combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the ''q''-difference polynomials or ''q''-harmonic polynomials are a

polynomial sequence In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in en ...

defined in terms of the ''q''-derivative. They are a generalized type of Brenke polynomial, and generalize the Appell polynomials. See also

Sheffer sequence In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are ...

Definition

The q-difference polynomials satisfy the relation :

\left(\frac \right)_q p_n(z) = 
\frac  = \frac  p_(z)= qp_(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: :

\fracp_n(z) = np_(z).

Generating function

The generalized

generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...

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

A(w)e_q(zw) = \sum_^\infty \frac w^n

where

e_q(t)

is the

q-exponential In combinatorial mathematics, a ''q''-exponential is a ''q''-analog of the exponential function, namely the eigenfunction of a ''q''-derivative. There are many ''q''-derivatives, for example, the classical ''q''-derivative, the Askey-Wilson ...

: :

e_q(t)=\sum_^\infty \frac=
\sum_^\infty \frac.

Here,

q!

is the

q-factorial In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ...

and :

(q;q)_n=(1-q^n)(1-q^)\cdots (1-q)

is the

q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ...

. The function

A(w)

is arbitrary but assumed to have an expansion :

A(w)=\sum_^\infty a_n w^n \mbox{ with } a_0 \ne 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.)'' Q-analogs Polynomials