# On the derivation of some asymptotic expressions involving combinatorics

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 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!

• I suggest that you ask the authors of the paper. No one is better than them to explain their calculations. Nov 24, 2023 at 17:04
• I agree. But I am not the official "reviewer" for this paper. I think MSE is a platform that also help. Following your reasoning then there is no meaning to have this site as there are plenty of questions asking for explain some details in some books, online notes, or journal papers; You can basically ask them to "ask the authors of the notes/books/papers", I doubt how many of them will try to ask the authors directly Nov 24, 2023 at 17:32
• I'm not insinuating that you should not ask this question here, but I sincerely believe that the authors of a paper are (most of the times) the most qualified people to answer a question about it. In fact, they publish their email because they expect to receive questions about their work. Nov 24, 2023 at 19:07
• Thank you for this reply. Nov 25, 2023 at 4:30