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We consider two positive, normalized measures dA(x) and dB(x) related by therelationship dA(x)=(C/(x+D))dB(x) or by dA(x) = (C/(x^2+E))dB(x) and dB(x) issymmetric. We show that then the polynomial sequences {a_{n}(x)}, {b_{n}(x)}orthogonal with respect to these measures are related by the relationshipa_{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 someattempts for extensions and generalizations. We also give some universalidentities involving polynomials {b_{n}(x)} and the sequence {{\kappa}_{n}}that have a form of Fourier series expansion of the Radon--Nikodym derivativeof one measure with respect to the other.

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