Some trace formulae involving the split sequences of a Leonard pair

TL;DR

This paper derives trace formulae involving the split sequences of Leonard pairs, which are special pairs of linear transformations with particular tridiagonal and diagonal matrix representations.

Contribution

It introduces new trace formulae for the split sequences of Leonard pairs, enhancing understanding of their algebraic structure and properties.

Findings

Derived explicit trace formulae for split sequences

Connected split sequences to trace functions

Provided new algebraic identities involving Leonard pairs

Abstract

Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy (i), (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a {\em Leonard pair} on $V$. In the literature on Leonard pairs there exist two parameter sequences called the first split sequence and the second split sequence. We display some attractive formulae for the first and second split sequence that involve the trace function.