A generalized Meijer transformation

In a series of papers [1-6], Kratzel studies a generalized version of the classical Meijer transformation with the Kernel function ( st ) ν η ( q , ν + 1 ; ( st ) q ) . This transformation is referred to as GM transformation which reduces to the classical Meijer transform when q = 1 . He also discussed a second generalization of the Meijer transform involving the Kernel function λ ν ( n ) ( x ) which reduces to the Meijer function when n = 2 and the Laplace transform when n = 1 . This is called the Meijer-Laplace (or ML) transformation. This paper is concerned with a study of both GM and ML transforms in the distributional sense. Several properties of these transformations including inversion, uniqueness, and analyticity are discussed in some detail.