q-difference polynomial

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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:

Generating function[edit]

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.)