On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures

Details On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures

2012-02-08

arxiv.org/abs/1202.1606v1

Mathematics

Classical Analysis and ODEs

Scientific paper

We consider two positive, normalized measures dA(x) and dB(x) related by the relationship dA(x)=(C/(x+D))dB(x) or by dA(x) = (C/(x^{2}+E))dB(x) and dB(x) is symmetric. We show that then the polynomials sequences {a_{n}(x)}, {b_{n}(x)} orthogonal with respect to these measures are related by the relationship a_{n}(x)=b_{n}(x)+{\kappa}_{n}b_{n-1}(x) or by a_{n}(x) = b_{n}(x) + {\lambda}_{n}b_{n-2}(x) for some sequences {{\kappa}_{n}} and {{\lambda}_{n}}. We present several examples illustrating this fact and also present some attempts for generalizations. We also give some universal identities involving polynomials {b_{n}(x)} and the sequence {{\kappa}_{n}}.

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