Boundary Problems Leading to Orthogonal Polynomials in Several Variables

where we take formally £_i(x)=0, po(x) — l, and have incorporated suitable constant factors in the pn{x). Conversely [2] we may pass from (1.2) to the existence of at least one spectral function \[/(x) with respect to which (1.1) holds. The position is much less clear in respect to orthogonal polynomials in several variables, for which work has been devoted mainly to the approach which starts with the orthogonality and to analogues of the classical polynomials. This field is surveyed in [3 ], where it is noted that there is a deficiency of results leading from recurrence relations to orthogonality. The purpose of this note is to indicate a genuine extension of (1.2) in the direction of several variables, preserving the usual oscillatory and orthogonal properties. We conclude by giving the relation between the polynomials to be defined and certain determinants of ordinary orthogonal polynomials, considered in a recent series of papers by Karlin and McGregor [4] ; in addition, we indicate the relation between this topic and the more general subject of simultaneous eigenvalue problems involving several parameters.