Recall the Voronoi summation formula for GL(2): if
f(x) is a Schwartz function which vanishes to infinite order at the
are the coefficients of a modular or Maass form for SL(2,
a, c relatively prime integers,
This formula has many analytic uses for dualizing sums of
coefficients (e.g. subconvexity, together with trace formulas).
It can be derived from the standard L-function (if a=0), and
from its twists (general a,c).
The usual proofs involve special
functions, but the final answer does not.
Is that avoidable?