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

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

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

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

Generating function

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

${\displaystyle A(w)e_{q}(zw)=\sum _{n=0}^{\infty }{\frac {p_{n}(z)}{[n]_{q}!}}w^{n}}$

where ${\displaystyle e_{q}(t)}$ is the q-exponential:

${\displaystyle e_{q}(t)=\sum _{n=0}^{\infty }{\frac {t^{n}}{[n]_{q}!}}=\sum _{n=0}^{\infty }{\frac {t^{n}(1-q)^{n}}{(q;q)_{n}}}.}$

Here, ${\displaystyle [n]_{q}!}$ is the q-factorial and

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

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

${\displaystyle A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}{\mbox{ with }}a_{0}\neq 0.}$

Any such ${\displaystyle A(w)}$ 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.)