Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials (original) (raw)

Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

Applicationes Mathematicae, 2002

Let {P k } be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients a k in f = k a k P k. A systematic use of the basic properties (including some nonstandard ones) of the polynomials {P k } results in obtaining a low order of the recurrence.

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

Numerical Algorithms, 2000

Let {P k } 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 a k [f ] in f = k a k [f ]P k. A systematic use of basic properties (including some nonstandard ones) of the polynomials {P k } 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.