Generalized sine equations. I
This is the first of a sequence of papers on functional equations of the form \[ f_ 1 (x+y )f_ 2 (x-y) = \sum^ n_{k=0} g_ k (x)h_ k (y). \] The present paper deals with the cases (i) \(n=1\), \(h_ 1 = id\), (ii) \(n = 2\), \(h_ k = h^ k\) \((k=0,1,2)\) with additive or multiplicative \(h\), and (iii) \(n\) arbitrary, one of the \(g_ k\)'s and one of the \(h_ k\)'s is identically 1 (both the \(\{g_ k\}\) and the \(\{h_ k\}\) are supposed to be linearly independent). In (i) and (iii) the values of the functions lie in an algebraically closed field of characteristic 0, while their domain is a divisible abelian group or an abelian group divisible by 2, respectively. In (ii) the range is \(\mathbb{C}\), the domain is \(\mathbb{R}\).
scientific article; zbMATH DE number 699776