Characterize the parameter values yielding nice orthogonal polynomials for the q-factorial moments

Characterize all values of the q-analogue parameter for which the corresponding q-factorial moment sequence has orthogonal polynomials with nice coefficients, establishing whether the two parameter values identified in equations (20) and (21) are the only such values.

Background

Section 3 studies q-analogues of factorial moments and shows that the q-factorial moment sequence has monic q-Laguerre orthogonal polynomials. The subsequent remark observes that another parameter specialization also produces a nice analogue with nice orthogonal polynomials.

The authors do not prove the apparent uniqueness of these two parameter values; they state only that they seem to be the only values for which the associated moment sequence has orthogonal polynomials with nice coefficients. Thus the unresolved issue is to determine all parameter values with this property.