A bilinear form relating two Leonard systems

Understand

Let $\Phi$, $\Phi'$ be Leonard systems over a field $\mathbb{K}$, and $V$, $V'$ the vector spaces underlying $\Phi$, $\Phi'$, respectively.

• In this paper, we introduce and discuss a balanced bilinear form on $V\times V'$.
• Such a form naturally arises in the study of $Q$-polynomial distance-regular graphs.
• We characterize a balanced bilinear form from several points of view.