How to recognize a Leonard pair

Abstract: Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\rightarrow V$ and $A^{*}: V\rightarrow V$ that satisfy (i) and (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 Leonard pair on $V$. In the literature, there are some parameters that are used to describe Leonard pairs called the intersection numbers ${a_{i}}{i=0}^{d}$, ${b{i}}{i=0}^{d-1}$, ${c{i}}{i=1}^{d}$, and the dual eigenvalues ${\theta^{*}{i}}{i=0}^{d}$. In this paper, we provide two characterizations of Leonard pairs. For the first characterization, the focus is on the ${a{i}}{i=0}^{d}$ and ${\theta^{*}{i}}{i=0}^{d}$. For the second characterization, the focus is on the ${b{i}}{i=0}^{d-1}$, ${c{i}}{i=1}^{d}$, and ${\theta^{*}{i}}_{i=0}^{d}$.