for e.g. for $\frac{1}{(1-ax)} = a^n$ or for $\frac{1}{(1-x)^2} = n+1$
generating function = $\frac{1}{(1-ax)^2}$
for e.g. for $\frac{1}{(1-ax)} = a^n$ or for $\frac{1}{(1-x)^2} = n+1$
generating function = $\frac{1}{(1-ax)^2}$
You give that $\frac{1}{(1-x)^2}$ is the generating function of $a_n=n+1$. That means that $ \frac{1}{(1-x)^2}=\sum_{k=0}^\infty (k+1)x^k\tag{1} $ Therefore, $ \frac{1}{(1-ax)^2}=\sum_{k=0}^\infty (k+1)(ax)^k\tag{2} $ Thus, $\frac{1}{(1-ax)^2}$ is the generating function for $a_n=(k+1)a^k.$
Method 1:
Use the general formula for the $k$-th derivative of a power (or establish through induction):
$\frac{\mathrm d^k}{\mathrm dx^k}\frac1{(1-ax)^2}=\frac{(k+1)!a^k}{(1-ax)^{k+2}}$
Method 2:
Since $\dfrac1{(1-ax)^2}=\dfrac1{(1-ax)}\cdot\dfrac1{(1-ax)}$, and $\dfrac1{(1-ax)}$ is the generating function for $a^k$, the coefficients of $\dfrac1{(1-ax)^2}$ are given by the autoconvolution of $a^k$; i.e. the sum
$\sum_{j=0}^k a^j a^{k-j}$
whose simplification I leave to you.
Another method. By the binomial theorem: $ (1 - u)^{-m} = \sum_{k \ge 0} \binom{-m}{k} (-u)^k $ Now we have, using Knuth's notation for falling/rising factorials: $m^{\underline{k}} = m (m - 1) \ldots (m - k + 1)$ and $m^{\overline{k}} = m (m + 1) \ldots (m + k - 1)$: $ \binom{-m}{k} = \frac{(-m)^{\underline{k}}}{k!} = \frac{(-1)^k m^{\overline{k}}}{k!} = \frac{(-1)^k (m + k - 1)^{\underline{k}}}{k!} = (-1)^k \binom{m + k - 1}{k} = (-1)^k \binom{k + m - 1}{m - 1} $ Note that this is just a $m - 1$-th degree polynomial in $k$.
So we have: $ (1 - u)^{-m} = \sum_{k \ge 0} \binom{k + m - 1}{m - 1} u^k $ The rest of the fun I leave to you.