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

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 m , convergeant vers des points irrationnels algébriques dans 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 (s)=±p n (1-s), et que tous leurs zéros sont situés sur la ligne critique (s)=1/2.

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 m , which converge to irrational algebraic points in 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)=±p n (1-s), and have zeros only on the critical line (s)=1/2.

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DOI : https://doi.org/10.5802/jtnb.985
Classification : 11B83,  11B39,  11J70,  33C45,  41A28
Mots clés : Special Sequences and Polynomials, Generalised Fibonacci Numbers, Orthogonal Polynomials, Vector Convergents
@article{JTNB_2017__29_2_369_0,
     author = {Mark W. Coffey and James L. Hindmarsh and Matthew C. Lettington and John D. Pryce},
     title = {On {Higher-Dimensional} {Fibonacci} {Numbers,} {Chebyshev} {Polynomials} and {Sequences} of {Vector} {Convergents}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {369--423},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {2},
     year = {2017},
     doi = {10.5802/jtnb.985},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.985/}
}
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, Tome 29 (2017) no. 2, pp. 369-423. doi : 10.5802/jtnb.985. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.985/

[1] George E. Andrews; Richard Askey; Ranjan Roy Special functions, Encyclopedia of Mathematics and Its Applications, Volume 71, Cambridge University Press, 1999, xvi+664 pages

[2] Mark W. Coffey Generalized raising and lowering operators for supersymmetric quantum mechanics (2015) (https://arxiv.org/abs/1501.06649)

[3] Mark W. Coffey; Matthew C. Lettington Mellin transforms with only critical zeros: Legendre functions, J. Number Theory, Volume 148 (2015) no. 2, pp. 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) (https://arxiv.org/abs/1510.05402)

[5] Harold Davenport The Higher Arithmetic. An introduction to the theory of numbers, Cambridge University Press, 2008, ix+239 pages

[6] Peter G. L. Dirichlet Lectures on Number Theory, History of Mathematics (Providence), Volume 16, American Mathematical Society, 1999, xx+275 pages

[7] Henry W. Gould Combinatorial identities. A standardized set of tables listing 500 binomial coefficient summations., Morgantown, 1972, viii+106 pages

[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, xvi++652 pages

[10] Wolfdieter Lang The field (2cosπ n), its Galois group, and length ratios in the regular n-gon (2012) (https://arxiv.org/abs/1210.1018v1)

[11] Matthew C. Lettington Fleck’s congruence, associated magic squares and a zeta identity, Funct. Approximatio, Comment. Math., Volume 45 (2011) no. 2, pp. 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., Volume 158 (2013) no. 1, pp. 1-31 | Article

[13] Wilhelm Magnus; Fritz Oberhettinger Formulas and Theorems for the Special Functions of Mathematical Physics, Chelsea Publishing Company, 1949, 172 pages

[14] John C. Mason; D. C. Handscomb Chebyshev polynomials, Chapman & Hall/CRC, 2003, xiv+341 pages

[15] John F. Rigby Equilateral triangles and the Golden Ratio, The Mathematical Gazette, Volume 72 (1988), pp. 27-30 | Article

[16] Theodore J. Rivlin Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, Pure and Applied Mathematics, John Wiley, 1990, xvi+249 pages

[17] Peter Steinbach Golden fields: A case for the heptagon, Math. Mag., Volume 70 (1997) no. 1, pp. 22-31 | Article

[18] Zhi-Wei Sun On the sum kr(modm) n k and related congruences, Isr. J. Math., Volume 128 (2002), pp. 135-156 | Article

[19] Zhi-Wei Sun; Daqing Wan On Fleck quotients, Acta Arith., Volume 127 (2007) no. 4, pp. 337-363 | Article

[20] David Surowski; Paul McCombs Homogeneous polynomials and the minimal polynomial of 2cos(2π/n), Missouri J. Math. Sci., Volume 15 (2003) no. 1, pp. 4-14

[21] Gabor Szegö Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Volume 23, American Mathematical Society, 1975, xii+432 pages

[22] Steven Vajda Fibonacci and Lucas Numbers, and the Golden Section, Ellis Horwood Books in Mathematics and its Applications, Halsted Press, 1989, 190 pages

[23] William Watkins; Joel Zeitlin The Minimal Polynomial of cos(2π/n), Am. Math. Mon., Volume 100 (1993) no. 5, pp. 471-474 | Article

[24] Carl S. Weisman Some congruences for binomial coefficients, Mich. Math. J., Volume 24 (1977), pp. 141-151 | Article