- #1
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Hi. The problem is as follows:
Let m and n be integers, we may assume that (if they are not equal), m is the smallest. Then
[tex]\sum_{i=0}^m \sum _{j=0}^n f((m+n)-2 (i+j)) = \sum_{i=0}^m \sum _{j=0}^{-2 i+m+n} f((m+n)-2 (i+j))[/tex]
for some sequence [itex]f(k)_k[/itex].
Anything you can think of, but probably just the standard manipulations on sums and some smart rewriting and/or separating different cases.
I have reduced from a given question to the above statement, and I'm quite positive that I haven't made a mistake in this -- so the given statement should be true. It doesn't really look true to me though, so I plugged this into Mathematica: it seems to hold for any two [itex]m \le n[/itex] I choose, but I just don't see how to prove this. I think it involves a manipulation with the double sum, because if I remove the outer sum and consider i a fixed number in the rest of the expression, it doesn't seem to hold anymore (I always need one or more of the f(n) to be zero, which they aren't in general).
So I think the given information now is necessary and sufficient (e.g. no extra information on f(n) or m and n should be needed).
Homework Statement
Let m and n be integers, we may assume that (if they are not equal), m is the smallest. Then
[tex]\sum_{i=0}^m \sum _{j=0}^n f((m+n)-2 (i+j)) = \sum_{i=0}^m \sum _{j=0}^{-2 i+m+n} f((m+n)-2 (i+j))[/tex]
for some sequence [itex]f(k)_k[/itex].
Homework Equations
Anything you can think of, but probably just the standard manipulations on sums and some smart rewriting and/or separating different cases.
The Attempt at a Solution
I have reduced from a given question to the above statement, and I'm quite positive that I haven't made a mistake in this -- so the given statement should be true. It doesn't really look true to me though, so I plugged this into Mathematica: it seems to hold for any two [itex]m \le n[/itex] I choose, but I just don't see how to prove this. I think it involves a manipulation with the double sum, because if I remove the outer sum and consider i a fixed number in the rest of the expression, it doesn't seem to hold anymore (I always need one or more of the f(n) to be zero, which they aren't in general).
So I think the given information now is necessary and sufficient (e.g. no extra information on f(n) or m and n should be needed).