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

摘要

We consider two positive, normalized measures dA(x) and dB(x) related by the relationship dA(x)=Cx+DdB(x) or by dA(x)=Cx2+EdB(x) and dB(x) is symmetric. We show that then the polynomial sequences an(x),bn(x) orthogonal with respect to these measures are related by the relationship an(x)=bn(x)+κnbn-1(x) or by an(x)=bn(x)+λnbn-2(x) for some sequences κn and 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) and the sequence κn that have a form of the Fourier series expansion of the Radon–Nikodym derivative of one measure with respect to the other.