Treffer: \((r,s)\)-exponentials

For any nonzero numbers \(r\) and \(s\), the authors define the \((r,s)\)-differentiation operator \[ (d_{r,s} f) (x)= (f(rx)- f(sx))/ x(r-x). \] The solutions of the equation \(d_{r,s} f=f\) are called \((r,s)\)-exponential functions. It is shown that, except for the case \(r= s=1\), there exist infinite number of linearly independent solutions of the equation and the structure of these solutions depends on the essential way on \(r\) and \(s\). As a rule, there exists an only unique solution regular in a neighborhood of \(x=0\) and equal to 1 at this point. Solutions are represented in the form of convergent series \(f(x)= x^\lambda \sum^\infty_{n= -\infty} c(n) x^n\). Such solutions exist only under additional conditions on \(r\) and \(s\). For other \(r\) and \(s\) solutions are constructed in integral form. These integral representations are given explicitly. If \(r