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Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials

Sergey M. Zagorodnyuk

Abstract

In this paper we study the generalized Bessel polynomials $y_n(x,a,b)$ (in the notation of Krall and Frink). Let $a>1$, $b\in(-1/3,1/3)\backslash\{ 0\}$. In this case we present the following positive continuous weights $p(θ) = p(θ,a,b)$ on the unit circle for $y_n(x,a,b)$: $$ 2πp(θ,a,b) = -1 + 2(a-1) \int_0^1 e^{-bu\cosθ} \cos(bu\sinθ) (1-u)^{a-2} du, $$ where $θ\in[0,2π]$. Namely, we have $$ \int_0^{2π} y_n(e^{iθ},a,b) y_m(e^{iθ},a,b) p(θ,a,b) dθ= C_n δ_{n,m},\qquad C_n\not=0,\ n,m\in\mathbb{Z}_+. $$ Notice that this orthogonality differs from the usual orthogonality of OPUC. Some applications of the above orthogonality are given.

Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials

Abstract

In this paper we study the generalized Bessel polynomials (in the notation of Krall and Frink). Let , . In this case we present the following positive continuous weights on the unit circle for : where . Namely, we have Notice that this orthogonality differs from the usual orthogonality of OPUC. Some applications of the above orthogonality are given.
Paper Structure (2 sections, 3 theorems, 96 equations)

This paper contains 2 sections, 3 theorems, 96 equations.

Table of Contents

  1. Introduction.
  2. Positive weights for Bessel polynomials.

Key Result

Theorem 1

Let $x_0$ be the unique number from the interval $[0,\pi/2]$ such that $2 \cos x_0 = e^{x_0}$. Let $a>1$, $b\in(-x_0,x_0)\backslash\{ 0 \}\supseteq (-1/3,1/3)\backslash\{ 0 \}$, be fixed numbers. The generalized Bessel polynomials $y_n(x,a,b)$ satisfy orthogonality relations (f1_14), where the posit

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 1
  • Proposition 1