The pseudo-cosine sequences of a distance-regular graph

Let Γ denote a distance-regular graph with diameter D ⩾ 3, valency k, and intersection numbers ai, bi, ci. By a pseudo-cosine sequence of Γ we mean a sequence of real numbers σ0, σ1, … , σD such that σ0 = 1 and ciσi−1 + aiσi + biσi + 1 = kσ1σi for 0 ⩽ i ⩽ D − 1. Let σ0, σ1, …, σD and ρ0, ρ1, … , ρD denote pseudo-cosine sequences of Γ. We say this pair of sequences is tight whenever σ0ρ0, σ1ρ1, …, σDρD is a pseudo-cosine sequence of Γ. In this paper, we determine all the tight pairs of pseudo-cosine sequences of Γ.