Glasgow Mathematical Journal

Research Article

Rational period functions of the modular group II

Marvin I. Knoppa1

a1 Temple University, Philadelphia

In the earlier article [7], I began the study of rational period functions for the modular group Γ(l) = SL(2, Z) (regarded as a group of linear fractional transformations) acting on the Riemann sphere. These are rational functions q(z) which occur in functional equations of the form

S0017089500004663_eqn1

where kxs2208Z and F is a function meromorphic in the upper half-plane xs210B, restricted in growth at the parabolic cusp ∞. The growth restriction may be phrased in terms of the Fourier expansion of F(z) at ∞:

S0017089500004663_eqn2

with some μxs2208Z. If (1.1) and (1.2) hold, then we call F a modular integral of weight 2k and q(z) the period of F.

(Received January 30 1980)

Footnotes

† This research was supported in part by National Science Foundation grant MCS-7903471.