ widetilde {g} _ {2k}}}}（t）= 0 $都没有质量。
Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. McCarthy \cite{mccarthy-pacific} initiated a study of hypergeometric functions in the $p$-adic setting. This function can be understood as $p$-adic analogue of Gauss' hypergeometric function, and also some kind of extension of Greene's hypergeometric function over $\mathbb{F}_p$. In this paper we investigate values of two generic families of McCarthy's hypergeometric functions denoted by ${_nG_n}(t)$, and ${_n\widetilde{G}_n}(t)$ for $n\geq3$, and $t\in\mathbb{F}_p$. The values of the function ${_nG_n}(t)$ certainly depend on whether $t$ is $n$-th power residue modulo $p$ or not. Similarly, the values of the function ${_n\widetilde{G}_n}(t)$ rely on the incongruent modulo $p$ solutions of $y^n-y^{n-1}+\frac{(n-1)^{n-1}t}{n^n}\equiv0\pmod{p}$. These results generalize special cases of $p$-adic analogues of Whipple's theorem and Dixon's theorem of classical hypergeometric series. We examine zeros of the functions ${_nG_n}(t)$, and ${_n\widetilde{G}_n}(t)$ over $\mathbb{F}_p$. Moreover, we look into the values of $t$ for which ${_nG_n}(t)=0$ for infinitely many primes. For example, we show that there are infinitely many primes for which ${_{2k}G_{2k}}(-1)=0$. In contrast, for $t\neq0$ there is no prime for which ${_{2k}\widetilde{G}_{2k}}(t)=0$.
