<s>[INST] Solution to curious infinite series How exactly does one find a closed form to:
$$ \sum_{i=0}^{\infty}\left[\frac{1}{i!}\left(\frac{e^2 -1}{2} \right)^i \prod_{j=0}^{i}(x-2j) \right]$$
When expanded it takes on the form
$$1 + \frac{e^2-1}{2}x + \frac{1}{2!} \left(\frac{e^2-1}{2} \right)^2x(x-2) + \frac{1}{3!} \left(\frac{e^2-1}{2} \right)^3x(x-2)(x-4)... $$
This doesn't appear to be similar to any type of Taylor Series I have seen before. 
 [/INST] Note that
$$\frac{1}{i!}\left(\frac{e^2-1}{2}\right)^i \prod_{j=0}^i (x-2j) = x \prod_{j=1}^i \frac{\frac{x}{2}-j}{j} (e^2-1)^i = x\binom{\frac{x}{2}-1}{i}(e^2-1)^i,$$
so
$$\sum_{i=0}^\infty \frac{1}{i!}\left(\frac{e^2-1}{2}\right)^i \prod_{j=0}^i (x-2j) = x\sum_{i=0}^\infty \binom{\frac{x}{2}-1}{i}(e^2-1)^i = x\left(1+(e^2-1)\right)^{\frac{x}{2}-1} = x e^{x-2}.$$
The clue is to recognise the binomial coefficient in $$\frac{1}{i!}\prod_{j=0}^i(x-2j).$$
If, as the expanded form indicates, the product is $\prod\limits_{j=0}^{i-1} (x-2j)$ instead of $\prod\limits_{j=0}^{i} (x-2j)$, we have
$$\frac{1}{i!}\prod_{j=0}^{i-1}\left(\frac{x}{2}-j\right) = \prod_{k=1}^i \frac{\frac{x}{2}+1-k}{k} = \binom{\frac{x}{2}}{i},$$
and
$$\sum_{i=0}^\infty \frac{1}{i!}\left(\frac{e^2-1}{2}\right)^i \prod_{j=0}^{i-1} (x-2j) = \sum_{i=0}^\infty \binom{\frac{x}{2}}{i}(e^2-1)^i = \bigl(1+(e^2-1)\bigr)^{\frac{x}{2}} = e^x.$$
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