Period Functions for Maass Wave Forms. I

This paper investigates a connection (first explored by the first author [Invent. Math. 127, No. 2, 271--306 (1997; Zbl 0922.11043)]) between Maass wave forms for \2(\ Z) and solutions to a certain three term functional equation. We list a special case of their results here.\ Theorem: Let s\∈ \ C with \(s)= ½. Then there is an isomorphism between the space of Maass cusp forms with eigenvalue s(1-s) on Γ and the space of real-analytic solutions of the three-term functional equation \ψ(x) = \ψ(x+1)+ (x+1)^-2s \ψ \(\+1\) on \ R+, which satisfy the growth condition \ψ(x)=\(1/x) \\ (x→ 0), \ \ψ(x)=\(1) \\ (x→ ∞).