A q-analogue of the 9-j symbols and their orthogonality

Abstract

We introduce a q-analogue of Wigner's 9-j symbols following the notational scheme used by Wilson in identifying the 6-j symbols with Racah polynomials, which eventually led Askey and Wilson to obtain a q-analogue of them, namely, the q-Racah polynomials. Most importantly, we prove the orthogonality of our analogues in complete generality, as well as derive an explicit polynomial expression for these new functions.

An Explicit Polynomial Expression for a q-Analogue of the 9- j Symbols

“…, then set τ = −1, and take the limit. However, as we remarked in [19], the results are equally valid for τ = +1, although the q → 1 limit is going to be different from F m,n (x, y) as given in (1.8). We should like to point out that, as in F m,n (x, y) of (1.11), the parameters a, b, c, d in R τ m,n (x, y|q) can also be complex, and therefore the numerical values ±1 of τ would automatically be implicit in the square root function (bcd) 1/2 .…”

“…Following the lead of Wilson [23], as well as our own experience in [11,19], we shall first replace a…”

“…For notational simplicity in the main results, we make a further replacement of the parameters a, b, c, d by −a/2, −b/2, −c/2, and −d/2, respectively, and, just as we did in [19], we shall introduce the function…”

“…Similar factorization occurs when we take the limit d → ∞. However, the main objective of this paper is to obtain a polynomial form of the q-analogue of F m,n (x, y) in (1.8) that we introduced in [19], namely,…”

“…We should like to point out that, as in F m,n (x, y) of (1.11), the parameters a, b, c, d in R τ m,n (x, y|q) can also be complex, and therefore the numerical values ±1 of τ would automatically be implicit in the square root function (bcd) 1/2 . However, for the sake of consistency of notation with our previous work in [19], we prefer to retain the τ symbol, with the understanding that only the principal value of (bcd) 1/2 is being considered here.…”