Szegő type polynomials and para-orthogonal polynomials

Szegő type polynomials with respect to a linear functional M for which the moments M [ t n ] = μ − n are all complex, μ − n = μ n and D n ≠ 0 for n ⩾ 0 , are considered. Here, D n are the associated Toeplitz determinants. Para-orthogonal polynomials are also studied without relying on any integral representation. Relation between the Toeplitz determinants of two different types of moment functionals are given. Starting from the existence of polynomials similar to para-orthogonal polynomials, sufficient conditions for the existence of Szegő type polynomials are also given. Examples are provided to justify the results.