(i) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^* is irreducible tridiagonal.

(ii) There exists a basis for V with respect to which the matrix representing A^* is diagonal and the matrix representing A is irreducible tridiagonal.We call such a pair a Leonard pair on V. Let (A, A^*) denote a Leonard pair on V. A basis for V is said to be standard for (A, A^*) whenever it satisfies (i) or (ii) above. A basis for V is said to be split for (A, A^*) whenever with respect to this basis the matrix representing one of A, A^* is lower bidiagonal and the matrix representing the other is upper bidiagonal. Let (A, A^*) and (B, B^*) denote Leonard pairs on V. We say these pairs are adjacent whenever each basis for V which is standard for (A, A^*) (resp. (B, B^*)) is split for (B, B^*) (resp. (A, A^*)). Our main results are as follows.

Theorem 1

There exist at most 3 mutually adjacent Leonard pairs on V provided the dimension of V is at least 2.

Theorem 2

Let (A, A^*), (B, B^*), and (C, C^*) denote three mutually adjacent Leonard pairs on V. Then for each of these pairs, the eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression.

Theorem 3

Let (A, A^*) denote a Leonard pair on V whose eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression. Then there exist Leonard pairs (B, B^*) and (C, C^*) on V such that (A, A^*), (B, B^*), and (C, C^*) are mutually adjacent.