Consider e.g. the following recurrence relation solution:

RSolve[a[m, n + 1] == a[m + 1, n], a[m, n], {m, n}]

{{a[m, n] -> C[1][m + n]}}

which states that the relation can be satisfied by a function of the sum of m and n.

Does Mathematica mean with this, that any solution to this recurrence relation must be of this type or does this only give one possible structure that solves it and there might be other ones still to be discovered?

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    $\begingroup$ I am pretty sure in this case, that Mathmatica has found all solutions... $\endgroup$ – Henrik Schumacher Aug 26 '19 at 22:04

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