On Algebraic Functions

TL;DR

This paper explores algebraic functions with Moebius-periodic rational coefficients, showing how they can be expressed using theta and eta functions, with applications to elliptic moduli and special series.

Contribution

It introduces a framework for representing algebraic functions with Moebius-periodic coefficients via theta and eta functions, connecting various classical special functions.

Findings

Functions take algebraic values under certain conditions

Representation via theta and eta functions is possible

Includes applications to elliptic moduli and continued fractions

Abstract

In this note we consider functions with Moebius-periodic rational coefficients. These functions under some conditions take algebraic values and can be recovered by theta functions and the Dedekind eta function. Special cases are the elliptic singular moduli, the Rogers-Ramanujan continued fraction, Eisenstein series and functions associated with Jacobi symbol coefficients.