One-sided L\'{e}vy stable distributions

In this paper, we show new representations of one-sided L\'{e}vy stable distributions for irrational L\'{e}vy indices of the type $\left(\frac{p}{q}\right)^{\frac{l_{2}}{l_{1}}}$ which are not covered in \cite{pg1} : for rational L\'{e}vy indices. Furthermore, other equivalent representations for a distribution of a rational L\'{e}vy index is described. We also give a simplest proof for the formulae which cover the cases for rational L\'{e}vy indices. Finally we introduce the concepts of L\'{e}vy smashing and L\'{e}vy-smashed gamma stochastic processes.