Monotonicity of zeros for a class of polynomials including hypergeometric polynomials

On monotonicity of zeros of paraorthogonal polynomials on the unit circle

“…6 We quote the English translation provided by Szüsz [40, Problem LXVI]. 7 These authors never mentioned the connection with the question posed by Turán. respect to a real parameter of eigenvalues of unitary upper Hessenberg matrices with positive subdiagonal elements. Our results can be considered as analogues of those presented by Ismail and Muldoon [21] (cf.…”

On co-polynomials on the real line

“…Notice that the existence of cases for which det(∂J n ( )/∂ ) = 0, mentioned at the beginning of the section, could imply strictly monotonicity of zeros. We recall that Corollary 2.3 was also perceived in [5] from the perturbation theory for symmetric matrices. Example 2.1.…”

“…decreasing) functions of . But for some cases related with COPRL, we can obtain more information on the behavior of zeros following a different approach recently proposed in [5].…”

“…In [5], the author studied the effect on OPRL when the Verblunsky coefficients {α n } n≥0 are perturbed in the following way:…”

“…Moreover, we will consider the finite composition of the above perturbations. In Section 2, we study some new inequalities for the zeros of COPRL by following the approach presented in [5] for the study of the monotonicity of zeros of a class of para-orthogonal polynomials on the unit circle including the Askey hypergeometric polynomials 2 F 1 (−n, a + bi; 2a; 1 − z), a, b ∈ R. In Section 3, we obtain a new structural relation based on a transfer matrix approach proposed recently in [4] for similar perturbations in the theory of orthogonal polynomials on the unit circle (OPUC, in short). Finally, in Section 3 we point out the connection with the OPUC.…”