I was hoping someone might be able to help me justify this sum equivalence I ran across in a proof. I'm sure it is something simple but never the less I am confused. I have the following for a function $f$ and operator $P_j$. $$ \sum_{j=-J}^{J-1} P_{j+1}f(x) - P_jf(x) = \sum_{j=-J+1}^{J}P_jf(x) - \sum_{j=J}^{J-1} P_j f(x) = P_J f(x) - P_{-J} f(x) $$ If you plug in say $J = 2, 3$ you can see that the end result is true, with little calculation. I am very confused with the middle statement it seems the first sum is just a re-indexing, but the "improper" sum $\sum_{j=J}^{J-1} P_j f(x)$ is confusing me. If someone could help me justify this equivalance I would be very grateful.
2026-03-25 17:24:47.1774459487
Equivalence of sums
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The first summation in the middle is really $$\displaystyle \sum_{j=-J}^{J-1} P_{j+1}f(x)$$ which can be written as $$\displaystyle \sum_{j=(-J)+1}^{(J-1)+1} P_{j}f(x)$$ or $$\displaystyle \sum_{j=-J+1}^{J} P_{j}f(x)$$