On d-orthogonality of the Sheffer systems associated to a convolution semigroup

In this note we investigate which Sheffer polynomials can be associated to a convolution semigroup of probability measures, usually induced by a stochastic process with stationary and independent increments. From a recent kind of d-orthogonality (d∈{2,3,…}), we characterize the associated d-orthogonal polynomials by the class of generating probability measures, which belongs to the natural exponential family with polynomial variance functions of exact degree 2d-1. This extends some results of (classical) orthogonality; in particular, some new sets of martingales are then pointed out. For each integer d⩾2 we completely illustrate polynomials with (2d-1)-term recurrence relation for the families of positive stable processes.