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

:$left(frac\; \{d\}\{dz\}\; ight)\_q\; p\_n(z)\; =\; frac\{p\_n(qz)-p\_n(z)\}\; \{qz-z\}\; =\; p\_\{n-1\}(z)$

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

:$frac\{d\}\{dz\}p\_n(z)\; =\; p\_\{n-1\}(z).$

Generating function

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

:$A(w)e\_q(zw)\; =\; sum\_\{n=0\}^infty\; p\_n(z)\; w^n$

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

:$A(w)=sum\_\{n=0\}^infty\; a\_n\; w^n\; mbox\{\; with\; \}\; a\_0\; e\; 0.$

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.)"