Paper detail

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

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 polynomial sequences {a_{n}(x)}, {b_{n}(x)} orthogonal with respect to these measures are related by the relationship a_{n}(x)=b_{n}(x)+κ_{n}b_{n-1}(x) or by a_{n}(x) = b_{n}(x) + λ_{n}b_{n-2}(x) for some sequences {κ_{n}} and {λ_{n}}. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {b_{n}(x)} and the sequence {κ_{n}} that have a form of Fourier series expansion of the Radon--Nikodym derivative of one measure with respect to the other.