Some trace formulae involving the split sequences of a Leonard pair
This paper studies so called Leonard pairs and their split sequences. Given a field \(K\) and a vector space \(V\) of finite positive dimension over \(K\), a Leonard pair is a pair \(A,\,A^{*}\) of linear transformations from \(V\) to itself that satisfies (i) there exists a basis \(\alpha\) for \(V\) such that \(\text{mat}_{\alpha,\alpha} A\) is irreducible tridiagonal and \(\text{mat}_{\alpha,\alpha}A^{*} = \text{diag}(\theta_0^{*},\theta_1^{*}, \dots,\theta_d^{*})\), (ii) there exists a basis \(\beta\) for \(V\) such that \(\text{mat}_{\alpha,\alpha} A^{*}\) is irreducible tridiagonal and \(\text{mat}_{\alpha,\alpha}A = \text{diag}(\theta_0,\theta_1, \dots, \theta_d)\). (A tridiagonal matrix is irreducible if each entry on the sub- and super-diagonal is non-zero.) The conditions (i), (ii) imply the existence of two bases \(\{u_i\}_0^d,\, \{v_i\}_0^d\) and scalars \(\{\psi_i\}_1^d, \,\{\phi_i\}_1^d\) such that \[ \begin{alignedat}{2} &Au_i= \theta_iu_i+u_{i+1}\quad (0\leq i\leq d-1),&&\qquad Au_d=\theta_du_d, \\ &A^{*}u_i= \varphi_iu_{i-1}+\theta_i^{*}u_i \quad (1\leq i\leq d),&&\qquad A^{*}u_0= \theta_0^{*}u_0,\\ &Av_i=\theta_{d-i}v_i + v_{i+1}\quad (0\leq i\leq d-1),&&\qquad Av_d= \theta_0v_d,\\ &A^{*}v_i=\phi_iv_{i-1}+\theta_i^{*}v_i \quad (1\leq i\leq d), &&\qquad A^{*}v_0 = \theta_0^{*}v_0. \end{alignedat} \] The sequence \(\{\varphi_i\}_1^d\) is called the first split sequence of the Leonard pair. For the background on Leonard pairs the reader is referred to 24 of the 25 references in the paper. After the introduction of a Leonard system and its parameter array (split sequences and certain sets of eigenvalues), formulae expressing the trace of certain polynomials in \(A,\,A^{*}\) in terms of the parameter sequences and, conversely, formulae expressing the parameter sequences in terms of quotients of traces of the polynomials are given. Finally formulae are exhibited that relate the first and second split sequences.
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