In this paper we give for generalized Bessel operators studied by M. I. Klyuchantsev a representation theory for solutions of the related heat equations. A

Abstract. Let ∆α be the Bessel operator with matricial coefficients defined on (0, ∞) bywhere α is a diagonal matrix and let q be an n × n matrix-valued function.In this work, we prove that there exists an isomorphism X on the space of even C ∞ , C n -valued functions which transmutes ∆α and (∆α + q). This allows us to define generalized translation operators and to develop harmonic analysis associated with (∆α + q). By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.

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Special functions of matrix argument arise in a diverse range of applications in harmonic analysis, number theory, multivariate statistics, quantum physics, and molecular chemistry. This paper presents series expansions for the Bessel functions associated with Jordan algebras of rank 2 or 3. Detailed information on the domains of convergence of these series are given where Horn's theorems for double and triple hypergeometric series are used. ᮊ 1999 Academic Press

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