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Change of basis for the tridiagonal pairs of type II

Nicolas Crampe, Julien Gaboriaud, Satoshi Tsujimoto

Abstract

We study tridiagonal pairs of type II. These involve two linear transformations $A$ and $A^\star$. We define two bases. In the first one, $A$ acts as a diagonal matrix while $A^\star$ acts as a block tridiagonal matrix, and in the second one, $A$ acts as a block tridiagonal matrix while $A^\star$ acts as a diagonal matrix. We obtain the change of basis coefficients between these two bases. The coefficients are special functions that are written as a nested product of polynomials that resemble Racah polynomials but involve shift operators in their expression.

Change of basis for the tridiagonal pairs of type II

Abstract

We study tridiagonal pairs of type II. These involve two linear transformations and . We define two bases. In the first one, acts as a diagonal matrix while acts as a block tridiagonal matrix, and in the second one, acts as a block tridiagonal matrix while acts as a diagonal matrix. We obtain the change of basis coefficients between these two bases. The coefficients are special functions that are written as a nested product of polynomials that resemble Racah polynomials but involve shift operators in their expression.

Paper Structure

This paper contains 25 sections, 8 theorems, 83 equations, 1 table.

Table of Contents

  1. Introduction
  2. Leonard pairs.
  3. Tridiagonal pairs.
  4. The goal.
  5. The bases.
  6. Multivariate special functions from univariate ones.
  7. Structure of the paper.
  8. Tridiagonal pairs of type II
  9. Split bases for tridiagonal pairs of type II
  10. Split basis
  11. The involution S
  12. Parameter constraints and irreducibility
  13. Tridiagonal relations
  14. Explicit diagonalization
  15. Diagonalization of A
  16. ...and 10 more sections

Key Result

Theorem 3.2

The operators $A$ and $A^\star$ given by eq:AAsVn satisfy the tridiagonal relations: Here $[X,Y]=XY-YX$ and $\{X,Y\}=XY+YX$.

Theorems & Definitions (14)

  • Definition 2.1: ItoTanabeetal2001
  • Definition 2.2
  • Example 3.1
  • Theorem 3.2
  • Proposition 4.1
  • Proposition 4.2
  • Remark 4.3
  • Proposition 4.4
  • Remark 4.5
  • Proposition 4.6
  • ...and 4 more