Six types of $E-$functions of the Lie groups O(5) and G(2)

TL;DR

This paper introduces new families of $E$-functions for the Lie groups O(5) and G(2), detailing their properties, orthogonality, and product decompositions, expanding the mathematical understanding of these functions.

Contribution

The paper presents previously unreported families of $E$-functions for O(5) and G(2), fully characterizing their properties and orthogonalities.

Findings

New $E$-function families for O(5) and G(2) are described.

Orthogonality properties are established for these functions.

Product decompositions are provided for the new families.

Abstract

New families of $E$-functions are described in the context of the compact simple Lie groups O(5) and G(2). These functions of two real variables generalize the common exponential functions and for each group, only one family is currently found in the literature. All the families are fully characterized, their most important properties are described, namely their continuous and discrete orthogonalities and decompositions of their products.