Multiplication of Power Series [Complex]

Question

I know that given two power series, $A$ and $B$, such that $A= \sum_{n=0}^{\infty} a_n x^n$ and $B= \sum_{n=0}^{\infty} b_n x^n$ then,

$AB= \sum_{n=0}^{\infty} (\sum_{i=0}^{n}a_i*b_{n-i})x^n$

But then I was wondering, if $A$ is of the form $A= \sum_{n=0}^{\infty} a_n x^{mn}$ and $B= \sum_{n=0}^{\infty} b_n x^n$ for some $m \in [2,3,...]$ then,

Will the product of A and B, be

$AB= \sum_{n=0}^{\infty} (\sum_{i=0}^{n}a_i*b_{n-i})x^{mn}$

Answer 1

Hint: In the first formula, where did the exponent on $x$ come from? (Further hint: it's the sum of the indices of $a$ and $b$) Why?

Have you tried a concrete example, with finite power series (i.e., polynomials) and $m = 2$? For instance, have you checked out the case

$$ A = 3x^2 + 2 x^4 + 11 x^6 \\ B = 1 + 3x + 2x^2 $$ by explicitly comparing your two formulas?

Answer 2

No, otherwise your power series would contain only powers with multiples of $m$ which is clearly wrong.

I don't see a "nice" closed form besides $$\sum_n \left(\sum_{mi+j=n} a_ib_j\right) x^n=\sum_n \left(\sum_{i=0}^{\lfloor n/m\rfloor} a_ib_{n-mi}\right) x^n$$