A NEW CHARACTERIZATION OF HOMOGENEOUS FUNCTIONS AND APPLICATIONS

Abstract

We present a new characterization of real homogeneous functions of a negative degree by a new counterpart of Euler’s homogeneous function theorem using quantum calculus and replacing the classical derivative operator by a $(p,q)$-derivative operator. As an application we study the solution of the Cauchy problem associated to the $(p,q)$-analogue of the Euler operator. Using this solution, a probabilistic interpretation is given in some details; more specifically, we prove that this solution is a stochastically continuous markovian transition operator. Finally, we study it’s associated subordinated stochastically markovian transition operator.