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**Rmercadojr** I've been trying to show that $\sum_{n=-\infty}^{\infty} q^{n(2n-1)} = \sum_{n=0}^{\infty} q^{n(n+1)/2}$ but I haven't been successful. I've managed to break the LHS into three pieces corresponding to the indices $-\infty$ to $-1$, $0$, and $1$ to $\infty$. Then I changed variables to get $1 + \sum_{n=1}^{\infty} q^{n(2n-1)} + \sum_{n=1}^{\infty} q^{n(2n+1)}$. I then added the two sums, but I can't find a way to manipulate it into the RHS.

I see how the LHS equals the RHS by expanding the terms, but I just haven't been able to show it algebraically. I found the identity $\sum_{n=0}^t f(2n) + \sum_{n=0}^t f(2n+1) = \sum_{n=0}^{2t+1} f(n)$. This seems like a very basic identity but I had never seen it. I haven't been able to find many examples of it being used either in books or online. Does anyone know a website where I can read more about it?

I'm sure that I can use this identity to solve my problem, but I haven't been able to. I will appreciate any help. Thank you.