The \(h\)-Laplace and \(q\)-Laplace transforms
For a given function \(x\) with complex values the \(h\)-Laplace transform is defined as \[ (h/(1+hz))\sum^\infty_{k=0}x(kh)/(1+ hz)^k, \] and the \(q\)-Laplace transform as \[ (q-1)\sum^\infty_{n=0} q^nx(q^n)/\prod^n_{k=0} (1+(q- 1)q^k z), \] where \(z\) is the complex variable in the image domain. Most properties of the usual Laplace transform are transferred to these two transforms. It is shown that all these transforms are special cases of a more general Laplace transform on time scales.
scientific article; zbMATH DE number 5673256