On Higher-Dimensional Fibonacci Numbers, Chebyshev Polynomials and Sequences of Vector Convergents
Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 369-423.

We study higher-dimensional interlacing Fibonacci sequences, 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 ${ℚ}^{m}$, which converge to irrational algebraic points in ${ℝ}^{m}$. 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 ${p}_{n}\left(s\right)=±{p}_{n}\left(1-s\right)$, and have zeros only on the critical line $\Re \left(s\right)=1/2$.

Nous étudions les suites de Fibonacci entrelacées multidimensionnelles, générées avec des fonctions de type Tchebychev ou des relations de récurrence $m$-dimensionnelles. Pour chaque nombre entier $m$, il y a une forme rationnelle et une forme entière de ces suites, et la suite entière peut être recouvrée en utilisant la structure de congruences modulo des nombres premiers des dénominateurs de la suite rationnelle.

À partir des suites, rationnelles ou entières, on construit des suites vectorielles dans ${ℚ}^{m}$, convergeant vers des points irrationnels algébriques dans ${ℝ}^{m}$. Les termes de la suite rationnelle peuvent être décrits par des récurrences simples, des polynômes trigonométriques, des polynômes binomiaux, des sommes de carrés, et aussi par sommes de quotients de puissances des diagonales signées du polygone régulier à $n$ côtés. Ces suites exhibent en plus une qualité de type « arc-en-ciel », et correspondent aux nombres de Fleck à indice négatif, ce qui amène à certaines identités combinatoires sur les coefficients binomiaux.

On montre que les familles de polynômes orthogonaux générateurs, qui définissent les relations de récurrence, sont divisibles par les polynômes minimaux de certains nombres algébriques, et on en déduit les récurrences linéaires du second ordre et les équations différentielles pour ces polynômes. De plus, on discute de plusieurs résultats concernant la formule de Christoffel-Darboux, la formule de Rodrigues et les opérateurs d’échelle. En outre, on démontre que les transformations de Mellin de ces polynômes satisfont une équation fonctionnelle de la forme ${p}_{n}\left(s\right)=±{p}_{n}\left(1-s\right)$, et que tous leurs zéros sont situés sur la ligne critique $\Re \left(s\right)=1/2$.

Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.985
Classification: 11B83, 11B39, 11J70, 33C45, 41A28
Keywords: Special Sequences and Polynomials, Generalised Fibonacci Numbers, Orthogonal Polynomials, Vector Convergents
Mark W. Coffey 1; James L. Hindmarsh 2; Matthew C. Lettington 2; John D. Pryce 2

2 School of Mathematics Cardiff University Senghennydd Road Cardiff, CF24 4AG, UK
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Mark W. Coffey; James L. Hindmarsh; Matthew C. Lettington; John D. Pryce. On Higher-Dimensional Fibonacci Numbers, Chebyshev Polynomials and Sequences of Vector Convergents. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 369-423. doi : 10.5802/jtnb.985. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.985/

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