K-Bessel functions associated to a 3-rank Jordan algebra

Bessel functions of matrix argument appeared as a subject of studies in the work of Herz [6]. One can find in the multivariate statistics literature some applications of these functions (see [11]). The main tool of Herz’s work was the Laplace transform and its inverse in the space of real symmetric matrices. He obtained several properties but a “good” differential system was lacking. This was the major contribution of Muirhead in [10] when he characterized them by a system of second-order partial differential equations and proved the uniqueness (up to multiplicative constant) of the solution which is analytic at 0. Later, Faraut and Travaglini [5] gave a generalization of these functions to a Jordan algebra. An extensive study was done in [2]. However, the explicit resolution of the Bessel-Muirhead system in general rank remains an open problem. Nevertheless, in [9] Mahmoud wrote down an explicit basis of the solutions in the rank 2 and 3 using series of one-variable Bessel functions. On the other hand, the K-Bessel function of matrix argument was defined earlier in Herz’s paper cited above and his conjecture was that there must be a linear relation between this kind of Bessel functions and the J’s one as known in the onevariable theory. The first result concerning this conjecture was established by the author in [3] for the rank two. In this paper, we continue our work and prove that a similar result for the K-Bessel function is also true when the Jordan algebra is of rank 3. In this case, there are four nonequivalent classes of real simple and Euclidean Jordan algebra: Herm(3,F) the algebra of 3× 3 Hermitian matrices, where F is the field of real, complex, quaternionic, or Cayley (octaves) numbers. In [3] we intended to perform a case-by-case calculation. In this way, a serious difficulty arises in the evaluation of some integral over the automorphism group of the Jordan algebra. However, a unified treatment is possible