Two linear transformations each tridiagonal with respect to an   eigenbasis of the other
  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 both conditions below:   (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible tridiagonal.   (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal, and the matrix representing $A$ is irreducible tridiagonal.   We call such a pair a Leonard pair on $V$. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. We discuss how Leonard systems correspond to the $q$-Racah and related polynomials from the Askey scheme. 
Paul Terwilliger