Q-difference polynomial

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.)"


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