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.
The q-difference polynomials satisfy the relation
where the derivative symbol on the left is the q-derivative. In the limit of , this becomes the definition of the Appell polynomials:
The generalized generating function for these polynomials is of the type of generating function for Brenke polynomials, namely
where is the q-exponential:
Here, is the q-factorial and
is the q-Pochhammer symbol. The function is arbitrary but assumed to have an expansion
Any such gives a sequence of q-difference polynomials.
- 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.)