What is the Generating Function for Bessel Functions?

[Mark Brewer]

Homework Statement

Show that the Bessel functions J_n(x) (where n is an integer) have a very nice generating function, namely,

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

Hint. Starting from the recurrence relation

J_n+1(x) + J_n-1(x) = (2n/x)J_n(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)

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

Any guidance would be helpful.

 

[Mark Brewer]

Thank you, Greg. You can bump this post.