# Motivation behind the appearance of Bessel functions in partial trace formulas

Bessel functions occur naturally on the Kloosterman side (or geometric side) of Petersson's formula and Kuznetsov's formula. Is there an intuitive explanation for their appearance? For instance, is it coincidence that Bessel functions arise when studying radially symmetric eigenfunctions of the Euclidean Laplacian (and the spectral side of Kuznetsov's formula is an average of eigenfunctions of the hyperbolic Laplacian)?

• Coincidence with what? Bessel functions are interesting, because they arise in connection with eignfunctions. Would you mind expanding your question? – András Bátkai Aug 17 '13 at 21:27
• See this question:mathoverflow.net/questions/105971/… – Stopple Aug 19 '13 at 23:03
• @AndrásBátkai: You don't find it coincidental that the eigenfunction of one type of Laplacian occurs in trace formulae involving another type of Laplacian? I'm asking whether there's a good reason for why Bessel functions "should" appear in these formulas. – Z. Afana Aug 20 '13 at 19:25