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 ) {\displaystyle q(k)=c_{d}(k)k^{d}+c_{d-1}(k)k^{d-1}+\cdots +c_{0}(k)} , where c i ( k ) {\displaystyle c_{i}(k)} is a periodic function with integral period. If c d ( k ) {\displaystyle c_{d}(k)} is not identically zero, then the degree of q {\displaystyle q} is d {\displaystyle d} . Equivalently, a function f : N → N {\displaystyle f\colon \mathbb {N} \to \mathbb {N} } is a quasi-polynomial if there exist polynomials p 0 , … , p s − 1 {\displaystyle p_{0},\dots ,p_{s-1}} such that f ( n ) = p i ( n ) {\displaystyle f(n)=p_{i}(n)} when i ≡ n mod s {\displaystyle i\equiv n{\bmod {s}}} . The polynomials p i {\displaystyle p_{i}} are called the constituents of f {\displaystyle f} .