oefficient in that sum".
This was the approach of Bessel and other 19th century mathematicians, but it is not very satisfying for some reason.  Perhaps history can help?
These functions were
studied by Daniel Bernoulli (the vibrations of a heavy chain, 1738), 
Euler (the vibrations of a membrane, 1764), Lagrange (planetary motion, 1770),
and Fourier (the motion of heat in a cylinder, 1822); Bessel studied them in the context of Kepler's equation, and wrote a monograph
about them in 1824. 
For an explanation