About integer-valued variants of the theta and 6j symbols

TL;DR

This paper introduces integer-valued functions gon and tet that mirror the properties of the theta and 6j symbols, simplifying their evaluation and revealing new relations, including connections to inverse Hilbert matrices.

Contribution

It presents the first systematic study of integer-valued variants of theta and 6j symbols, with explicit formulas and novel relations, enhancing computational and theoretical understanding.

Findings

gon and tet share key properties with theta and 6j symbols

Explicit formulas and evaluation procedures are provided

New relations between gon and inverse Hilbert matrices are established

Abstract

These notes contain essentially a rewriting of several properties of two well-known quantities, the so-called theta symbol (or triangular symbol), which is rational, and the 6j symbol, which is usually irrational, in terms of two related integer-valued functions called gon and tet. Existence of these related integer-valued avatars, sharing most essential properties with their more popular partners, although a known fact, is often overlooked. The properties of gon and tet are easier to obtain, or to formulate, than those of the corresponding theta and 6j symbols, both in the classical and quantum situations. Their evaluation is also simpler (the paper displays a number of explicit formulae and evaluation procedures that may speed up some computer programs). These two integer-valued functions are unusual, in that their properties do not appear to be often discussed in the literature, but their features reflect those of related real-valued functions discussed in many places. Some of the properties that we shall discuss seem however to be new, in particular several relations between the function gon and the inverse Hilbert matrices.