(Feller Vol.1, P.323-324) Recurrent event occurs if a success run of length $r$ occurs. The author shows how to derive the distribution of recurrence times approximately. Let $F(s)$ be the generating function of this distribution, and the author shows that $$F(s) = \frac{p^rs^r(1-ps)}{1-s + qp^rs^{r+1}}.$$ The author is trying to derive $f_n$ (the probability that the first recurrence occurs at the $n$th trial; $F(s) = \sum_{n=0}^\infty f_n s^n$), using the method of partial fractions. To do this, the author claims that "the equation in the display clearly shows that the denominator has a unique positive root $s=x$". I don't understand this statement. How do we know that $1-s+qp^rs^{r+1}$ has a unique positive root?
2026-04-11 22:00:51.1775944851
$1-s + qp^rs^{r+1}$ has a unique positive root $s=x$.
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