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A general Voronoï summation formula for the (metaplectic) double cover of   $\operatorname{GL}_2$   is derived via the representation theoretic framework à la Ichino–Templier. The identity is also formulated classically and used to establish Voronoï summation formulae for half-integral weight modular forms and Maaß forms.

Voronoï Summation for Half-Integral Weight Automorphic Forms