[Paper Review] Fibonacci Modules and Multiple Fibonacci Sequences

Abdul Rauf Nizami|ArXiv.org|Oct 22, 2008

Advanced Mathematical Theories and Applications3 references3 citations

TL;DR

This paper introduces double Fibonacci sequences over modules, generalizing classical Fibonacci sequences to two-dimensional recurrence relations. It establishes that the space of such sequences forms a free module of rank 4, isomorphic to the tensor product of two Fibonacci modules, and derives rational generating functions and diagonal properties for these sequences.

ABSTRACT

Double Fibonacci sequences are introduced and they are related to operations with Fibonacci modules. Generalizations and examples are also discussed.

Motivation & Objective

• To generalize Fibonacci sequences to two-dimensional recurrence relations over modules.
• To define and study double Fibonacci sequences satisfying independent linear recurrences in two indices.
• To establish isomorphisms between tensor products of Fibonacci modules and double Fibonacci sequence modules.
• To derive rational generating functions for double Fibonacci sequences.
• To investigate structural properties such as diagonal monotonicity and recurrence relations along diagonals.

Proposed method

• Define double Fibonacci sequences $(x_{n,k})$ satisfying $x_{n+2,k} = a x_{n+1,k} + b x_{n,k}$ and $x_{n,k+2} = c x_{n,k+1} + d x_{n,k}$ for $n,k \geq 0$.
• Use the $ $-module structure of sequences and shift operators $H$ and $V$ to define the $ [H,V]$-module of double sequences.
• Prove that $ _{ }(a,b) igotimes_{ } _{ }(c,d) o _{ igotimes }^{[2]}((a,b) igotimes (c,d))$ is a natural $ [H,V]$-module isomorphism.
• Derive the rational generating function $G(t,s) = q(t)^{-1} r(s)^{-1} ig[ ext{linear combination of initial terms} ig]$, where $q(t) = 1 - at - bt^2$, $r(s) = 1 - cs - ds^2$.
• Generalize to multiple Fibonacci sequences in $p$ indices using higher-order recurrences and tensor products of $p$ Fibonacci modules.
• Establish a diagonal recurrence: $ab x_{n,k+3} + (a^2 + b)c x_{n+1,k+2} = a(c^2 + d)x_{n+2,k+1} + cd x_{n+3,k}$ under condition $a^2 d = b c^2$.

Experimental results

Research questions

• RQ1 How can Fibonacci sequences be generalized to two-dimensional arrays satisfying independent linear recurrences in each index?
• RQ2 What is the algebraic structure of the module of double Fibonacci sequences over a commutative ring $ $?
• RQ3 Is there a natural isomorphism between the tensor product of two Fibonacci modules and the module of double Fibonacci sequences?
• RQ4 What is the form of the generating function for double Fibonacci sequences?
• RQ5 Under what conditions do double Fibonacci sequences exhibit alternating monotonicity along diagonals?

Key findings

• The module $ _{ }^{[2]}((a,b) igotimes (c,d))$ of double Fibonacci sequences is isomorphic to $ _{ }^{[2]}(a,b) igotimes_{ } _{ }^{[2]}(c,d)$, and is a free $ $-module of rank 4.
• An explicit basis for $ _{ }^{[2]}((a,b) igotimes (c,d))$ is given by the sequences $P_{i}^{[n]}(a,b) P_{j}^{[k]}(c,d)$ for $(i,j) otin igrace{0,1igrace}^2$.
• The generating function of a double Fibonacci sequence is rational and explicitly given by $G(t,s) = q(t)^{-1} r(s)^{-1} ig[ x_{0,0}(1-at)(1-cs) + x_{1,0}t(1-cs) + x_{0,1}(1-at)s + x_{1,1}ts ig]$, where $q(t) = 1 - at - bt^2$, $r(s) = 1 - cs - ds^2$.
• For the special case $(a,b) = (c,d) = (1,1)$, the diagonal property $x_{n,k+3} - x_{n+3,k} = 2(x_{n+2,k+1} - x_{n+1,k+2})$ holds for all integer sequences in $ _{ }^{[2]}(1,1)$.
• The diagonal recurrence $ab x_{n,k+3} + (a^2 + b)c x_{n+1,k+2} = a(c^2 + d)x_{n+2,k+1} + cd x_{n+3,k}$ holds whenever $a^2 d = b c^2$.
• The module $ _{ }^{[p]}( extbf{a}^{(1)}, extbf{a}^{(2)}, extbf{a}^{(3)}, extbf{a}^{(4)})$ of multiple Fibonacci sequences in $p$ indices is free of rank $D = d_1 d_2 imes imes d_p$.