Self-Similar Cauchy Problems and Generalized Mittag-Leffler Functions

By observing that the fractional Caputo derivative of order α ∈ (0, 1) can be expressed in terms a multiplicative convolution operator, we introduce and study class such operators which also have same self-similarity property as derivative. We proceed by identifying subclass is bijection with set Bernstein functions provide several representations their eigenfunctions, corresponding function, generalize Mittag-Leffler function. Each eigenfunction turns out to Laplace transform right-inverse non-decreasing self-similar Markov process associated via so-called Lamperti mapping this Resorting spectral theoretical arguments, investigate generalized Cauchy problems, defined these operators. In particular, both stochastic representation, inverse processes, an explicit given functions, solution problems. This work could seen an-in depth analysis specific class, one property, general increasing processes introduced [15].