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

We consider two positive, normalized measures dA(x)dA(x) and dB(x)dB(x) related by the relationship dA(x)=Cx+DdB(x) or by dA(x)=Cx2+EdB(x) and dB(x)dB(x) is symmetric. We show that then the polynomial sequences {an(x)},{bn(x)}an(x),bn(x) orthogonal with respect to these measures are related by the relationship an(x)=bn(x)+κnbn-1(x)an(x)=bn(x)+κnbn-1(x) or by an(x)=bn(x)+λnbn-2(x)an(x)=bn(x)+λnbn-2(x) for some sequences {κn}κn and {λn}λn. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {bn(x)}bn(x) and the sequence {κn}κn that have a form of the Fourier series expansion of the Radon–Nikodym derivative of one measure with respect to the other.