Four types of special functions of G_2 and their discretization

TL;DR

This paper explores four families of special functions derived from the Lie group G2, analyzing their properties, orthogonality, discretization, and how their products decompose into sums, including novel functions not previously documented.

Contribution

It introduces two new families of special functions related to G2, compares their properties with known functions, and studies their orthogonality and product decompositions.

Findings

All four families are orthogonal over a finite region.

Discretely orthogonal when sampled at lattice points.

Products decompose into finite sums of the functions.

Abstract

Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G2, are compared and described. Two of the four families (called here C- and S-functions) are well known, whereas the other two (S^L- and S^S-functions) are not found elsewhere in the literature. It is shown explicitly that all four families have similar properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space, and they are discretely orthogonal when their values, sampled at the lattice points F_M \subset F, are added up with a weight function appropriate for each family. Products of ten types among the four families of functions, namely CC, CS, SS, SS^L, CS^S, SS^L, SS^S, S^SS^S, S^LS^S and S^LS^L, are completely decomposable into the finite sum of the functions. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.