Quasi-polynomial

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