Parallelogram-law-type identities
The parallelogram law in the complex plane is \(2(|a|^2+ |b|^2)= |a+ b|^2+ |a- b|^2\); it relates the lengths of the diagonals of a parallelogram with vertices \(0\), \(a\), \(b\) and \(a+ b\) to the lengths of its sides. In three theorems, the author proves several generalizations of this formula; they all refer to arbitrary complex numbers \(z_1,\dots, z_n\) (instead of \(a\), \(b\)) together with a fixed set \(A= \{\alpha_1,\dots, \alpha_n\}\) of complex numbers with \(\alpha_1+\cdots+ \alpha_n= 0\). Special choices of \(n\) and \(A\) lead to formulas involving sums of roots of unity, trigonometric functions, binomial coefficients, and permutations over the symmetric and the alternating groups (of degree \(n\)).
scientific article; zbMATH DE number 803707