Discrete $g$-Mellin transform and discrete g-Mellin convolution in probability theory

Definition of discrete $g$-Mellin transform and discrete $g$-Mellin convolution are given. The property of pseudo-linearity of discrete $g$-Mellin transform and some algebraic properties, such as commutativity, associativity, associativity with scalar pseudo-multiplication, and distributivity of convolution over pseudo-addition, for discrete $g$-Mellin convolution are proven. Some illustrative examples of discrete $g$-Mellin transform and discrete $g$-Mellin convolution are given.