Definition. The discrete q-Hermite I polynomials are Al-Salam-Carlitz I polynomials with a = -1 :
Orthogonality.
Recurrence relation.
Normalized recurrence relation.
where
q-Difference equation.
Forward shift operator.
or equivalently
Backward shift operator.
or equivalently
where
Rodrigues-type formula.
Generating function.
Remark. The discrete q-Hermite I polynomials are related to the discrete q-Hermite II polynomials defined by (3.29.1) in the following way :
References. [13], [17], [67], [83], [98], [193], [208], [283].
