Orthogonal Polynomials Associated with Complementary Chain Sequences

Abstract

Abstract. Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. Using the relation between these complementary chain sequences and the corresponding Verblunsky coefficients, the para-orthogonal polynomials and the associated Szegő polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric functions and the other involving Carathéodory functions are also provided. A connection between these two illustrations by means of complementary chain sequences is also observed.

Rational spectral transformation of continued fractions associated to a perturbed $R_I$ type recurrence relations

“…, n ≥ 0. It is important to remark that this situation is same as discussed in [1] when η = 1. In fact, the co-dilated chain sequence discussed above is the complementary chain sequence for the case η = 1 in (6.8).…”

“…It was also shown in[1, Section 4] that the polynomialsrn+1 (z) = ((1 + i βn+1 )z + (1 − i βn+1 ))r n (z) − 4a n+1 zr n−1 (z),with initial conditions r 0 = 1 andr 1 = (1 + i β1 )z + (1 − i β1 ), are palindromic polynomials rn (z) = z n + ω (η) (z n−1 + . .…”

Vietoris type theorem related to positivity of trigonometric polynomials

“…In [5], an illustration of Orthogonal polynomials on the unit circle are provided in terms of the Gaussian hypergeometric function 2 F 1 (a, b; c; z). Taking lead from the results given in [5] we provide the following result.…”