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

Stanislaw Lewanowicz — 2002

Applicationes Mathematicae

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.

Minimal projective operators

Stanisław Lewanowicz — 1979

Mathematica Applicanda

The author reviews results (without proofs) from the theory of minimal projective operators. As he remarks, an excellent introduction to this theory is the survey paper by E. W. Cheney and K. H. Price [Approximation theory (Proc. Sympos., Lancaster, 1969), pp. 261–289, Academic Press, London, 1970; MR0265842]. The author is motivated by a number of papers in this topic published after 1970, bringing essentially new results, e.g., an existence theorem for the minimal operators in the class of all...

Linearization of the product of orthogonal polynomials of a discrete variable

Saïd BelmehdiStanisław LewanowiczAndré Ronveaux — 1997

Applicationes Mathematicae

Let P k be any sequence of classical orthogonal polynomials of a discrete variable. We give explicitly a recurrence relation (in k) for the coefficients in P i P j = k c ( i , j , k ) P k , in terms of the coefficients σ and τ of the Pearson equation satisfied by the weight function ϱ, and the coefficients of the three-term recurrence relation and of two structure relations obeyed by P k .

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