What is "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(k)k + c(k)k + ⋯ + c(k), where c(k) is a periodic function with integral period. If c(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, …, p such that f(n) = p(n) when $n \equiv i \bmod s$. The polynomials p are called the constituents of f.