Bessel function explain this step

[rem]

bessel function please explain this step

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.

Homework Equations

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

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.

 

[huyen_vyvy]

can't interpret your question, what does ** mean? and _?

 

[Kurdt]

I'd imagine ** means to the power of and _ means a subscript.

$$\sum_{n=j}^\infty \left(\frac{-a}{4}\right)^{\frac{n}{n!}}(2n_{n+j})=(-1)^j e^{\frac{-a}{2}} I\left(\frac{a}{2}\right)$$

That seems to be the first equation.

 

[Kurdt]

Well I have no idea where to start answering the question. Bessel's equation has the following general form.

$$x^2y''+xy'+(x^2-\nu^2)y=0$$

This equation can be solved using the method of Frobenius. this is basically where we substitute the following series for the y's.

$$\sum_{m=0}^\infty a_mx^{m+r}=y(x)$$

Following the method of Frobenius you can gain the general solution for nu being integer and general nu. The modified Bessel function turns out to be the solution to the modified Bessel equation:

$$x^2y''+xy'-(x^2+\nu^2)y=0$$

So presumably your question will have involved an equation of the form of the modified Bessel function. Without any more information there is not much else anyone can tell you about this situation. They could probably tell you more about Bessel functions since I have only touched on the basics.