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

[math]\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]_qp_{n-1}(z) }[/math]

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

[math]\displaystyle{ \frac{d}{dz}p_n(z) = np_{n-1}(z). }[/math]

Generating function

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

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

where [math]\displaystyle{ e_q(t) }[/math] is the q-exponential:

[math]\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}. }[/math]

Here, [math]\displaystyle{ [n]_q! }[/math] is the q-factorial and

[math]\displaystyle{ (q;q)_n=(1-q^n)(1-q^{n-1})\cdots (1-q) }[/math]

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

[math]\displaystyle{ A(w)=\sum_{n=0}^\infty a_n w^n \mbox{ with } a_0 \ne 0. }[/math]

Any such [math]\displaystyle{ A(w) }[/math] 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.)