ABSTRACT
Laguerre polynomials, defined on the space of real symmetric matrices, appeared for the first time in the work of C.S. Herz. A remarkable application in multivariate statistics was done by A.T. James, A.G. Constantine, and R.J. Muirhead. Later and in the general case of Jordan algebra, the first author proved that spherical functions of some Gel’fand pair can be expressed by those polynomials, thus generalizing a result from A. Koranyi. Using Laplace transform and only in the cases of real symmetric and Hermitian matrices of rank, F. Ricci and A.T. Vignati proposed a set of r differential operators of order 2,4, …,2r. This chapter proposes, in the case of a rank two Jordan algebra, a set of two differential operators of order two, each for a vector of Laguerre polynomials. The technical tools are the integration by parts and the expression of the derivative of an ultraspherical polynomial in terms of ultraspherical polynomials.
