Balanced Leonard Pairs

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. By a Leonard pair on $V$ we mean an ordered pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following two conditions: (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. Let $v^*_0, ..., v^*_d$ (resp. $v_0, ..., v_d$) denote a basis for $V$ that satisfies (i) (resp. (ii)). For $0 \leq i \leq d$ let $a_i$ denote the coefficient of $v^*_i$, when we write $A v^*_i$ as a linear combination of $v^*_0, ..., v^*_d$, and let $a^*_i$ denote the coefficient of $v_i$, when we write $A^* v_i$ as a linear combination of $v_0..., v_d$. In this paper we show $a_0=a_d$ if and only if $a^*_0=a^*_d$. Moreover we show that for $d \geq 1$ the following are equivalent: (i) $a_0=a_d$ and $a_1=a_{d-1}$; (ii) $a^*_0=a^*_d$ and $a^*_1=a^*_{d-1}$; (iii) $a_i=a_{d-i}$ and $a^*_i=a^*_{d-i}$ for $0 \leq i \leq d$. We say $A$, $A^*$ is balanced whenever (i)--(iii) hold. We say $A$, $A^*$ is essentially bipartite (resp. essentially dual bipartite}) whenever $a_i$ (resp. $a^*_i$) is independent of $i$ for $0 \leq i \leq d$. Observe that if $A$, $A^*$ is essentially bipartite or dual bipartite, then $A$, $A^*$ is balanced. For $d \neq 2$ we show that if $A$, $A^*$ is balanced then $A$, $A^*$ is essentially bipartite or dual bipartite.