Notation for summing over all cyclic permutations of three indices

Question

I have an expression which depends on three indices. For example, something like $\cos(\alpha_1 +\alpha_2-\alpha_3)$. Say I want to write the sum $$\cos(\alpha_1 +\alpha_2-\alpha_3) + \cos(\alpha_3 +\alpha_1-\alpha_2) + \cos(\alpha_2 +\alpha_3-\alpha_1)$$ so that I get sum over all cyclic permutations of $1,2,3$. Is there a nice way to write such a sum with sigma notation? I could not come up with anything which was not very clumsy to write or read. I thought about maybe writing the sum over $i,i+1,i+2$ mod $3$, but this does not fit exactly what I want (for instance, since I cannot take the mod inside the sum, since it could contain real numbers).

Thanks in advance.

Answer 1

We can write this expression using cyclic sum notation \begin{align*} \color{blue}{\sum_{\mathrm{cyc}}\cos(\alpha_1+\alpha_2-\alpha_3)}&=\cos(\alpha_1+\alpha_2-\alpha_3)+\cos(\alpha_3+\alpha_1-\alpha_2)\\ &\qquad+\cos(\alpha_2+\alpha_3-\alpha_1) \end{align*} replacing the letters $\alpha_1,\alpha_2,\alpha_3$ systematically in a cyclic manner.