The Pseudo Cosine Sequences of a Distance-Regular Graph

Let $G$ denote a distance-regular graph with diameter $D\ge 3$, valency $k$, and intersection numbers $a_i$, $b_i$, $c_i$. By a {\it pseudo cosine sequence} of $G$ we mean a sequence of real numbers $s_0,s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_D$ such that $s_0=1$ and $c_i{s}_{i-1}+a_i{s}_i+b_i{s}_{i+1}=k{s}_1{s}_i$ for $0\le i\le D-1$. Let $s_0,s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_D$ and $p_0,p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_D$ denote pseudo cosine sequences of $G$. We say this pair of sequences is {\it tight} whenever $s_0 p_0, s_1 p_1, …, s_D p_D$isapseudocosinesequenceof$G$. In this paper, we determine all the tight pairs of pseudo cosine sequences of $G$.