A generalization of Meijer’s G function motivated by probability

Докладчик: Dmitrii Karp

Резюме: Meijer’s G function appeared in probability in 1930-ies even before Meijer’s work as the density of the product of independent beta variables and as the density of the likelihood ratio test for samples drawn from many classical probability distributions. It is defined by a contour integral of the appropriate ratio of gamma products (sometimes reducible to the inverse Mellin transform). Replacing the Euler gamma function by the Barnes double gamma function in this contour integral, we introduce a new family of special functions which we call Mellin-Barnes’ K function. It is a quite general family which contains Meijer’s G function (thus also all classical hypergeometric functions pFq), as well as several new functions that appeared recently in the study of random processes and the fractional fenomena. For example, the running supremum of an alpha-stable Levy process has density expressible by K function as well as does the exponential functionals of the hypergeometric process. In the talk, we will discuss the definition of the new function, its basic analytic and transformation properties and relations to the functions found in the literature. We further define a generalization of the Kilbas-Saigo function introduced earlier to solve in closed form certain classes of integral and differential equations of fractional order and observe that both the original and the generalized Kilbas-Saigo functions are special cases of our new function.

The talk is based on a joint work with Alexey Kuznetsov (York University, Toronto, Canada).