While trying to solve the differential equation $$ a \frac{\partial^2y}{\partial a^2} + \frac{\partial y}{\partial a} -a y = e^{a\cos\phi}\sin\phi\;\;\;,\;\;\; y(0,\phi) = \phi $$
during a calculation I was doing, I found an intriguing infinite series form of the solution: $$ y(a,\phi) = \phi\sum_{k=-\infty}^\infty I_k(a)\operatorname{sinc}(k\phi). $$ This looks like the kind of thing that would have a "known" form, but no amount of massaging seemed to get Mathematica to give an answer (the above is in fact the result of said massaging--the initial form was much more complicated). Anyone have any ideas?