The classical Petersson formula gives a formula for $$\sum_{c>0} \frac{S(m, n, c)}{c} J(m, n, c)$$ for $J$ an explicit Bessel function. Is there a similar formula existing for different weights or functions, for instance $$\sum_{c>0} \frac{S(m, n, c)}{c^k} J(m, n, c)$$ for integers (or real numbers) $k$?
2026-03-27 07:57:05.1774598225
Petersson formula with powers
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