Consider the following two equations: $$ \begin{cases} A_k(n+1)-A_k(n)=\beta \displaystyle \frac{ A_{k-1}(n)- A_k(n)}{\alpha+ n} + \delta_{k, \beta} \\ \\ \displaystyle \frac{\partial{B_k(t)}}{\partial t}= \beta \frac{B_{k-1}(t)- B_k(t)}{\alpha+ t} + \delta_{k ,\beta} \end{cases} \\ \\ $$ The first equation is a two-dimensional difference equation (in $k$ and $n$ domains), and the second equation is its time-continuous analog. In the first equation, $n$ and $k$ are nonnegative integers. In the second equation, $k$ is nonnegative integer and $t\geq 0$ is continuous. The parameter $\beta$ is a positive integer, and $\delta_{\cdot ,\cdot}$ is the Kronecker delta function. Also, $\alpha>0$ is a constant. At $t=n=0$, we have $A_k(0)=B_k(0)$ for all $k$.
Note that, although $B_k(t)$ is continuous in time, we can look at its values at integer times. Consider $t=n$, where $n>0$ is an integer. Can we find an upper bound on $|A_k(n)-B_k(n)|$ as a function of $n$?