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Symmetry groups at the one-parameter and zero-parameter levels

Determine the symmetry groups governing the terminating transformations for the one-parameter families (continuous big q- and q^{-1}-Hermite polynomials, represented by terminating 5W4^{-3} and 5W4^{3} series) and the zero-parameter families (continuous q- and q^{-1}-Hermite polynomials, represented by terminating 4W3^{-4} and 4W3^{4} series), given that only the orders (each equal to 2) are currently identified.

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Background

After analyzing higher-parameter levels, the paper observes that at the one-parameter level (continuous big q- and q{-1}-Hermite) and the zero-parameter level (continuous q- and q{-1}-Hermite) there are only two representations in each case, implying groups of order 2 by counting transformations. However, the precise identification of these symmetry groups is left unresolved.

Clarifying the exact group structure (e.g., whether these are cyclic groups of order 2 or arise as parts of larger symmetry constructions) would complete the symmetry-group picture for these terminal levels of the q-Askey scheme.

References

In the one parameter case which corresponds to the continuous big $q$ and big $q{-1}$-Hermite polynomials and the terminating ${}_5W_4{-3}$ and ${}_5W_43$, the orders are 2 since there are two separate representations each with one one possible rearrangement for each. In the zero parameter case which corresponds to the continuous $q$ and $q{-1}$-Hermite polynomials and the terminating ${}_4W_3{-4}$ and ${}_4W_34$, the orders are 2 as there are two representations only one possible rearrangement for each. It is not yet clear what the symmetry groups for these might be.

— Terminating representations, transformations and summations for the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials (2508.08162 - Cohl et al., 11 Aug 2025) in Section 7: The symmetry group q-Askey scheme