Fourier Analysis in Finite Nilpotent Groups

Abstract

Let s denote the ordinary Fibonacci sequence taken modulo a natural number m. This is a bi-infinite periodic sequence (or loop) indexed by the integers. Thus s = (s _i) and s ₀ = 0,s ₁ = 1. The shortest period of this sequence is called the fundamental period, and it will be denoted by k. We sometimes refer to this quantity as Wall’s Number [15]. In the course of investigating recurrences in finite nilpotent groups, the authors have been drawn into investigating the sums of monomials (manufactured from s) where the range of summation is a fundamental period. As is well-known, ∑ ^k−1₀ s _i = 0. The fundamental periods of loops associated with recurrence relations in finite nilpotent groups can be governed by sums far more complex than this. Indeed, the authors have been driven to show that sums such as

$$ \sum\limits_{{r = 0}}^{{k - 1}} {\sum\limits_{{j = 0}}^{{r - 1}} {\sum\limits_{{i = 0}}^{{j - 1}} {s_r^2{s_j}s_i^2{s_{{r - j - 1}}}{s_{{j - i - 1}}}{{\left( { - 1} \right)}^{{r + 1}}} + \sum\limits_{{i = 0}}^{{k - 1}} {\sum\limits_{{i = 0}}^{{j - 1}} {{s_j}\left( {\begin{array}{*{20}{c}} {{s_j}} \\ 2 \\ \end{array} } \right){s_{{j - i - 1}}}s_i^2{{\left( { - 1} \right)}^{{j + 1}}} + \sum\limits_{{i = 0}}^{{k - 1}} {s_i^2\left( {\begin{array}{*{20}{c}} {{s_i}} \\ 3 \\ \end{array} } \right){{\left( { - 1} \right)}^{{i + 1}}}} } } } } } $$

(1)

actually vanish. The reader wishing for the full and intricate details should consult [1] or [2]. The group theoretic implications of these calculations are summarized in [3]. The techniques for handling multiple sums are similar in spirit to those demonstrated here, but the manipulations are considerably longer.

The work of the first author was supported by Ataturk University, Erzurum, Turkey