Bessel functions (idea)

The Bessel functions may be defined by means of the generating function:
g(x,t)=e^x/2*(t-1/t) = J_ntⁿ
Equating coefficients of t^n on both sides we obtain the series expansion of the Bessel functions:
J_n = sum((-1)^k * (x/2)^(n+2k)/(k!(n+k)!))

Differentiating the generating function w.r.t t we get
(1+t^(-2))g(x,t)=sum(n*J_ntⁿ)
equating coefficients of t^n on both sides we obtain the recursion relation
(2n/x)*J_n = J_n-1 + J_n+1 Differentiating the generating function w.r.t x gives
2*J'_n = J_n-1 - J_n+1

These recursion relations may now be used to prove that J_n satisfies Bessel's differential equation:
x²J_n + xJ_n+(x² - n²)J_n = 0 The generating function may be used to prove many other properties. For exampe parity
J_n(-x)=(-1)^{nn(x) J_-n(x) = (-1)nn(x)}

^{If Bessel's differential equation is divided by x it becomes self-adjoint and thus we expect its eigenfunctions to be orthogonal(with a weighting function of x). Orthogonality for the Bessel functions is rather special. Here is not J_n and J_m which are orthogonal but J_p(a_n) and J_p(a_m) which are orthogonal where a_n and a_m are two zeroes of J_p. Thus we may expand any function in a series of zeroes of one Bessel function.}

^{Arfken is a good reference for Special functions in general. Of course the Bible of special functions is the book by Abramowitz.}