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) and (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 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, , . 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, , . 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.