A characterization of “classical” d-orthogonal polynomials
Journal of Approximation Theory
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
Le´vy-Sheffer and IID-Sheffer polynomials with applications to stochastic integrals
Journal of Computational and Applied Mathematics
K terms recurrence relations and polynomial variance functions of the Kth degree
Journal of Computational and Applied Mathematics - Special issue on orthogonal polynomials, special functions and their applications
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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.