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# On higher-dimensional Fibonacci numbers, Chebyshev polynomials and sequences of vector convergents

 Coffey, Mark W., Hindmarsh, James L., Lettington, Matthew C. and Pryce, John D. 2017. On higher-dimensional Fibonacci numbers, Chebyshev polynomials and sequences of vector convergents. Journal de Theorie des Nombres de Bordeaux 29 (2) , pp. 369-423. 10.5802/jtnb.985

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## Abstract

We study higher-dimensional interlacing Fibonacci sequen\-ces, generated via both Chebyshev type functions and $m$-dimensional recurrence relations. For each integer $m$, there exist both rational and integer versions of these sequences, where the underlying prime congruence structures of the rational sequence denominators enables the integer sequence to be recovered. From either the rational or the integer sequences we construct sequences of vectors in $\mathbb{Q}^m$, which converge to irrational algebraic points in $\mathbb{R}^m$. The rational sequence terms can be expressed as simple recurrences, trigonometric sums, binomial polynomials, sums of squares, and as sums over ratios of powers of the signed diagonals of the regular unit $n$-gon. These sequences also exhibit a rainbow type'' quality, and correspond to the Fleck numbers at negative indices, leading to some combinatorial identities involving binomial coefficients. It is shown that the families of orthogonal generating polynomials defining the recurrence relations employed, are divisible by the minimal polynomials of certain algebraic numbers, and the three-term recurrences and differential equations for these polynomials are derived. Further results relating to the Christoffel-Darboux formula, Rodrigues' formula and raising and lowering operators are also discussed. Moreover, it is shown that the Mellin transforms of these polynomials satisfy a functional equation of the form $p_n(s)=\pm p_n(1-s)$, and have zeros only on the critical line $\Re (s)=1/2$.

Item Type: Article Publication Published Mathematics Q Science > QA Mathematics 2118-8572 8 August 2016 6 June 2016 10 Mar 2020 21:54 http://orca.cf.ac.uk/id/eprint/93651

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