Variations on a Theorem of Fine & Wilf

  • Authors:

  • Filippo Mignosi;Jeffrey Shallit;Ming-wei Wang

  • Affiliations:

  • -;-;-

  • Venue:
  • MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
  • Year:
  • 2001

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Abstract

In 1965, Fine & Wilf proved the following theorem: if (fn)n≥0 and (gn)n≥0 are periodic sequences of real numbers, of periods h and k respectively, and fn = gn for 0 ≤ n ≤ h+k-gcd(h, k), then fn = gn for all n ≥ 0. Furthermore, the constant h + k - gcd(h, k) is best possible. In this paper we consider some variations on this theorem. In particular, we study the case where fn ≤ gn instead of fn = gn. We also obtain a generalization to more than two periods.