How Do Bessel Functions Relate to Fourier Transforms in SHM Problems?

[rem]

bessel function please explain

1. Homework Statement

summation limits (n=j to infinity) (-a/4)**n/n!(2n_
n+j)
=(-1)**j e**(-a/2) I(a/2) where j>=1 the rest are constants and I is summation index
i was just solving a SHM problem involving Fourier transform in which this happens to be one of the steps involving the solution. i got this solution from mathematica it seems it's a modified bessel function of 1st kind.can anyone please explain this.i know nothing about bessel function and my basics in mathematics is bit shaky.

2. Homework Equations
iv(x)=summation limits 0 to infinity.(1/s!(s+1)!)*(x/2)^(2s+v)


3. The Attempt at a Solution

i read book by arfken and others but still can't understand.now it's more confusing.i got so confused with this step i can no longer remember the actual problem.

 

[Kurdt]

This is a double post. the exact same question with some responses was posted here:

https://www.physicsforums.com/showthread.php?t=176728