I am trying to teach myself more about generating functions and how they "generate" sequences. Specifically I am trying to learn more about how to go from sequence to function and I would like some practice problems that you think would be helpful for someone just learning about generating functions? Thanks.
For the sake of context, an example.
Let $a,b,r_0,r_1$ be fixed real numbers. Then define for $n\in\Bbb N_0$, $$r_{n+2}=ar_{n+1}+br_n$$ and $$R(x)=\sum_{n\geq0}r_nx^n$$ Multiplying both sides of the recurrence by $x^{n+2}$ and summing from $n=0$ to $\infty$, $$R(x)-r_0-r_1x=axR(x)-ar_0x+bx^2R(x)$$ $$R(x)=\frac{r_0+(r_1-ar_0)x}{1-ax-bx^2}$$
Currently the most advanced thing like this I've ever done is show that $$\sum_{n\geq1}H_{n}^{(k)}s^n=\frac{\mathrm{Li}_k(s)}{1-s}$$ Where $$H_n^{(k)}:=\sum_{i=1}^{n}\frac1{i^k}$$ and $$\mathrm{Li}_k(s):=\sum_{n\geq1}\frac{s^n}{n^k}$$ Are respectively the harmonic numbers of order $k$, and the $k$-th polylogarithm function.
Hit me!