Recurrence Relations for the Coefficients in Series Expansions with Respect to Semi-Classical Orthogonal Polynomials

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Let {Pk} be a sequence of the semi-classical orthogonal polynomials. Given a function f satisfying a linear second-order differential equation with polynomial coefficients, we describe an algorithm to construct a recurrence relation satisfied by the coefficients ak[f] in f = ∑kak[f]Pk. A systematic use of basic properties (including some nonstandard ones) of the polynomials {Pk} results in obtaining a recurrence of possibly low order. Recurrences for connection or linearization coefficients related to the first associated generalized Gegenbauer, Bessel-type and Laguerre-type polynomials are given explicitly.

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