Can You Help Me With This Markov Chain Question? For a birth and death process with birth rates, $\lambda_i$ and death rates $\mu_i$ $(i=0,1,2...)$ respectively. Show that the transition probabilities, $P_{i,j}(t)$ satisfy the following differential equations 
$P'_{0,j}(t) = -\lambda_{0} P_{0,j}(t) +\lambda_{0} P_{1,j}(t)$
$P'_{i,j}(t) = \mu_{i} P_{i-1,j}(t)-(\lambda_{i} +\mu_{i}) P_{i,j}(t) +\lambda_{i}P_{i+1,j}(t)$ for $i \geq 1$ 
Here is what I have so far: 
First of all, the subscripts are different than a normal birth and death process. So I think the transition matrix is the following 
$P_{i,j}(t) = $\begin{matrix}  
              -\lambda_{0} & \lambda_{0} & 0 & 0 & ... \\
               \mu_{0}   &    -(\lambda_{1}+\mu_{0}) & \lambda_{1} & 0  &...\\ 
               0 &  \mu_{1} & -(\lambda_{2} + \mu_{1}) & \lambda_{2} &0 &...\\
               0 & 0 & \mu_{2} & -(\lambda_{3} + \mu_{2} ) & \lambda_{3} &...\\
      \end{matrix}
I know from the Chapman-Kolomogrov Equations, I can somehow manipulate them in order to get the differential equations but I am not exactly sure how to do that. 
Thank you for all of your inputs. Much help is appreciated. 
 A: We begin with the Chapman-Kolmogorov equations. For small $h$, we have
\begin{eqnarray*}
P_{i,j}(t+h) &=& \sum_{k=0}^{\infty} P_{i,k}(h) P_{k,j}(t) \\
&& \\
&=& P_{i,i-1}(h) P_{i-1,j}(t) + P_{i,i}(h) P_{i,j}(t) + P_{i,i+1}(h) P_{i+1,j}(t) + o(h) \\
&& \qquad\text{since the probability of transition in time $h$ from $i$ to} \\
&& \qquad\text{anyhere other than $i-1,\;i,\;i+1$ equals $o(h)$.} \\
&& \\
&=& \mu_i h P_{i-1,j}(t) + (1-\lambda_i h-\mu_i h) P_{i,j}(t) + \lambda_i h P_{i+1,j}(t) + o(h) \\
&& \\
\therefore\quad \dfrac{P_{i,j}(t+h) - P_{i,j}(t)}{h} &=& \mu_i P_{i-1,j}(t) - (\lambda_i + \mu_i) P_{i,j}(t) + \lambda_i P_{i+1,j}(t) + o(h) \\
&& \\
\therefore\quad P_{i,j}^{'}(t) &=& \mu_i P_{i-1,j}(t) - (\lambda_i + \mu_i) P_{i,j}(t) + \lambda_i P_{i+1,j}(t) \\
&& \qquad\text{taking the limit as $h\to0$.}
\end{eqnarray*}
For the particular case of $i=0$, where $\mu_i=0$, this equation becomes
\begin{eqnarray*}
P_{0,j}^{'}(t) &=& - \lambda_0 P_{0,j}(t) + \lambda_0 P_{1,j}(t).
\end{eqnarray*}
