There exist several different characterizations of the class of quadratic natural exponential families onR, two of which use orthogonal polynomials. In Feinsilver (1986), the polynomials result from the derivation of the probability densities while Meixner (1934) adopts an exponential generating function. In this paper, we consider multidimensional extensions of their results which still yield quadratic or simple quadratic natural exponential families.