Abstract
The 6j-symbols for representations of the q-deformed algebra of polynomials on mathrm {SU}(2) are given by Jackson’s third q-Bessel functions. This interpretation leads to several summation identities for the q-Bessel functions. Multivariate q-Bessel functions are defined, which are shown to be limit cases of multivariate Askey–Wilson polynomials. The multivariate q-Bessel functions occur as 3nj-symbols.
Highlights
It is well known that Wigner’s 6 j-symbols for the SU(2) group are multiples of hypergeometric orthogonal polynomials called the Racah polynomials
6 j-symbols for the SU(2) quantum group can be expressed in terms of q-Racah polynomials, which are q-hypergeometric orthogonal polynomials
Properties of 6 j-symbols such as summation formulas and orthogonality relations lead to properties of specific families of orthogonal polynomials, see e.g., [21,22, Chaps. 8, 14]
Summary
It is well known that Wigner’s 6 j-symbols for the SU(2) group are multiples of hypergeometric orthogonal polynomials called the Racah polynomials. We consider 6 j-symbols for representations of the q-deformed algebra of polynomials on SU(2). The 6 j-symbols for tensor products of three infinite-dimensional representations can be expressed in terms of Jackson’s third q-Bessel functions [8]. We consider three fundamental identities for 6 j-symbols (see e.g., [1]): Racah’s backcoupling identity, the Biedenharn–Elliott identity and the hexagon identity These identities are obtained by decomposing 3- or 4-fold tensor product representations in several ways. We obtain the fundamental identities for 6 j-symbols, leading to summation formulas for the q-Bessel functions. We show that these multivariate q-Bessel functions occur as 3n j-symbols, and use this interpretation to find a summation formula. We will use the following generating function to identify the 6 j-symbols with q-Bessel functions.
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