Holomorphic solutions of an inhomogeneous Cauchy equation (Q1822698)

The equation (1) \(\phi (x+y)-\phi (x)-\phi (y)=f(x)f(y)h(x+y)\) is considered, where \(\phi\), f, and h are unknown functions holomorphic in a neighborhood of the origin. The authors find all solutions. The special case of (1) with \(f=h\) had been solved previously by the first author and \textit{I. Fenyö} [Rend. Mat. Appl. VII., Ser. 5(1988), No.3/4, 387-392 (1985; Zbl 0666.39005)]. Normalizations allow one to assume that \(\phi '(0)=0\), and that exactly one of the following four cases holds. A theorem corresponds to each case. (I) \(f(0)=1\), (II) \(f(0)=0\) and \(h(0)=0\), (III) \(f(0)=0\), \(h(0)=1\), and \(h^{(3)}(0)=-f''(0)h''(0),\) and (IV) \(f(0)=0\), \(h(0)=1\), and \(h^{(3)}(0)\neq -f''(0)h''(0).\) For case (I), the solutions are \(f(x)=e^{\sigma x},\) \(h(x)=e^{-\sigma x},\) \(\phi (x)=-1\), for some complex \(\sigma\). The solutions in case (II) are obtained by reducing (1) to the equation solved in the paper cited. In this case, the solution triples (f,h,\(\phi)\) must be of one of three different forms; one of the forms involves the Jacobi elliptic functions while the other two forms involve only elementary functions. The two remaining cases are studied in detail. In each, the solution triples (f,h,\(\phi)\) must be of one of two different forms, one elementary and the other involving elliptic functions.