Suppose we have the master equation $$\partial_{t}P_{n}(t)=a(n+1)nP_{n+1}(t)+b(n-1)P_{n-1}(t)-[an(n-1)+bn]P_{n}(t)$$ where $a$ and $b$ are constants. I would like to take the continuum limit of this equation to obtain the form $$\partial_{t}P(n,t)=-\partial_{n} \{[bn-an(n-1)]P(n,t)\}+\frac{1}{2}\partial_{n}^2\{[bn+an(n-1)]P(n,t)\}$$ However for some reason I cannot get this to work. I have tried taylor expanding the master equation to second order using $$P_{n+1}(t)=P(n,t)+\partial_{n}P(n,t)+\frac{1}{2}\partial_{n}^{2}P(n,t)+...$$ $$P_{n-1}(t)=P(n,t)-\partial_{n}P(n,t)+\frac{1}{2}\partial_{n}^{2}P(n,t)+...$$ and collecting terms in each derivative of $P(n,t)$. Then I expand the derivatives in the second equation and collect terms of each kind. However they don't seem to match (the algebra is long so I won't bother posting it here). So is there something I am missing? Thanks for any help :)
Update: so I believe the above approach may be a little off. I think I should first set $$P_{n}(t)\rightarrow P(x,t)$$ $$n\rightarrow x$$ $$n+1\rightarrow x+\Delta x$$ $$n-1 \rightarrow x-\Delta x$$ and then taylor expand in $\Delta x$. I should then keep terms only up to order $\Delta x^2$ and at the end set $\Delta x=1$ and $x=n$. This improves things a little, however the two results are still slightly different.