# Bessel Generating Function

1. Nov 13, 2015

### Mark Brewer

1. The problem statement, all variables and given/known data
Show that the Bessel functions Jn(x) (where n is an integer) have a very nice generating function, namely,

G(x,t) := ∑ from -∞ to ∞ of tn Jn(x) = exp {(x/2)((t-T1/t))},

Hint. Starting from the recurrence relation

Jn+1(x) + Jn-1(x) = (2n/x)Jn(x),

show that G(x,t) satisfies the differential equation (t+1/t)G(x,t) = (2t/x)∂G/∂t. Partially integrate this equation and fix the unknown function of x by the requirement that the coefficient of t0 be J0(x)

3. The attempt at a solution
This is the setup I have so far. I'm not sure if this is the correct way to go. If it is please let me know so I can continue down the pipe line.

∫xe^((x/2)(t-1/t))∂t = ∫((2t^2)/(t^2)+1)∂G

2. Nov 19, 2015