Leonard pairs, parameter arrays representing them, or the Askey-Wilson relations conveniently.

This paper presents convenient normalizations of Leonard pairs and their Askey-Wilson relations. We generally assume that the dimension of the underling vector space is at least 4, and use Terwilliger’s classification [@TerwLTparr] (or [@terwgen Section 35]) of parameter arrays representing Leonard pairs. For parameter arrays of the $q$-type, we present two normalizations: one that is close to Terwilliger’s general expressions in [@TerwLTparr], and one where Askey-Wilson coefficients are normalized most attractively. For other parameter arrays, we give one normalization. This work is more of bookkeeping kind than of deep research. Examples of Askey-Wilson relations for normalized Leonard pairs are given in [@TerwVid], [@hjalmarpaul]. Indirectly, Askey-Wilson relations for Leonard pairs arising from certain distince regular graphs are computed in [@Cur2hbipT], [@Go].

We note that Terwilliger’s classification of parameter arrays by certain families of orthogonal polynomials from the Askey-Wilson scheme can be largely imitated to categorize Leonard pairs and Askey-Wilson relations; see Sections \[lsparar\] and \[classifics\] below. We have the same types of Leonard pairs and of Askey-Wilson relations, except that the quantum $q$-Krawtchouk and affine $q$-Krawtchouk types are merged.

The paper is organized as follows. In the next section we 