Hyperbolic Fourier coefficients of Poincar\'e series

TL;DR

This paper extends classical Fourier coefficient formulas for Poincaré series by expressing hyperbolic Fourier coefficients in terms of hypergeometric series and generalized Kloosterman sums, enabling numerical computation.

Contribution

It introduces a new explicit formula for hyperbolic Fourier coefficients of Poincaré series using hypergeometric functions and generalized Kloosterman sums, broadening the classical theory.

Findings

Derived explicit formulas for hyperbolic Fourier coefficients

Connected hyperbolic Kloosterman sums to lattice points on hyperbolas

Enabled numerical computation of hyperbolic Fourier coefficients

Abstract

Poincar\'e in 1911 and Petersson in 1932 gave the now classical expression for the parabolic Fourier coefficients of holomorphic Poincar\'e series in terms of Bessel functions and Kloosterman sums. Later, in 1941, Petersson introduced hyperbolic and elliptic Fourier expansions of modular forms and the associated hyperbolic and elliptic Poincar\'e series. In this paper we express the hyperbolic Fourier coefficients of Poincar\'e series, of both parabolic and hyperbolic type, in terms of hypergeometric series and Good's generalized Kloosterman sums. In an explicit example for the modular group, we see that the hyperbolic Kloosterman sum corresponds to a sum over lattice points on a hyperbola contained in an ellipse. This allows for numerical computation of the hyperbolic Fourier coefficients.