The \(h\)-Laplace and \(q\)-Laplace transforms. (English) Zbl 1188.44008

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.


44A55 Discrete operational calculus
44A10 Laplace transform
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