Some trace formulae involving the split sequences of a Leonard pair

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i), (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V . Let diag(θ0, θ1, . . . , θd) denote the diagonal matrix referred to in (ii) above and let diag(θ∗ 0 , θ ∗ 1 , . . . , θ ∗ d ) denote the diagonal matrix referred to in (i) above. It is known that there exists a basis u0, u1, . . . , ud for V and there exist scalars φ1, φ2, . . . , φd in K such that Aui = θiui + ui+1 (0 ≤ i ≤ d − 1), Aud = θdud, A ∗ui = φiui−1 + θ ∗ i ui (1 ≤ i ≤ d), A ∗u0 = θ ∗ 0u0. The sequence φ1, φ2, . . . , φd is called the first split sequence of the Leonard pair. It is known that there exists a basis v0, v1, . . . , vd for V and there exist scalars φ1, φ2, . . . , φd in K such that Avi = θd−ivi + vi+1 (0 ≤ i ≤ d − 1), Avd = θ0vd, A ∗vi = φivi−1 + θ ∗ i vi (1 ≤ i ≤ d), A∗v0 = θ ∗ 0v0. The sequence φ1, φ2, . . . , φd is called the second split sequence of the Leonard pair. We display some attractive formulae for the first and second split sequence that involve the trace function. 1 Leonard pairs and Leonard systems Throughout this paper K will denote an arbitrary field. We begin by recalling the notion of a Leonard pair. We will use the following notation. A square matrix X is called tridiagonal whenever each nonzero entry lies on either the diagonal, the subdiagonal, or the superdiagonal. Assume X is tridiagonal. Then X is called irreducible whenever each entry on the subdiagonal is nonzero and each entry on the superdiagonal is nonzero. Definition 1.1 [14] Let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A : V → V that satisfy (i), (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. Note 1.2 It is a common notational convention to use A to represent the conjugatetranspose of A. We are not using this convention. In a Leonard pair A, A the linear transformations A and A are arbitrary subject to (i) and (ii) above. We refer the reader to [1], [2], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25] for background on Leonard pairs. We especially recommend the survey [23]. When working with a Leonard pair, it is convenient to consider a closely related object called a Leonard system. To prepare for our definition of a Leonard system, we recall a few concepts from linear algebra. Let d denote a nonnegative integer and let Matd+1(K) denote the K-algebra consisting of all d+ 1 by d + 1 matrices that have entries in K. We index the rows and columns by 0, 1, . . . , d. For the rest of this paper we let A denote a Kalgebra isomorphic to Matd+1(K). Let V denote a simple A-module. We remark that V is unique up to isomorphism of A-modules, and that V has dimension d+1. Let v0, v1, . . . , vd denote a basis for V . For X ∈ A and Y ∈ Matd+1(K), we say Y represents X with respect to v0, v1, . . . , vd whenever Xvj = ∑d i=0 Yijvi for 0 ≤ j ≤ d. Let A denote an element of A. We say A is multiplicity-free whenever it has d+ 1 mutually distinct eigenvalues in K. Let A denote a multiplicity-free element in A. Let θ0, θ1, . . . , θd denote an ordering of the eigenvalues of A, and for 0 ≤ i ≤ d put