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The Pascal Matrix, Commuting Tridiagonal Operators and Fourier Algebras

W. Riley Casper, Ignacio Zurrian

Abstract

We consider the (symmetric) Pascal matrix, in its finite and infinite versions, and prove the existence of symmetric tridiagonal matrices commuting with it by giving explicit expressions for these commuting matrices. This is achieved by studying the associated Fourier algebra, which as a byproduct, allows us to show that all the linear relations of a certain general form for the entries of the Pascal matrix arise from only three basic relations. We also show that pairs of eigenvectors of the tridiagonal matrix define a natural eigenbasis for the binomial transform. Lastly, we show that the commuting tridiagonal matrices provide a numerically stable means of diagonalizing the Pascal matrix.

The Pascal Matrix, Commuting Tridiagonal Operators and Fourier Algebras

Abstract

We consider the (symmetric) Pascal matrix, in its finite and infinite versions, and prove the existence of symmetric tridiagonal matrices commuting with it by giving explicit expressions for these commuting matrices. This is achieved by studying the associated Fourier algebra, which as a byproduct, allows us to show that all the linear relations of a certain general form for the entries of the Pascal matrix arise from only three basic relations. We also show that pairs of eigenvectors of the tridiagonal matrix define a natural eigenbasis for the binomial transform. Lastly, we show that the commuting tridiagonal matrices provide a numerically stable means of diagonalizing the Pascal matrix.
Paper Structure (9 sections, 11 theorems, 84 equations, 1 figure)

This paper contains 9 sections, 11 theorems, 84 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bispectrality and Fourier algebras
  3. Bispectral functions
  4. Fourier algebras
  5. Matrices commuting with Pascal's matrix
  6. Proof of the Main Theorem
  7. A dual commuting pair
  8. Spectra and the binomial transform
  9. Numerical diagonalization of the Pascal matrix

Key Result

Theorem 2.2

The function $\psi: \mathbb{N}\times\mathbb{N}\rightarrow \mathbb{R}$ defined by is bispectral and satisfies

Figures (1)

  • Figure 1: Comparison of the $\ell^2$-norm errors in eigenvectors of $T_N$, calculated from diagonalizing $T_N$ versus diagonalizing $J_N$ for $N=15$.

Theorems & Definitions (25)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 3.1
  • proof
  • ...and 15 more