Let's consider generating function $$F(x) = (1+x)^r = \sum_{n=0} \binom{r}{n} x^n$$ And another generating function $$G(x) = (1+x^2)^r = \sum_{n=0} \binom{r}{n}x^{2n}$$ Please note those 2 functions differs only on $x^k$
After that let's take definition of binomial $$\binom{r}{n} = \frac{r^{\underline{n}}}{n!}$$ and apply it to our function $F(x)$ and $G(x)$ first.
$$F(x)=(1+x)^r=\sum_{n=0}^\infty r^{\underline n} \frac{x^n}{n!}$$ $$G(x)=(1+x^2)^r=\sum_{n=0}^\infty r^{\underline n} \frac{x^{2n}}{n!}$$
Whole question is about multiplying those functions.
To multiply $F(x)$ and $G(x)$ we have to have two function of form $\sum_{n=0}^\infty a_n x^n$, but our G(x) has form $\sum_{n=0}^\infty a_n x^{2n}$ So idea here is to make it right form, obviously.
Here i have doubts: should i treat this function as exponential generating function or ordinary generating function?
Because in first case it's seems like i'm iterating over sequence $\langle r^{\underline n} \rangle$ And in second case, if i want to make "correct" ordinary generating function i have to take $x=\sqrt{x}$ and apply it to function, it will give me obviously $G(\sqrt{x})=G_2(x)=(1+x)^r=\sum_{n=0}^\infty x^n$, but now $F(x)G(x) \neq F(x)G_2(x)$, so it doesnt seem like a good solution.
On the side note, $$F(x)G(x) = \sum_{n=0}^\infty ( \sum_{k=0}^n \binom{r}{k} \binom{r}{n-2k}) x^n$$
So my question is: how to work with such generating function? How to multiply them and how to change them to "right" form?
(Second side question : can we multiply our functions if both are in form $\sum_{n=0}^\infty a_n x^{kn}$ for some k being an integer/real number?)