Quasi-polynomial

In mathematics, a quasi-polynomial (sometimes called 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.

==Definition== A quasi-polynomial is a function q defined on \mathbb{Z} of the form q(n) = c_d(n) n^d + c_{d-1}(n) n^{d-1} + \cdots + c_0(n), where each c_i(n) is a periodic function with integral period. If c_d(n) is not identically zero, then the degree of q is d, and any common period of c_0(n), c_1(n), \dots, c_d(n) is a period of q. The minimal such period (sometimes simply called the period or the quasi-period of q) is the least common multiple of the periods of c_0(n), c_1(n), \dots, c_d(n).