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

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

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

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

Generating function

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

$A(w)e_q(zw) = \sum_{n=0}^\infty \frac{p_n(z)}{[n]_q!} w^n$

where $e_q(t)$ is the q-exponential:

$e_q(t)=\sum_{n=0}^\infty \frac{t^n}{[n]_q!}= \sum_{n=0}^\infty \frac{t^n (1-q)^n}{(q;q)_n}.$

Here, $[n]_q!$ is the q-factorial and

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

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

$A(w)=\sum_{n=0}^\infty a_n w^n \mbox{ with } a_0 \ne 0.$

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

References

• A. Sharma and A. M. Chak, "The basic analogue of a class of polynomials", Riv. Mat. Univ. Parma, 5 (1954) 325-337.
• Ralph P. Boas, Jr. and R. Creighton Buck, Polynomial Expansions of Analytic Functions (Second Printing Corrected), (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263. (Provides a very brief discussion of convergence.)