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Period functions for Maass wave forms. I. (English) Zbl 1061.11021
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 $$\text{SL}_2(\mathbb Z)$$ and solutions to a certain three term functional equation. We list a special case of their results here.
Theorem: Let $$s\in \mathbb C$$ with $$\text{Re}(s)= \frac{1}{2}$$. Then there is an isomorphism between the space of Maass cusp forms with eigenvalue $$s(1-s)$$ on $$\Gamma$$ and the space of real-analytic solutions of the three-term functional equation $\psi(x) = \psi(x+1)+ (x+1)^{-2s} \psi \biggl(\frac{x}{x+1}\biggr)$ on $$\mathbb R_+$$, which satisfy the growth condition $\psi(x)=\text{o}(1/x) \;(x\rightarrow 0), \qquad \psi(x)=\text{o}(1) \;(x\rightarrow \infty).$

##### MSC:
 11F37 Forms of half-integer weight; nonholomorphic modular forms 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F72 Spectral theory; trace formulas (e.g., that of Selberg)
##### Keywords:
period functions; Maass cusp forms; functional equation
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