Two Linear Transformations each Tridiagonal with Respect to an Eigenbasis of the other; Comments on the Parameter Array

Let K denote a field. Let d denote a nonnegative integer and consider a sequence p = (θi, θ ∗ i , i = 0...d;φj , φj , j = 1...d) consisting of scalars taken from K. We call p a parameter array whenever: (PA1) θi 6= θj, θ ∗ i 6= θ ∗ j if i 6= j, (0 ≤ i, j ≤ d); (PA2) φi 6= 0, φi 6= 0 (1 ≤ i ≤ d); (PA3) φi = φ1 ∑i−1 h=0 θh−θd−h θ0−θd + (θ i − θ ∗ 0)(θi−1 − θd) (1 ≤ i ≤ d); (PA4) φi = φ1 ∑i−1 h=0 θh−θd−h θ0−θd + (θ i − θ ∗ 0)(θd−i+1 − θ0) (1 ≤ i ≤ d); (PA5) (θi−2 − θi+1)(θi−1 − θi) , (θ i−2 − θ ∗ i+1)(θ ∗ i−1 − θ ∗ i ) −1 are equal and independent of i for 2 ≤ i ≤ d − 1. In [13] we showed the parameter arrays are in bijection with the isomorphism classes of Leonard systems. Using this bijection we obtain the following two characterizations of parameter arrays. Assume p satisfies PA1, PA2. Let A,B,A, B denote the matrices in Matd+1(K) which have entries Aii = θi, Bii = θd−i, A∗ii = θ ∗ i , B ∗ ii = θ ∗ i (0 ≤ i ≤ d), Ai,i−1 = 1, Bi,i−1 = 1, A ∗ i−1,i = φi, B ∗ i−1,i = φi (1 ≤ i ≤ d), and all other entries 0. We show the following are equivalent: (i) p satisfies PA3–PA5; (ii) there exists an invertible G ∈ Matd+1(K) such that G AG = B and GAG = B; (iii) for 0 ≤ i ≤ d the polynomial i ∑ n=0 (λ − θ0)(λ − θ1) · · · (λ − θn−1)(θ ∗ i − θ ∗ 0)(θ ∗ i − θ ∗ 1) · · · (θ ∗ i − θ ∗ n−1) φ1φ2 · · ·φn is a scalar multiple of the polynomial i ∑ n=0 (λ − θd)(λ − θd−1) · · · (λ − θd−n+1)(θ ∗ i − θ ∗ 0)(θ ∗ i − θ ∗ 1) · · · (θ ∗ i − θ ∗ n−1) φ1φ2 · · · φn . We display all the parameter arrays in parametric form. For each array we compute the above polynomials. The resulting polynomials form a class consisting of the q-Racah, qHahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, affine q-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, Bannai/Ito, and Orphan polynomials. The Bannai/Ito polynomials can be obtained from the q-Racah polynomials by letting q tend to −1. The Orphan polynomials have maximal degree 3 and exist for char(K) = 2 only. For each of the polynomials listed above we give the orthogonality, 3-term recurrence, and difference equation in terms of the parameter array. ∗