K-bessel Functions in Two Variables

The Bessel-Muirhead hypergeometric system (or 0 F 1-system) in two variables (and three variables) is solved using symmetric series, with an explicit formula for coefficients , in order to express the K-Bessel function as a linear combination of the J-solutions. Limits of this method and suggestions for generalizations to a higher rank are discussed. 1. Introduction. The Bessel functions (of the first kind) defined on the space of real symmetric matrices appeared in the work of James [5] as an ingredient in the expression of some densities in multivariate statistics. At the same time, more systematic treatment was done by Herz [4]. In [8], Muirhead proved that they are solutions of a system of differential operators which will be designated here as Bessel-Muirhead operators following [6]. We can see [1, 3] for the generalization of this set of functions to a Jordan algebra. In what follows, we explicitly write down a fundamental set of solutions when the rank equals 2 or 3. Our approach is slightly different from [7] in the final form of the coefficients. Then (and this is our main result), we express the K-Bessel function defined in this context as a linear combination of the J-solutions in the rank-2 case, so answering a question in [4].