Power Series Identity for Bessel Functions

[matematikuvol]

Homework Statement

Show
$$e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=-\infty}J_n(x)t^n$$

Homework Equations

$$J_k(x)=\sum^{\infty}_{n=0}\frac{(-1)^n}{(n+k)!n!}(\frac{x}{2})^{2n+k}$$

The Attempt at a Solution

Power series product
$$(\sum^{\infty}_{n=0}a_n)\cdot (\sum^{\infty}_{n=0} b_n)=\sum^{\infty}_{n=0}c_n$$
where
$$\sum^n_{i=0}a_ib_{n-i}$$
$$(\sum^{\infty}_{n=0}\frac{1}{n!}(\frac{x}{2})^nt^n)\cdot (\sum^{\infty}_{n=0} \frac{(-1)^n}{n!}(\frac{x}{2})^nt^{2i-n})=\sum^{\infty}_{n=0}c_n$$
where
$$c_n=\sum^{n}_{i=0}\frac{(-1)^{n-i}}{i!(n-i)!}(\frac{x}{2})^nt^{2i-n}$$
So
$$e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=0}\sum^{n}_{i=0} \frac {(-1)^{n-i}}{i!(n-i)!}(\frac{x}{2})^nt^{2i-n}$$
I don't see how to get from here
$$e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=-\infty}J_n(x)t^n$$

 

[haruspex]

You'll need to extract t from the inner sum. To do that, you need to change the summation variables so that only one of them (the outer sum variable) shows up in t's exponent. It's a bit messy because it can take either sign, so it might be best to handle the ranges 2i>=n, 2i<n separately.

 

[matematikuvol]

Is there some other way to do it. Easier? Here I go from Taylor to Laurent series.

 

[haruspex]

It's probably not that hard. Substitute for n with n=2i-k. Your double sum is over n>=i, so that becomes i>=k. Sum over i first, and see if that produces J_k(x). I doubt there's an easier way.