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, … , θ_d) 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, … , u_d for V and there exist scalars ?₁, ?₂, … , ?_d in K such that Au_i = θ_iu_i + u_i+1 (0 ? i ? d − 1), Au_d = θ_du_d, , . The sequence ?₁, ?₂, … , ?_d is called the first split sequence of the Leonard pair. It is known that there exists a basis v₀, v₁, … , v_d for V and there exist scalars ?₁, ?₂, … , ?_d in K such that Av_i = θ_d−iv_i + v_i+1 (0 ? i ? d − 1),Av_d = θ₀v_d, , . The sequence ?₁, ?₂, … , ?_d 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.