Definition. The discrete q-Hermite II polynomials are Al-Salam-Carlitz II polynomials with a = -1 :
Orthogonality.
where
Recurrence relation.
Normalized recurrence relation.
where
q-Difference equation.
Forward shift operator.
or equivalently
Backward shift operator.
or equivalently
Rodrigues-type formula.
Generating functions.
Remark. The discrete q-Hermite II polynomials are related to the discrete q-Hermite I polynomials defined by (3.28.1) in the following way :
