My questions come from the supplementary material in a recent preprint Nonequilibrium statistical mechanics of money/energy exchange models. My first question comes from page 35. Specifically, suppose that $N,M$ are positive integers and define $\mathcal{N}_{N,M}$ recursively through the relation (for $N \geq 3$) $$\mathcal{N}_{N,M} = 1+2\sum_{\ell=1}^{M-1} \frac{1-1/\mathcal{N}_{N-1,M}}{1+1/\mathcal{N}_{N-1,M-\ell}}\prod_{k=1}^{\ell-1}\frac{1-1/\mathcal{N}_{N-1,M-k}}{1+1/\mathcal{N}_{N-1,M-k}} +(1-1/\mathcal{N}_{N-1,M})\prod_{k=1}^{M-1}\frac{1-1/\mathcal{N}_{N-1,M-k}}{1+1/\mathcal{N}_{N-1,M-k}}\tag{1}\label{1}$$ starting with $\mathcal{N}_{2,M} = 2M$. It is claimed that when $M \gg 1$, $\mathcal{N}_{N,M}$ will behave like $$\mathcal{N}_{N,M} = \frac{2M}{N-1} + \mathcal{O}\left(\frac{1}{M}\right) \tag{2}\label{2}.$$ However, the authors of the paper did not provide a rigorous argument for the asymptotic relation \eqref{2}, so I am wondering if someone can help in this regard.
My second question comes from page 35 in the aforementioned paper again. Assume that we have a probability mass function (depending on $N$ and $M$) defined by $$p^{N,M}_m = \frac{N-1}{2M}\left\{\begin{aligned} &1,&\quad m = 0,\\ &\frac{2(m-M)(N-2(M+1))}{M(N+2(M-m-1))}\frac{(N/2-M)_{m-1}}{(2-N/2-M)_{m-1}},&\quad 0<m<M, \\ &\frac{(1+M-N/2)(N/2-M)_{M-1}}{M(2-N/2-M)_{M-1}},&\quad m = M, \end{aligned} \right. ~~+ \mathcal{O}\left(\frac{1}{M^3}\right),$$ where $(a)_n = a(a+1)\cdots (a+n-1)$ denotes the Pochhammer symbol. It is stated (without detailed computations unfortunately) that $$\sum_{m=0}^M m^k p^{N,M}_m = \frac{k!M^k}{(N)_k} + \mathcal{O}\left(M^{k-1}\right) \tag{3}\label{3}$$ for each $k \in \mathbb{N}_+$. May I know how can we show the asymptotic relation \eqref{3} rigorously? Any help is greatly appreciated!