Extending the Meijer G-Function

By replacing the Euler gamma function by the Barnes double gamma function in the definition of the Meijer G-function, we introduce a new family of special functions, which we call K-functions. This is a very general class of functions, which includes as special cases Meijer G-functions (thus also all hypergeometric functions \( _p F_q\) $_p^{}{F}_q$ ) as well as several new functions that appeared recently in the literature. Our goal is to define the K-function, study its analytic and transformation properties and relate it to several functions that appeared recently in the study of random processes and the fractional Laplacian. We further introduce a generalization of the Kilbas–Saigo function and show that it is a special case of K-function.