𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Polynomials with Odd Orthogonal Multiplicity

✍ Scribed by Alan G.B. Lauder

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
296 KB
Volume
4
Category
Article
ISSN
1071-5797

No coin nor oath required. For personal study only.

✦ Synopsis

Let the orthogonal multiplicity of a monic polynomial g over a field % be the number of polynomials f over %, coprime to g and of degree less than that of g, such that all the partial quotients of the continued fraction expansion of f/g are of degree 1. Polynomials with positive orthogonal multiplicity arise in stream cipher theory, part of cryptography, as the minimal polynomials of the initial segments of sequences which have perfect linear complexity profiles. This paper focuses on polynomials which have odd orthogonal multiplicity; such polynomials are characterized and a lower bound on their orthogonal multiplicity is given. A special case of a conjecture on rational functions over the finite field of two elements with partial quotients of degree 1 or 2 in their continued fraction expansion is also proved.

πŸ“œ SIMILAR VOLUMES

Semiclassical Multiple Orthogonal Polyno
Semiclassical Multiple Orthogonal Polynomials and the Properties of Jacobi–Bessel Polynomials
✍ A.I. Aptekarev; F. MarcellΓ‘n; I.A. Rocha πŸ“‚ Article πŸ“… 1997 πŸ› Elsevier Science 🌐 English βš– 429 KB

This paper deals with Hermite Pade polynomials in the case where the multiple orthogonality condition is related to semiclassical functionals. The polynomials, introduced in such a way, are a generalization of classical orthogonal polynomials (Jacobi, Laguerre, Hermite, and Bessel polynomials). They

A Lower Bound for Orthogonal Polynomials
A Lower Bound for Orthogonal Polynomials with an Application to Polynomial Hypergroups
✍ R. Szwarc πŸ“‚ Article πŸ“… 1995 πŸ› Elsevier Science 🌐 English βš– 173 KB

We give a lower bound for solutions of linear recurrence relations of the form \(z a_{n}=\sum_{k=n-N}^{n+N} \alpha_{k, n} a_{k}\), whenever \(z\) is not in the \(P^{P}\)-spectrum of the corresponding banded operator. In particular if \(P_{n}\) are polynomials orthonormal with respect to a measure \(

On Some Classes of Polynomials Orthogona
On Some Classes of Polynomials Orthogonal on Arcs of the Unit Circle Connected with Symmetric Orthogonal Polynomials on an Interval
✍ Alexei Zhedanov πŸ“‚ Article πŸ“… 1998 πŸ› Elsevier Science 🌐 English βš– 404 KB

Starting from the Delsarte Genin (DG) mapping of the symmetric orthogonal polynomials on an interval (OPI) we construct a one-parameter family of polynomials orthogonal on the unit circle (OPC). The value of the parameter defines the arc on the circle where the weight function vanishes. Some explici

Strong asymptotics of orthogonal polynom
Strong asymptotics of orthogonal polynomials with respect to exponential weights
✍ P. Deift; T. Kriecherbauer; K. T-R McLaughlin; S. Venakides; X. Zhou πŸ“‚ Article πŸ“… 1999 πŸ› John Wiley and Sons 🌐 English βš– 344 KB

We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e -Q(x) dx on the real line, where Q(x) = βˆ‘ 2m k=0 q k x k , q 2m > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem

Relative Asymptotics for Orthogonal Poly
Relative Asymptotics for Orthogonal Polynomials with a Sobolev Inner Product
✍ F. Marcellan; W. Vanassche πŸ“‚ Article πŸ“… 1993 πŸ› Elsevier Science 🌐 English βš– 413 KB

We investigate orthogonal polynomials for a Sobolev type inner product \(\langle f, g\rangle=(f, g)+\lambda f^{\prime}(c) g^{\prime}(c)\), where \((f, g)\) is an ordinary inner product in \(L_{2}(\mu)\) with \(\mu\) a positive measure on the real line. We compare the Sobolev orthogonal polynomials w