Definitions

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 type of Brenke polynomial, and generalize the Appell polynomials. See also Sheffer sequence.

Definition

The q-difference polynomials satisfy the relation

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

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

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

Generating function

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

$A\left(w\right)e_q\left(zw\right) = sum_\left\{n=0\right\}^infty p_n\left(z\right) w^n$

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

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

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

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

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

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

Any such $A\left(w\right)$ 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.)
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