Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Laguerre constellation of classical orthogonal Polynomials

Roberto S. Costas-Santos

TL;DR

This work introduces the Laguerre constellation (LC) as the collection of classical orthogonal polynomial sequences whose Pearson-type equation $\mathscr D(\phi(x)\mathbf u)=\psi(x)\mathbf u$ has $\deg \phi=1$ or $\deg \phi^*=1$. It develops a unified framework showing that LC polynomials satisfy a second-order difference (Sturm–Liouville) structure and a corresponding structure-type relation, enabling a comprehensive characterization of LC families. The paper identifies the six principal LC families—Laguerre, Charlier, Meixner, big $q$-Laguerre, little $q$-Laguerre/Wall, and Stieltjes-Wigert—and provides extensive, explicit operator identities, Rodrigues-type formulas, and shift relations for each, along with inter-family connections via parameter transformations. A central contribution is a characterization theorem stating the equivalence between belonging to the Laguerre constellation and satisfying the associated Sturm–Liouville/discrete difference equations, complemented by a detailed catalog of identities (Recurrence, Rodrigues, shift, and difference relations) for every family. The work also emphasizes computational verification, including Wolfram Mathematica implementations, to validate the intricate relations across the LC families, underscoring the practical utility of the LC classification in algebraic and computational orthogonal-polynomial theory.

Abstract

A linear functional $\bf u$ is classical if there exist polynomials, $φ$ and $ψ$, with $°φ\le 2$, $°ψ=1$, such that ${\mathscr D}\left(φ(x) {\bf u}\right)=ψ(x){\bf u}$, where ${\mathscr D}$ is a certain differential, or difference, operator. The polynomials orthogonal with respect to the linear functional ${\bf u}$ are called {\sf classical orthogonal polynomials}. In the theory of orthogonal polynomials, a correct characterization of the classical families is of great interest. In this work, on the one hand, we present the Laguerre constellation, which is formed by all the classical families for which $°φ=1$, obtaining for all of them new algebraic identities such as structure formulas, orthogonality properties as well as new Rodrigues formulas; on the other hand, we present a theorem that characterizes the classical families belonging to the Laguerre constellation.

The Laguerre constellation of classical orthogonal Polynomials

TL;DR

This work introduces the Laguerre constellation (LC) as the collection of classical orthogonal polynomial sequences whose Pearson-type equation has or . It develops a unified framework showing that LC polynomials satisfy a second-order difference (Sturm–Liouville) structure and a corresponding structure-type relation, enabling a comprehensive characterization of LC families. The paper identifies the six principal LC families—Laguerre, Charlier, Meixner, big -Laguerre, little -Laguerre/Wall, and Stieltjes-Wigert—and provides extensive, explicit operator identities, Rodrigues-type formulas, and shift relations for each, along with inter-family connections via parameter transformations. A central contribution is a characterization theorem stating the equivalence between belonging to the Laguerre constellation and satisfying the associated Sturm–Liouville/discrete difference equations, complemented by a detailed catalog of identities (Recurrence, Rodrigues, shift, and difference relations) for every family. The work also emphasizes computational verification, including Wolfram Mathematica implementations, to validate the intricate relations across the LC families, underscoring the practical utility of the LC classification in algebraic and computational orthogonal-polynomial theory.

Abstract

A linear functional is classical if there exist polynomials, and , with , , such that , where is a certain differential, or difference, operator. The polynomials orthogonal with respect to the linear functional are called {\sf classical orthogonal polynomials}. In the theory of orthogonal polynomials, a correct characterization of the classical families is of great interest. In this work, on the one hand, we present the Laguerre constellation, which is formed by all the classical families for which , obtaining for all of them new algebraic identities such as structure formulas, orthogonality properties as well as new Rodrigues formulas; on the other hand, we present a theorem that characterizes the classical families belonging to the Laguerre constellation.
Paper Structure (13 sections, 15 theorems, 51 equations, 1 figure, 1 table)

This paper contains 13 sections, 15 theorems, 51 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic concepts and results
  4. Definition of the operators in $\mathbb P$ and $\mathbb P^*$
  5. The Laguerre constellation
  6. The families
  7. The Laguerre poynomials
  8. The Charlier polynomials
  9. The Meixner polynomials
  10. The big $q$-Laguerre polynomials
  11. The little $q$-Laguerre/Wall polynomials
  12. The Stieltjes-Wigert polynomials
  13. Code in wolfram Mathematica${}^\circledR$ to check the identities

Key Result

Lemma 6

Let ${\bf u}\in \mathbb P^*$ be a quasi-definite classical functional, let $(p{\ThisStyle{\raisebox{-2\LMpt}{$\SavedStyle_$}}}n)$ be the polynomial sequence orthogonal with respect to ${\bf u}$. If $(p{\ThisStyle{\raisebox{-2\LMpt}{$\SavedStyle_$}}}n)$ belong to the LC then, there exists a numerical

Figures (1)

  • Figure 1: Relations between the families in the Laguerre constellation. The gray lines are the particular cases. The black lines are the limiting cases.

Theorems & Definitions (25)

  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 5
  • Lemma 6
  • Remark 7
  • Lemma 8
  • Remark 9
  • Lemma 10
  • Remark 12
  • ...and 15 more