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I have a question on the partial difference equation

$$f(n+1, k) = (k+1) f(n,k) + (n+1-k)f(n,k-1)$$

where $(k, n) \in \mathbb{Z}^2$.

It is well known, that under some boundary conditions this equation generates Eulerian numbers.

I did not find any information about the general solution of this PDE. Is it known?

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  • $\begingroup$ This question is similar to: Difference equation $A(n,x)=p(x)A(n-1,x-1)+q(x)A(n-1,x)$. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. $\endgroup$
    – Sam Hopkins
    Commented Aug 18 at 23:28
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    $\begingroup$ This equation is different to one you have given a link to, because the coefficients in the equation I wrote do not solely depend on one variable. $\endgroup$
    – Oleksandr Liubimov
    Commented Aug 18 at 23:32
  • $\begingroup$ Apologies, I see. But I'll still leave the link to that other question since it seems possibly relevant. $\endgroup$
    – Sam Hopkins
    Commented Aug 19 at 0:15
  • $\begingroup$ Thank you for the link! $\endgroup$
    – Oleksandr Liubimov
    Commented Aug 19 at 0:29
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    $\begingroup$ It's an instance of the Knuth-Graham-Patashnik recurrence. Check out this paper. $\endgroup$
    – Max Alekseyev
    Commented Aug 19 at 4:51

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