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)=Cx+DdB(x) or by dA(x)=Cx^2+EdB(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)[email protected]"nb"n"-"1(x) or by a"n(x)=b"n(x)[email protected]"nb"n"-"2(x) for some sequences @k"n and @l"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 @k"n that have a form of the Fourier series expansion of the Radon-Nikodym derivative of one measure with respect to the other.

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