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

Abstract

This article contains essentially a rewriting of several properties of two well-known quantities, the so-called theta symbol (or triangular symbol), which is rational, and the 6 j 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 6 j symbols in both classical and quantum situations. Their evaluation is also simpler (this paper displays a number of explicit formulas 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 inverse Hilbert matrices.