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Orthogonal polynomials and quadratic transformations. (English) Zbl 0936.42012

Two problems concerning quadratic transformation of orthogonal polynomials are studied. Let \(P_n(x)\) \((n \geq 0)\) be a sequence of monic orthogonal polynomials and \(Q_n(x)\) \((n \geq 0)\) a sequence of monic polynomials such that the polynomials with even degree are given by \(Q_{2n}(x)=P_n(T(x))\), with \(T(x)\) a quadratic monic polynomial. The first problem is to find necessary and sufficient conditions such that \(Q_n(x)\) \((n\geq 0)\) is also a sequence of monic orthogonal polynomials, to find the relation between the moment functionals of these two systems of orthogonal polynomials, and to characterize the positive definite case (when orthogonality is with respect to a positive measure). This problem is studied in Section 2 and the analysis is based on kernel polynomials for a specific parameter \(c\). For the second problem the polynomials of odd degree are given by \(Q_{2n+1}(x) = (x-a)P_n(T(x))\), with \(a\) a fixed number. This problem is studied in Section 3 and uses co-recursive polynomials. The results are illustrated by three examples (Section 4).

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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