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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, 29 no. 2 (2017), p. 369-423, doi: 10.5802/jtnb.985
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Class. Math.: 11B83, 11B39, 11J70, 33C45, 41A28
Keywords: Special Sequences and Polynomials, Generalised Fibonacci Numbers, Orthogonal Polynomials, Vector Convergents

Résumé - Abstract

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 $\mathbb{Q}^m$, 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 $\Re (s)=1/2$.


[1] George E. Andrews, Richard Askey & Ranjan Roy, Special functions, Encyclopedia of Mathematics and Its Applications 71, Cambridge University Press, 1999
[2] Mark W. Coffey, “Generalized raising and lowering operators for supersymmetric quantum mechanics”,, 2015
[3] Mark W. Coffey & Matthew C. Lettington, Mellin transforms with only critical zeros: Legendre functions, J. Number Theory 148 (2015), p. 507-536 Article
[4] Mark W. Coffey & Matthew C. Lettington, “On Fibonacci Polynomial Expressions for Sums of $m$-th Powers, their implications for Faulhaber’s Formula and some Theorems of Fermat”,, 2015
[5] Harold Davenport, The Higher Arithmetic. An introduction to the theory of numbers, Cambridge University Press, 2008  MR 2462408
[6] Peter G. L. Dirichlet, Lectures on Number Theory, History of Mathematics (Providence) 16, American Mathematical Society, 1999
[7] Henry W. Gould, Combinatorial identities. A standardized set of tables listing 500 binomial coefficient summations., Morgantown, 1972
[8] Radosław Grzymkowski & Roman Wituła, Calculus Methods in Algebra, Part One, WPKJS, 2000, in Polish
[9] Thomas Koshy, Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics, John Wiley, 2001  MR 1855020
[10] Wolfdieter Lang, “The field $\mathbb{Q}(2 \cos \frac{\pi }{n})$, its Galois group, and length ratios in the regular n-gon”,, 2012
[11] Matthew C. Lettington, Fleck’s congruence, associated magic squares and a zeta identity, Funct. Approximatio, Comment. Math. 45 (2011), p. 165-205 Article
[12] Matthew C. Lettington, A trio of Bernoulli relations, their implications for the Ramanujan polynomials and the special values of the Riemann zeta function, Acta Arith. 158 (2013), p. 1-31 Article
[13] Wilhelm Magnus & Fritz Oberhettinger, Formulas and Theorems for the Special Functions of Mathematical Physics, Chelsea Publishing Company, 1949
[14] John C. Mason & D. C. Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, 2003
[15] John F. Rigby, Equilateral triangles and the Golden Ratio, The Mathematical Gazette 72 (1988), p. 27-30 Article
[16] Theodore J. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, Pure and Applied Mathematics, John Wiley, 1990
[17] Peter Steinbach, Golden fields: A case for the heptagon, Math. Mag. 70 (1997), p. 22-31 Article
[18] Zhi-Wei Sun, On the sum $\sum _{k\equiv r\pmod {m}}\binom{n}{k}$ and related congruences, Isr. J. Math. 128 (2002), p. 135-156 Article
[19] Zhi-Wei Sun & Daqing Wan, On Fleck quotients, Acta Arith. 127 (2007), p. 337-363 Article
[20] David Surowski & Paul McCombs, Homogeneous polynomials and the minimal polynomial of $2\cos {(2\pi /n)}$, Missouri J. Math. Sci. 15 (2003), p. 4-14
[21] Gabor Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications 23, American Mathematical Society, 1975
[22] Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section, Ellis Horwood Books in Mathematics and its Applications, Halsted Press, 1989
[23] William Watkins & Joel Zeitlin, The Minimal Polynomial of $\cos {(2\pi /n)}$, Am. Math. Mon. 100 (1993), p. 471-474 Article
[24] Carl S. Weisman, Some congruences for binomial coefficients, Mich. Math. J. 24 (1977), p. 141-151 Article