Association Schemes with Multiple Q-polynomial Structures^*

It is well known that an association scheme χ = (X, {R_i}0 ≤ i≤ d) with k₁ > 2 has at most two P-polynomial structures. The parametrical condition for an association scheme to have two P-polynomial structures is also known. In this paper, we give a similar result for Q-polynomial association schemes. In fact, if d > 5, then we obtain exactly the same parametrical conditions for the dual intersection numbers or Krein parameters.