Normal solutions of difference equations, Euler?s functions and spirals

Abstract

We consider the linear first order difference equation $$ f(x + 1) = \alpha (x) \cdot f(x) $$ for the unknown function \(f:I \subseteq \mathbb{R} \to \mathbb{C}.\) We present, under some conditions on the given function α, a product representation of a specific solution of the considered difference equation that can be understood as the normal or principal solution in the concept of N. E. Norlund, F. John, W. Krull and others. This normal solution is characterized by its asymptotic behavior near infinity. In that way we get a characterization of e.g. the gamma function, the q-gamma function and of several classical spirals.