How to Multiply Two Infinite Series Correctly? From my readings on the wikipedia, I was able to gather that the product of two infinite series $\sum_{i=0}^{\infty} a_{i}$ and $\sum_{j=0}^{\infty} b_{j} $ is outlined by the Cauchy Product. The cauchy product formula is explicitly shown below, $$ \sum_{i=0}^{\infty} a_i \sum_{j=0}^{\infty} b_j = \sum_{i=0}^{\infty} \sum_{j=0}^{i} a_{j} b_{i-j}. $$ 
However, I've saw this one Youtube video where multiplying two infinite series does not follow the Cauchy Product. 
Example 1: $$\sum_{n=0}^{\infty} H_{n}(x) \frac{t^n}{n!} \, \sum_{m=0}^{\infty} H_{m}(x) \frac{s^m}{m!} = \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{H_{n}(x) H_{m}(x)}{n! \, m!} t^n s^m $$
Picture 1: Involving the product of two generating functions. The youtube link to the video in Picture 1 is https://youtu.be/X7nlQFWv7bE?t=1m51s .
Is Example 1 valid and if so, why is it valid? Shouldn't the person have applied the Cauchy Product? Any resources that can point me in the right direction is appreciated.
 A: Quoting that very same Wikipedia article:

Convergence and Mertens' theorem
Let $(a_n)_n$ and $(b_n)_n$ be real or complex sequences. It was proved by Franz Mertens that, if the series $\sum_{n=0}^\infty a_n$ converges to $A$ and $\sum_{n=0}^\infty b_n$ converges to $B$, and at least one of them converges absolutely, then their Cauchy product converges to $AB$.
It is not sufficient for both series to be convergent; if both sequences are conditionally convergent, the Cauchy product does not have to converge towards the product of the two series [...]

(emphasis mine). The article then proceeds to show a counterexample in the case where the two series are only conditionally convergent.
In short: do not forget the assumptions.

Now, why is Example 1 valid? Well, you can definitely write, for all $N,M$,
$$
\sum_{n=0}^N H_n(x) \frac{t^n}{n!} \cdot 
\sum_{m=0}^M H_m(x) \frac{s^m}{m!}
 = \sum_{n=0}^N\sum_{m=0}^M H_n(x)H_m(x) \frac{s^m}{m!}\frac{t^n}{n!}
$$
Now, if both the left and the right expression converge when $N,M\to \infty$, you can get the identity. One typically has to prove it's the case, though.

Edit: a thing that may not be explicit: if you are asking whether it is possible to have two different expressions (Cauchy product and other) for the same product, well, yes. It is possible to have two expressions that look different, yet are equal.
