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

{(\frac{d}{d z})}_{q} p_{n} (z) = \frac{p_{n} (q z) - p_{n} (z)}{q z - 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 $q \to 1$ , this becomes the definition of the Appell polynomials:

\frac{d}{d z} p_{n} (z) = n p_{n - 1} (z) .

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

A (w) e_{q} (z w) = \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}) \dots (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} with a_{0} \neq 0 .

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

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.)