Quasi-elliptic functions

Abstract

Abstract. We study certain generalizations of elliptic functions, namely quasi-elliptic functions.> 0, such that ( + 1 ) = ( ), ( + 2 ) = ( ) for each ∈ C. In the case = = 0 mod 2 this is a classical theory of elliptic functions. A class of quasi-elliptic functions is denoted by ℰ. We show that the class ℰ is nontrivial. For this class of functions we construct analogues ℘ , of ℘ and Weierstrass functions. Moreover, these analogues are in fact the generalizations of the classical ℘ and functions in such a way that the latter can be found among the former by letting = 0 and = 0. We also study an analogue of the Weierstrass function and establish connections between this function and ℘ as well as . Let , ∈ C * , | | < 1. A meromorphic in C * function is said to be -loxodromic of multiplicator if for each ∈ C * ( ) = ( ). We obtain telations between quasi-elliptic and -loxodromic functions.