
Q-difference polynomial

In combinatorial mathematics, the q"-difference polynomials or q"-harmonic polynomials are a polynomial sequence defined in terms of the "q"-derivative. They are a generalized type of Brenke polynomial, and generalize the Appell polynomials. See also Sheffer sequence.
Definition.
The q-difference polynomials satisfy the relation
where the derivative symbol on the left is the q-derivative. In the limit of formula_2, this becomes the definition of the Appell polynomials:
Generating function.
The generalized generating function for these polynomials is of the type of generating function for Brenke polynomials, namely
where formula_5 is the q-exponential:
Here, formula_7 is the q-factorial and
is the q-Pochhammer symbol. The function formula_9 is arbitrary but assumed to have an expansion
Any such formula_9 gives a sequence of q-difference polynomials.