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.