TITLE: What's the summation of formal series $\sum_{n\geq0}\binom{n\delta}{n}x^n$？
QUESTION [6 upvotes]: $\delta$ is a positive number. Is this Taylor expansion of some function?

REPLY [8 votes]: The Bürmann-Lagrange theorem gives that
$$\sum_{n\geq 0} {n\delta \choose n} t^n = \frac{1}{1-\delta t(1+z)^{\delta -1}}=\frac{1+z}{1+(1-\delta) z}$$
where $z=z(t)=\sum_{n\geq 1} \frac{1}{n}{n\delta \choose n-1}t^n$ is the solution of $z=t(1+z)^\delta$ $\big($i.e.
$z(t)$ is the local inverse at $0$ of  $z \mapsto \frac{z}{(1+z)^\delta}\big)$.
See e.g problem 216 in section 3 of Polya/Szegő, Problems and Theorems in Analysis I.