This question is an extension to Finding UMVUE for $p^t$ associated with a negative binomial distribution, I followed exactly the same strategy to compare the terms but got this weird expression when I try to find $E(g(x))=var(X)=\frac{r(1-p)}{p^2}$. I do not know how to deal with r, and it seems if I want to start from j=r, the right-hand side will have terms available starts from $q^{j+1}$. For finding $E(g(x))=log(p)$, I don't have any clue yet.
$$ \sum_{j=r}^{\infty} g(j)\left(\begin{array}{c} j-1 \\ r-1 \end{array}\right) q^{j}=\frac{q^{r}}{(1-q)^{r-t}}=r\sum_{k=r+2}^{\infty}\left(\begin{array}{c} k-1 \\ r+1 \end{array}\right) q^{k-1} \quad, \forall q \in(0,1) $$
Followed what was suggested I think I have a good understanding of how to find g(*) which gives $E(g(x))=p^t$, but it seems I still cannot proceed to the second and third part of the question. Could someone please give me some clue for the third part of the question, and possibly, point out where I have made mistakes for the second one?
The original question is attached below.
Let $X$ be a random variable having the negative binomial distribution with $$ P(X=x)=\left(\begin{array}{c} x-1 \\ r-1 \end{array}\right) p^{r}(1-p)^{x-r}, x=r, r+1 \ldots $$ where $p \in(0,1)$ and $r$ is a known positive integer.
Find the UMVUE of $p^{t},$ where $t$ is a positive integer and $t<r$.