Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Periodicity in quasipolynomial convolution. | Periodicity in quasipolynomial convolution Thomas Zaslavsky Department of Mathematical Sciences Binghamton University of SUNY Binghamton NY 13902-6000 . zaslav@ Submitted Jul 15 2004 Accepted Nov 12 2004 Published Nov 22 2004 Mathematics Subject Classifications Primary 39A12 44A35 Secondary 05A15 11D04 11D45 15A18 52B20 Keywords quasipolynomial convolution period degree circulant matrix null space semimagic square Frobenius coin problem. Abstract The leading term of a convolution of quasipolynomials with periods p and q is periodic with period gcd p q smaller than expected. The degree of the convolution is usually d e 1 we characterize the exceptions. To do this we need to characterize the null space of a circulant matrix. We wish to point out a simple yet unexpected property of quasipolynomial calculus. A quasipolynomial is a function of positive integers that is given by a cyclically repeating sequence of polynomials that is d f t X at k tk for t 1 2 3 . k 0 where the coefficient at k is a periodic function of t for each k. Suppose at k cycles with period pk then p lcm p0 pi --- pd is the period of f meaning that f t ft t where f1 f2 . . . is a sequence of polynomials ft p ft for all t and p is the smallest positive number for which that is so. The degree of f is the largest degree of any ft. A quasipolynomial function extends to all integers but we shall not need that fact. It is well known that a quasipolynomial is a function of positive integers whose generating function Gf x X f t xt t i Research supported by the SGPNR. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2004 R11 1 is a rational function with denominator 1 xp d 1 and with numerator of the form x x where deg p d 1 . See 7 Section slight adjustments are needed because Stanley s quasipolynomials are defined for t 0. We are interested in concocting a new function F by what might be called discrete integration or in general convolution. Here is a simple example 1 F t f t q f t
