Quasi polynomial

In mathematics, a quasi-polynomial (pseudo-polynomial) is a generalization of polynomials. While the coefficients of a polynomial come from a ring, the coefficients of quasi-polynomials are instead periodic functions with integral period. Quasi-polynomials appear throughout much of combinatorics as the enumerators for various objects.

A quasi-polynomial can be written as q ( k ) = c d ( k ) k d + c d − 1 ( k ) k d − 1 + ⋯ + c 0 ( k ) , where c i ( k ) is a periodic function with integral period. If c d ( k ) is not identically zero, then the degree of q is d . Equivalently, a function f : N → N is a quasi-polynomial if there exist polynomials p 0 , … , p s − 1 such that f ( n ) = p i ( n ) when n ≡ i mod s . The polynomials p i are called the constituents of f .