Let
\mathbb K {\mathbb K}
denote a field. Let it d denote a nonnegative integer and consider a sequence p=(
qi, q*i,i=0...d; jj, fj,j=1...d)\theta_i, \theta^*_i,i=0...d; \varphi_j, \phi_j,j=1...{\it d})
consisting of scalars taken from
\mathbb K {\mathbb K}
. We call
p a
parameter array whenever: (PA1)
qi ¹ qj, q*i ¹ q*j\theta_i \not=\theta_j, \; \theta^*_i\not=\theta^*_j
if
i ¹ ji\not=j
are equal and independent of
i for
2 £ i £ d-12 \leq i \leq d-1
. In Terwilliger,
J. Terwilliger, Linear Algebra Appl., Vol. 330(2001) p. 155 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 and PA2. Let
A, B,A^*, B^* denote the matrices in
Matd+1{Mat}_{{\it d}+1}
(
\mathbb K {\mathbb K}
) which have entries
Aii=
\mathbb K {\mathbb K}
) such that
G–1AG=
B and
G–1A*G=
B*; (iii) for 0
$ \sum_{n=0}^i \frac{
(\lambda-\theta_0) (\lambda-\theta_1) \cdots (\lambda-\theta_{n-1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\varphi_1\varphi_2\cdots \varphi_n}$ \sum_{n=0}^i \frac{
(\lambda-\theta_0) (\lambda-\theta_1) \cdots (\lambda-\theta_{n-1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\varphi_1\varphi_2\cdots \varphi_n}
is a scalar multiple of the polynomial
$\sum_{n=0}^i \frac{ (\lambda-\theta_d) (\lambda-\theta_{d-1}) \cdots
(\lambda-\theta_{d-n+1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1)
\cdots (\theta^*_i-\theta^*_{n-1}) } {\phi_1\phi_2\cdots \phi_n}.$\sum_{n=0}^i \frac{ (\lambda-\theta_d) (\lambda-\theta_{d-1}) \cdots
(\lambda-\theta_{d-n+1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1)
\cdots (\theta^*_i-\theta^*_{n-1}) } {\phi_1\phi_2\cdots \phi_n}.
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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,
q-Hahn, 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 (
\mathbb K {\mathbb 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.