Quasi-elliptic functions

Abstract

We study certain generalizations of elliptic functions, namely quasi-elliptic functions. Let $p = e^{i\alpha},$ $q = e^{i\beta},$ $\alpha,\, \beta \in \mathbb{R}.$ A meromorphic in $\mathbb{C}$ function $g$ is called quasi-elliptic if there exist $\omega_1, \omega_2 \in \mathbb{C}^{*},$ $\mathrm{Im} \frac{\omega_2}{\omega_1} > 0,$ such that $g(u+\omega_1)=pg(u)$, $g(u+\omega_2)=qg(u)$ for each $u\in\mathbb{C}$. In the case $\alpha = \beta = 0 \mod 2\pi$ this is a classical theory of elliptic functions. A class of quasi-elliptic functions is denoted by $\mathcal{QE}.$ We show that the class $\mathcal{QE}$ is nontrivial. For this class of functions we construct analogues $\wp_{\alpha \beta}$, $\zeta_{\alpha \beta}$ of $\wp$ and $\zeta$ Weierstrass functions. Moreover, these analogues are in fact the generalizations of the classical $\wp$ and $\zeta$ functions in such a way that the latter can be found among the former by letting $\alpha=0$ and $\beta=0$. We also study an analogue of the Weierstrass $\sigma$ function and establish connections between this function and $\wp_{\alpha \beta}$ as well as $\zeta_{\alpha \beta}$. Let $q, p \in\mathbb{C}^*,$ $|q|<1.$ A meromorphic in $\mathbb{C^{*}}$ function $f$ is said to be $p$-loxodromic of multiplicator $q$ if for each $z \in \mathbb{C}^{*}$ $f(qz) = pf(z).$ We obtain telations between quasi-elliptic and $p$-loxodromic functions.