Bessel systems for Jordan algebras of rank 2 and 3 (Q1301866)

The paper is devoted to the study of Bessel functions of matrix argument arising from Bessel systems associated with Jordan algebras. It is known in the literature that the dimension of the space of solutions of the Bessel system of rank \(r\) is less or equal to \(2^r\). It was conjectured that this dimension is exactly \(2^r\). In this paper positive answers to this conjecture are given in the cases of ranks \(r=2\) and \(r=4\). In this study double and triple series expansions of the forms \(\sum _{p,q=0}^\infty a^\delta _{pq}t_1^{p-\delta ( 2q+\nu +d/2)} t^q_2\) and \(\sum _{p,q=0}^\infty a^{\delta ,\delta '}_{pq}t_1^{p-\delta ( 2q+\nu +d/2)} t^q_2\), \(\delta =0,\;1\), for Bessel functions are obtained. A study of the domains of convergence of these series (using Horn's method) is developed. The motivation for the investigations of this paper is the hope to obtain analogous results for the higher rank Bessel systems by means of an inductive method.