A bilinear form relating two Leonard systems

Let Φ, Φ^′ be Leonard systems over a field , 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.