Charakterisierung des Cotangens mit Replikativität
The function \(x\to \cot (\pi x)\) \((0<x<1)\) is a solution of the system \[ f(x)=(1/k)\sum^{k-1}_{i=0}f((x+i)/k),\quad (k=1,2,3,...) \] of functional equations. We show that any continuous solution of this system is of the form \(f(x)=a\cdot \cot (\pi x)+b\) with some constants a and b. The result contains a new characterization of the cotangent function by means of simple properties. Furthermore we give some information concerning the modified system \[ u(k)f(x)+v(k)=(1/k)\sum^{k- 1}_{i=0}f((x+i)/k),\quad (k\in S), \] which can be used to characterize by the same method Gauss' Psi function and Hurwitz' Zeta function \(x\to \zeta (s,x)\) (Re(s)\(\geq 1\), \(s\neq 1)\).
scientific article; zbMATH DE number 3946798
scientific article