K terms recurrence relations and polynomial variance functions of the Kth degree

The aim of this paper is to investigate the recurrence relations obtained for some polynomials associated with natural exponential families, via their variance functions. In the case of polynomial variance functions of the Kth degree we obtain a recurrence relation with K terms. A few examples are indicated and the orthogonality of such polynomials is characterized (this characterization is due to Pommeret (Test 5 (1996) 77–111)). Moreover, some of the recent results related to Lie algebras, Lévy processes and Martingales, are reviewed in a more general setting.