Suppose that a real $c:=\lambda t>0$ and a natural $k$ are fixed, whereas $n:=N_0\to\infty$.Take any real $m>0$. Then$$f(k,t)=g_m(k,t)+h_m(k,t),$$where$$g_m(k,t):=e^{-c}\sum_{0\le j<m}\frac{c^j}{j!}\,\binom{n+j}k,$$$$h_m(k,t):=e^{-c}\sum_{j\ge m}\frac{c^j}{j!}\,\binom{n+j}k.$$For each $j$, $\binom{n+j}k\sim n^k/k!$, whence$$g_m(k,t)\sim P_m\frac{n^k}{k!},$$where$$P_m:=e^{-c}\sum_{0\le j<m}\frac{c^j}{j!}.$$Also, $\binom{n+j}k\le(n+j)^k/k!=O(n^k+j^k)$ for all $j$, whence$$h_m(k,t)=O\Big((1-P_m)n^k+\sum_{j\ge0}\frac{c^j}{j!}\,j^k\Big)=O\Big((1-P_m)n^k+1\Big)=O\big((1-P_m)n^k\big).$$Letting now $m\to\infty$ and noting that then $P_m\to1$, we conclude that$$f(k,t)\sim\frac{n^k}{k!}.$$
↧
