I'm a bit lost in trying to take the product of a power series between two functions $f(x) = \frac{1}{(1-x)^k}$ and $g(x)=\frac{1}{(1-x^r)^k}$.
I know both can be expanded to, \begin{align*} \frac{1}{(1-x)^k} = \sum_{n=0}^{\infty}\binom{n+k-1}{k-1}x^n \\ \frac{1}{(1-x^r)^k} = \sum_{n=0}^{\infty}\binom{n+k-1}{k-1}x^{rn}. \end{align*}
This is where I get a bit lost. Since the exponents for $x$ are different, it seems like the Cauchy product is not applicable. Is there a further step I'm missing?
$$ \eqalign{ & \sum\limits_{0\, \le \,n} {\left( \matrix{ n + k - 1 \cr k - 1 \cr} \right)x^{\,n} } \sum\limits_{0\, \le \,m} {\left( \matrix{ m + k - 1 \cr k - 1 \cr} \right)x^{r\,m} } = \cr & = \sum\limits_{0\, \le \,m} {\sum\limits_{0\, \le \,n} {\left( \matrix{ n + k - 1 \cr k - 1 \cr} \right)\left( \matrix{ m + k - 1 \cr k - 1 \cr} \right)x^{r\,m + n} } } = \cr & = \sum\limits_{0\, \le \,s} {\left( {\sum\limits_{0\, \le \,m\, \le \,\left\lfloor {s/r} \right\rfloor } {\left( \matrix{ s - rm + k - 1 \cr k - 1 \cr} \right)\left( \matrix{ m + k - 1 \cr k - 1 \cr} \right)} } \right)x^s } = \cr & = \sum\limits_{0\, \le \,s} {\left( {\sum\limits_{0\, \le \,m\,\left( { \le \,\left\lfloor {s/r} \right\rfloor } \right)} {\left( \matrix{ s - rm + k - 1 \cr s - rm \cr} \right)\left( \matrix{ m + k - 1 \cr m \cr} \right)} } \right)x^s } \cr} $$