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I'm having trouble interpreting the following result of RSolve to this difference equation:

RSolve[n1 F[n1+1, n2] == (n1+n2) F[n1, n2] + F[n1-1, n2+1] - F[n1, n2+1], 
  F[n1, n2], n1]

enter image description here

The result makes no sense to me, since it is expressed as a sum over K[1] starting at K[1] = 0 causing the factorial (-1 + K[1])! to diverge.

Is there some special interpretation of this factorial?

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    $\begingroup$ Note that the "solution" includes the function F on the right-hand-side, so it's not really a solution. In any case, your recurrence relation does not really define F for all n1,n2, so it cannot really be solved. Not sure how to interpret the output by MMA though. $\endgroup$ – AccidentalFourierTransform May 6 '18 at 21:46

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