Homogeneous functions: New characterization and applications

Positive homogeneous functions on R of a negative degree are characterized by a new counterpart of the Euler’s homogeneous function theorem using quantum calculus and replacing the classical derivative operator by Jackson derivative. As application we start by characterizing the harmonic functions associated to Jackson derivative. Then, the solution of the Cauchy problem associated to the analogue of the Euler operator is given. Using this solution we study the associated ν-potential. Its Markovianity property is treated. Keywords: Homogeneous functions, Euler’s theorem, Quantum calculus, Cauchy problem, Markovian semigroup