From orthogonal polynomials on the unit circle to functional equations via generating functions (Q2790615)

On the unit circle that has certain specific Schur parameters under prescribed conditions, orthogonal polynomials are explored. Section 1 is brief yet exhaustive for including some very basic concepts and terminologies, namely, the set of monic polynomials orthogonal on the complex unit circle with respect to a certain measure, called Dirichlet measure, the sequence of a particular kind that uniquely defines a set of orthogonal polynomials on the unit circle, which is called a sequence of Schur parameters. Three term recurrence relations are defined. It is made explicit that the orthogonal polynomials that are considered in the present paper, corresponds to a certain sequence. For certain values of alpha, the sequence, defined by equation (1.2), corresponds to Geronimus polynomials, whereas for a certain value of c, it corresponds to Rogers-Szegő polynomials. For both alpha and c to be equal to zero, the sequence corresponds to the standard Lebesgue measure on the unit circle (the authors for the sake of brevity and in the context of the present paper, have preferred calling it Lebesgue polynomials). The paper concerns the investigation of a family of orthogonal polynomials whose extreme cases are the families mentioned. Section 2 investigates two generating functions under different notations, and one of them, equation (1.4), is represented as a sum of two q-hypergeometric functions which investigates the extremal cases of Geronimus and Rogers-Szegő polynomials. Section 3 describes the formulation of the second generating function, which is shown to obey a functional differential equation of the pantograph type and so it can be expanded in Dirichlet series, which eventually leads to its explicit representation in terms of the q-Bessel functions. Having employed the same method and terminologies, in Section 4, generating functions for orthogonal polynomials of the second kind are derived. Since the generating functions of two kinds, given in equations (1.4) and (1.5), are complementary, in Section 5 we notice the derivation of the underlying Caratheodory function and of the orthogonality measure. It is claimed that the construction of the orthogonal polynomials is formal, owing to the fact that the concept of investigation begins from Schur parameters and are used in the recurrence relation (1.1) rather than to commence from an orthogonality measure.