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

${\displaystyle \left({\frac {d}{dz))\right)_{q}p_{n}(z)={\frac {p_{n}(qz)-p_{n}(z)}{qz-z))={\frac {q^{n}-1}{q-1))p_{n-1}(z)=[n]_{q}p_{n-1}(z)}$

where the derivative symbol on the left is the q-derivative. In the limit of ${\displaystyle q\to 1}$, this becomes the definition of the Appell polynomials:

${\displaystyle {\frac {d}{dz))p_{n}(z)=np_{n-1}(z).}$

Generating function

The generalized generating function for these polynomials is of the type of generating function for Brenke polynomials, namely

${\displaystyle A(w)e_{q}(zw)=\sum _{n=0}^{\infty }{\frac {p_{n}(z)}{[n]_{q}!))w^{n))$

where ${\displaystyle e_{q}(t)}$ is the q-exponential:

${\displaystyle e_{q}(t)=\sum _{n=0}^{\infty }{\frac {t^{n)){[n]_{q}!))=\sum _{n=0}^{\infty }{\frac {t^{n}(1-q)^{n)){(q;q)_{n))}.}$

Here, ${\displaystyle [n]_{q}!}$ is the q-factorial and

${\displaystyle (q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots (1-q)}$

is the q-Pochhammer symbol. The function ${\displaystyle A(w)}$ is arbitrary but assumed to have an expansion

${\displaystyle A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}{\mbox{ with ))a_{0}\neq 0.}$

Any such ${\displaystyle A(w)}$ gives a sequence of q-difference polynomials.