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Three-dimensional Riordan arrays and bivariate Laguerre polynomials

Nikolai A. Krylov

TL;DR

This work develops a 3-D Riordan array framework to unify Laguerre polynomials and their multivariate generalizations. It expresses Laguerre polynomials in one variable as layers and products of a signed 3-D Pascal-type Riordan array, yielding explicit exponential generating functions, and extends the construction to bivariate Laguerre polynomials with a double-indexed array representation and the compact EGF $\sum_{n,m\ge0} L_{n,m}(x,y) \frac{s^n}{n!} \frac{t^m}{m!} = \frac{e^{-(s x + t y)/(1-s-t)}}{1-s-t}$. The paper also provides explicit summation formulas and irreducibility results for these polynomials, illustrating the efficacy of 3-D Riordan products in analyzing multivariate orthogonal families. Overall, the approach offers a unified algebraic method to generate, manipulate, and obtain generating functions for Laguerre polynomials and their multivariate extensions, with potential applications in combinatorics and special function theory.

Abstract

We show how to represent various families of Laguerre polynomials by the three-dimensional Riordan arrays, and use the fundamental theorem of Riordan arrays to obtain the corresponding exponential generating functions.

Three-dimensional Riordan arrays and bivariate Laguerre polynomials

TL;DR

This work develops a 3-D Riordan array framework to unify Laguerre polynomials and their multivariate generalizations. It expresses Laguerre polynomials in one variable as layers and products of a signed 3-D Pascal-type Riordan array, yielding explicit exponential generating functions, and extends the construction to bivariate Laguerre polynomials with a double-indexed array representation and the compact EGF . The paper also provides explicit summation formulas and irreducibility results for these polynomials, illustrating the efficacy of 3-D Riordan products in analyzing multivariate orthogonal families. Overall, the approach offers a unified algebraic method to generate, manipulate, and obtain generating functions for Laguerre polynomials and their multivariate extensions, with potential applications in combinatorics and special function theory.

Abstract

We show how to represent various families of Laguerre polynomials by the three-dimensional Riordan arrays, and use the fundamental theorem of Riordan arrays to obtain the corresponding exponential generating functions.

Paper Structure

This paper contains 3 sections, 3 theorems, 72 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. 3-D Riordan arrays and Laguerre polynomials
  3. Bivariate Laguerre polynomials irreducible over $\mathbb Q$

Key Result

Theorem 1

For all $n,m\in\mathbb N_0$, the infinite 2-D matrix of bivariate Laguerre polynomials with integer coefficients $\bigl[L_{n,m}(x,y)\bigr]_{n,m\geq 0}$ equals the following formal product of two 3-D Riordan arrays and the infinite 2-D matrix $\bigl[x^j\bigr]_{j,k\geq 0}$.

Figures (1)

  • Figure 1: Exponential 3-D Riordan array $\left[\frac{1}{1-t},\frac{-t}{1-t},\frac{1}{1-t}\right]$

Theorems & Definitions (6)

  • Theorem 1
  • proof : Proof of the Theorem.
  • Corollary 1
  • proof
  • Theorem 2
  • proof