p-Elliptic functions

Анотація

We investigate $p$-elliptic functions (meromorphic in $\mathrm{ℂ}$ functions satisfying the conditions $g(u+{\omega }_1)=g\left(u\right)$, $g(u+{\omega }_2)=pg\left(u\right)$, ${\omega }_1, {\omega }_2\in \mathrm{ℂ}$, $p\in \mathrm{ℂ}\backslash \left\{0\right\}$, $Im\frac{{\omega }_2}{{\omega }_1}>0$. In the case $p=1$ this is the classical theory of elliptic functions. $p$-Elliptic functions generate so-called $p$-loxodromic functions and vice versa. We  generalize the elliptic Weierstrass $\wp$-function and find the corresponding $p$-loxodromic function in the case $\left|p\right|=1$.