Hyperbolic Fourier coefficients of Poincaré series

Poincaré (Ann Fac Sci Toulouse Sci Math Sci Phys 3:125–149, 1912 ) and Petersson (Acta Math 58(1):169–215, 1932 ) gave the now classical expression for the parabolic Fourier coefficients of holomorphic Poincaré series in terms of Bessel functions and Kloosterman sums. Later, in 1941, Petersson intro...

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Bibliographic Details
Published in:The Ramanujan journal 2016-11, Vol.41 (1-3), p.465-518
Main Authors: O’Sullivan, Cormac, Taylor, Karen
Format: Article
Language:English
Subjects:
Bessel functions
Coefficients
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Fourier series
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
Numerical analysis
Sums
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Summary:Poincaré (Ann Fac Sci Toulouse Sci Math Sci Phys 3:125–149, 1912 ) and Petersson (Acta Math 58(1):169–215, 1932 ) gave the now classical expression for the parabolic Fourier coefficients of holomorphic Poincaré 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 Poincaré series. In this paper, we express the hyperbolic Fourier coefficients of Poincaré 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.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-016-9772-6