Coffey, Mark W., Hindmarsh, James, Lettington, Matthew C. and Pryce, John D. 2017. On higherdimensional Fibonacci numbers, Chebyshev polynomials and sequences of vector convergents. Journal de Theorie des Nombres de Bordeaux 29 (2) , pp. 369423. 10.5802/jtnb.985 
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Abstract
We study higherdimensional 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 threeterm recurrences and differential equations for these polynomials are derived. Further results relating to the ChristoffelDarboux 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 $p_n(s)=\pm p_n(1s)$, and have zeros only on the critical line $\Re (s)=1/2$.
Item Type:  Article 

Date Type:  Published Online 
Status:  Published 
Schools:  Mathematics 
Subjects:  Q Science > QA Mathematics 
Publisher:  Institut de Mathématiques de Bordeaux 
ISSN:  21188572 
Date of First Compliant Deposit:  8 August 2016 
Date of Acceptance:  6 June 2016 
Last Modified:  28 Oct 2019 15:46 
URI:  http://orca.cf.ac.uk/id/eprint/93651 
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 On higherdimensional Fibonacci numbers, Chebyshev polynomials and sequences of vector convergents. (deposited 18 Aug 2016 10:15) [Currently Displayed]
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