The general solution of the generalized Schilling's equation (Q1207300)

The author extends a result which he presented at the 1991 Koninki (Poland) International Conference on Functional Equations and Inequalities (ICFEI). As generalization of ``Schilling's functional equation'', which arose from Physics, he offers the general solution of the equation \(Af(qx)=B(f(x+1)+Cf(x-1)+Df(x)\) \((x\in\mathbb{R})\), where \(A,D\in\mathbb{R}\), \(B,C\in\mathbb{R}\backslash\{0\}\), \(q\in[0,1]\) are otherwise arbitrary constants, and notes that it is determined by its values on [0,1].