K-Bessel functions associated to 3-rank Jordan algebra
Hacen Dib
TL;DR
Extends the Rank-2 Bessel–Muirhead analysis to rank-3 Euclidean Jordan algebras, expressing the K-Bessel function as a finite linear combination of rank-3 J‑solutions after a rank-reduction argument. It builds a rank-3 algebraic framework via Cayley–Hamilton relations, spectral decomposition, and Peirce grading, to obtain determinant identities and reduction formulas. The authors derive explicit coefficients a_ν^j and b_ν^j, in terms of Gamma factors, giving K_ν^{[3]}(x) as a sum of J_ν^{[3,j]} and t_3^{-ν} J_{-ν}^{[3,j]}, with consistency to the rank-2 limit. The work provides integral representations and boundary behavior linking to the Ω_3 gamma function, offering concrete tools for explicit computations in higher-rank Jordan-algebra harmonic analysis.
Abstract
Using Bessel-Muirhead system, we can express the K-bessel function defined on a Jordan algebra as linear combination of the J-solutions. We determine explicitly the coefficients when the rank of this Jordan algebra is three after a reduction to the rank two. The main tools are some algebraic identities developed for the occasion.