Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas

Abstract

When a nontrivial measure µ on the unit circle satisfies the symmetry dµ(e i(2π−θ) ) = −dµ(e iθ ) then the associated OPUC, say S n , are all real. In this case, in [12], Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials {zS n (z) + S * n (z)} and {zS n (z) − S * n (z)} satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval [−1, 1]. The same authors, in [13], have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently in [10] and then [8] the extension associated with the para-orthogonal polynomials zS n (z)− S * n (z) was thoroughly explored, especially from the point of view of the three term recurrence, and chain sequences play an important part in this exploration. The main objective of the present manuscript is to provide the theory surrounding the extension associated with the para-orthogonal polynomials zS n (z) + S * n (z) for any nontrivial measure on the unit circle. Like in [10] and [8] chain sequences also play an important role in this theory. Examples and applications are also provided to justify the results obtained.

Complementary Romanovski-Routh polynomials: From orthogonal polynomials on the unit circle to Coulomb wave functions

“…with respect to the measure µ (b) given above. The polynomials R n (b; ·) and the associated orthogonal polynomials Φ n (b; ·), in addition to have been studied in [31], they have been used as examples in a sequence of papers [5,7,8,10,12,18,19], without knowing anything about their connection to the CRR polynomials. The results obtained in [18] are focused on the three term recurrence of the type (1.7) and the associated generalized eigenvalue problem (with these respect see also [17] and [40]).…”

Quadrature rules from a RII type recurrence relation and associated quadrature rules on the unit circle

“…With specific choices of {ρ n } n≥0 , such that |ρ n | = 1, n ≥ 0, the resulting sequences of para-orthogonal polynomials {Ψ n (µ; ρ n−1 , z)} n≥1 can be made to satisfy nice three term recurrence relations (see, for example, [4] and [8]).…”

“…The choice τ 1 = I(µ)/I(µ) follows from [4,Thm. 4.2], by observing that the polynomials {R n } n≥0 given by R 0 (z) = 1 and…”

“…Proof. The proof of this theorem follows from results established in [4] and [8]. To sketch the direction behind the proof, we consider the monic para-orthogonal polynomials Ψ n+1 (ν ǫ ; τ n , z), n ≥ 0, where τ n = Φ n (ν ǫ ; 1)/Φ * n (ν ǫ ; 1), n ≥ 0.…”

Christoffel formula for kernel polynomials on the unit circle

“…1 In what follows we will be mainly interested in the case when all z j 's are pairwise distinct (or equivalently, when all zeros of G 2m are simple). 4) and either one of the following three conditions is satisfied: 6) or p j (0) = 0 for j = 1, . .…”