Polynomials and Second Order Linear Recurrences

TL;DR

This paper explores the polynomial representations of second-order linear recurrences, proving their nonexistence for certain generalized Fibonacci sequences using Pell's equation techniques and novel identities.

Contribution

It introduces methods to prove the nonexistence of polynomials for specific recurrence relations and explicitly finds polynomials for some cases, advancing understanding of Diophantine representations.

Findings

Explicit polynomials found for certain recurrences

Proved nonexistence of polynomials for generalized Fibonacci sequences

Developed new identities and techniques for recurrence analysis

Abstract

One of the most interesting results of the last century was the proof completed by Matijasevich that computably enumerable sets are precisely the diophantine sets [MRDP Theorem, 9], thus settling, based on previously developed machinery, Hilbert's question whether there exists a general algorithm for checking the solvability in integers of any diophantine equation. In this paper we describe techniques to prove the nonexistence of polynomials in two variables for some simple generalizations of the Fibonacci sequence (explicit diophantine representation of Fibonacci numbers were known from Jones' polynomial whose positive values have the same range as that of Fibonacci numbers), and we believe similar techniques exist for the primes. In this paper we mainly show the following results: (1) using one of the many techniques known for solving the Pell's equation, namely the solution in an extended number system, we prove the existence and explicitly find the polynomials for the recurrences of the form $e(n)=ae(n-1)+e(n-2)$ with starting values of 0 and 1 in particular, and for any arbitrary starting values, in the process defining a concept of fundamental starting numbers, (2) we prove a few identities that seem to be quite interesting and useful, (3) we use these identities in a novel way to generate systems of equations of certain rank deficiency using which we disprove for the first time the existence of any polynomial in 2 variables for the generalized recurrence of the form $e(n)=ae(n-1)+be(n-2)$