of vector spaces $\sigma:V\mapsto W$ such that $\sigma
A\sigma^{-1}=B$ and $\sigma A^*\sigma^{-1}=B^*$. We say that $(A,A^*)$ and $(B,B^*)$ are [*isomorphic*]{} if there is an isomorphism of Leonard pairs from $(A,A^*)$ to $(B,B^*)$.

\[defpa\] [@TerwLTparr] By a [*parameter array*]{} over ${{\Bbb K}}$, of diameter $d$, we mean a sequence $$\label{paraar}
(\theta_0,\theta_1,\ldots,\theta_d;\;\theta_0^*,\theta_1^*,\ldots,\theta_d^*;\;
\varphi_1,\ldots,\varphi_d;\;\phi_1,\ldots,\phi_d)$$ of scalars taken from ${{\Bbb K}}$, that satisfy the following conditions:

1.  $\theta_i\neq\theta_j$ and $\theta_i^*\neq\theta_j^*$ if $i\neq j$, for $0\le i,j\le d$.

2.  $\varphi_i\neq 0$ and $\phi_i\neq 0$, for $1\le i\le d$.

3.  $\displaystyle\varphi_i=\phi_1\sum_{j=0}^{i-1}
    \frac{\theta_j-\theta_{d-j}}{\theta_0-\theta_d}+\left(\theta^*_i-\theta_0^*\right)
    \left(\theta_{i-1}-\theta_d\right)$, for $1\le i\le d$.

4.  $\displaystyle\phi_i=\varphi_1\sum_{j=0}^{i-1}
    \frac{\theta_j-\theta_{d-j}}{\theta_0-\theta_d}+\left(\theta^*_i-\theta_0^*\right)
    \left(\theta_{d-i+1}\!-\theta_0\right)$, for $1\le i\le d$.

5.  The expressions $$\frac{\theta_{i-2}-\theta_{i+1}}{\theta_{i-1}-\theta_i},\qquad
    \frac{\theta^*_{i-2}-\theta^*_{i+1}}{\theta^*_{i-1}-\theta^*_i}$$ are equal and independent of $i$, for $2\le i\le d-1$.

To get a Leonard pair from parameter array (\[paraar\]), one must choose a basis for $V$ and define the two linear transformations by the following matrices (with respect to that basis): $$\label{split1}
\left(\begin{array}{ccccc} \theta_0 \\ 1 & \theta_1 \\ & 1 & \theta_2 \\
& & \ddots & \ddots \\ & & & 1 & \theta_d \end{array}\right),\qquad
\left(\begin{array}{ccccc} \theta^*_0 & \varphi_1 \\ & \theta^*_1 &
\varphi_2 \\ & & \theta^*_2 & \ddots \\
& & & \ddots  & \varphi_d \\ & & & & \theta^*_d \end{array}\right).$$ 