We assume $m>1$. We study a slightly different problem: $k$ being a fixed strictly positive integer, we assume that there are $k$ individuals in generation 0. We denote by $W_n$ the number of individuals in the $n$-th generation and define $u_n$ as the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ for the first time at rank $n$: $$u_n=P\left((W_n=k)\cap\left(\bigcap_{i=1}^{n-1}(W_i\neq k)\right)\right)$$ For $n$ and $r$ non-zero natural integers, $u_n^{(r)}$ is the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ for the $r$-th time at rank $n$.
Verify that the series $\sum_{n\geqslant 1}u_n s^n$ and $\sum_{n\geqslant 1}u_n^{(r)}s^n$ converge when $s\in[-1,1]$.

We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. Let $m$ be a strictly positive integer. We say that a random variable $Y : \Omega \rightarrow \mathbb { N }$ admits a finite moment of order $m$ if $Y$ admits a finite expectation, that is, if the series $\sum n ^ { m } P ( Y = n )$ converges. We then call the moment of order $m$ of $Y$ the real number $$\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$$
Show that if $Y : \Omega \rightarrow \mathbb { N }$ is a random variable associated with a generating function $G _ { Y }$ of radius strictly greater than 1, then $Y$ admits a finite moment of all orders.

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$.
Show that the power series $\sum \lambda_{k} t^{k}$ has a radius of convergence $\rho(X)$ greater than or equal to $\mathbb{P}(X = 0)$. For all real $t$ in $]-\rho(X), \rho(X)[$, we set $$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and $H_X$ denotes its auxiliary power series.
For $t \in ]-\rho(X), \rho(X)[$, show $G_{X}^{\prime}(t) = H_{X}^{\prime}(t) G_{X}(t)$, then $G_{X}(t) = \exp\left(H_{X}(t)\right)$.

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ is the unique real sequence such that for all $k \in \mathbb{N}^{*}$, $k\mathbb{P}(X = k) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}(X = k-j)$.
Show that the power series $\sum \lambda_{k} t^{k}$ has a radius of convergence $\rho(X)$ greater than or equal to $\mathbb{P}(X = 0)$. For all real $t$ in $]-\rho(X), \rho(X)[$, we set $$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, with auxiliary power series $H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$ defined for $t \in ]-\rho(X), \rho(X)[$.
For $t \in ]-\rho(X), \rho(X)[$, show $G_{X}^{\prime}(t) = H_{X}^{\prime}(t) G_{X}(t)$, then $G_{X}(t) = \exp\left(H_{X}(t)\right)$.

We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned} & \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\ & \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n } \end{aligned}$$ Show that the power series defining $F$ and $G$ have radius of convergence greater than or equal to 1. Justify then that the functions $F$ and $G$ are defined and of class $C^{\infty}$ on $]-1,1[$.
Show that $G$ is defined and continuous on $[-1,1]$ and that $$G ( 1 ) = P ( R \neq + \infty ) .$$