Structure relations for orthogonal polynomials on the unit circle

Structure relations for orthogonal polynomials with respect to Hermitian linear functionals are studied. Firstly, we prove that semi-classical orthogonal polynomials satisfy structure relations of the following type: , where s1,s2,r1,r2 are integers (specified in the text), is the reversed polynomial of Pn, , and βn,k,γn,k,αn,k,ηn,k are complex numbers. Then, we study the semi-classical character of sequences of orthogonal polynomials {Rn},{Pn}, connected through a structure relation of the following type: , where the integers s1,s2,r1,r2 satisfy some natural conditions specified in the text.