Cyclic sum

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A cyclic sum is a summation that cycles through all the values of a function and takes their sum, so to speak.

Rigorous definition

Consider a function $f(a_1, a_2, \ldots, a_n)$. The cyclic sum $\sum_{cyc} f(a_1, a_2, \ldots, a_n)$ is equal to

\[f(a_1, a_2, \ldots, a_n) + f(a_2, a_3, \ldots, a_n, a_1) + f(a_3, a_4, \ldots, a_n, a_1, a_2) + \ldots + f(a_n, a_1, a_2, \ldots, a_{n-1}).\]

Note that not all permutations of the variables are used; they are just cycled through.


A cyclic sum is often specified by having the variables to cycle through underneath the sigma, as follows: $\sum_{a,b,c}\frac{ab}{cd}.$ Note that a cyclic sum need not cycle through all of the variables.

A cyclic sum is also sometimes specified by $\sum_{cyc}$. This notation implies that all variables are cycled through.

See also