A bilinear form relating two Leonard systems

Let Φ, Φ′ be Leonard systems over a field K, and V, V′ the vector spaces underlying Φ, Φ′, respectively. In this paper, we introduce and discuss a balanced bilinear form on V×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.