×
Mustonen, Seppo; Haukkanen, Pentti; Merikoski, Jorma

Some polynomials associated with regular polygons. (English) Zbl 1316.11020

Acta Univ. Sapientiae, Math. 6, No. 2, 178-193 (2014).
Summary: Let \(G_n\) be a regular \(n\)-gon with unit circumradius, and \(m =\lfloor\frac{n}{2}\rfloor,\mu=\lfloor\frac{n-1}{2}\rfloor\). Let the edges and diagonals of \(G_n\) be \(e_{n1} < \dots < e_{nm}\). We compute the coefficients of the polynomial \[ (x - e^2_{n1})\cdots (x - e^2_{n{\mu}}). \] They appear to form a well-known integer sequence, and we study certain related sequences, too. We also compute the coefficients of the polynomial \[ (x - s^2_{n1}) \dots (x - s^2_{nm}), \] where \[ s_{ni}=\cot\frac{(2i-1)\pi}{2n},\quad i=1,\dots,m. \] We interpret \(s_{n1}\) as the sum of all individual edges and diagonals of \(G_n\) divided by \(n\). We also discuss the interpretation of \(s_{n2},\dots,s_{nm}\), and present a conjecture on expressing \(s_{n1},\dots, s_{nm}\) using \(e_{n1},\dots, e_{nm}\).

MSC:

11B83 Special sequences and polynomials
11C08 Polynomials in number theory
15B36 Matrices of integers
51M20 Polyhedra and polytopes; regular figures, division of spaces

Keywords:

polynomial; regular polygon; eigenvalue

Software:

OEIS
PDFBibTeX XMLCite
\textit{S. Mustonen} et al., Acta Univ. Sapientiae, Math. 6, No. 2, 178--193 (2014; Zbl 1316.11020)
Full Text: DOI
OA License

Online Encyclopedia of Integer Sequences:

Triangle of coefficients of Chebyshev’s S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).
Irregular triangle read by rows of the initial floor(n/2) + 1 coefficients of 1/C(x)^n, where C(x) is the g.f. of the Catalan sequence (A000108).
Riordan array ((1+x^2)/(1-x)^2, -x/(1-x)^2).
Triangle by rows with row n formed by coefficients of the characteristic polynomial of the n X n tridiagonal matrix with m_{i,i} = 2 for i=1..n, m_{i,i-1} = m_{i,i+1} = -1 for i=2..n-1, and m_{1,2} = m_{n,n-1} = -2.

References:

[1] N. D. Cahill, J. R. D’Errico, J. P. Spence, Complex factorization of the Fibonacci and Lucas numbers, Fibonacci Quart., 41 (2003), 13{19.;} · Zbl 1056.11005
[2] H. M. S. Coxeter, Regular Convex Polytopes, Cambridge U. Pr., 1974.; · Zbl 0296.50009
[3] S. Mustonen, Lengths of edges and diagonals and sums of them in regular polygons as roots of algebraic equations, 2013, 44 pp. http://www.survo.fi/papers/Roots2013.pdf;
[4] The On-Line Encyclopedia of Integer Sequences (OEIS). http://oeis.org/;
[5] D. E. Rutherford, Some continuant determinants arising in physics and chemistry, I, Proc. Royal Soc. Edinburgh, 62A (1947), 229{236.;} · Zbl 0030.00501
[6] D. Y. Savio, E. R. Suryanarayan, Chebychev polynomials and regular polygons, Amer. Math. Monthly, 100 (1993), 657{661.;} · Zbl 0789.33003
[7] E. W. Weisstein, Sine, MathWorld {A Wolfram Web Resource. <http://mathworld.wolfram.com/Sine.html>;}
[8] E. W. Weisstein, Tangent, MathWorld {A Wolfram Web Resource. http://mathworld.wolfram.com/Tangent.html;}
[9] A. M. Яглом, И. М. Яглом, Элементарный вывод формул Вал­лиса, Лейбница и Эйлера для числа я, Успехи Матем. Наук, 8 (1953), 181-187. (А. М. Yaglom, I. М. Yaglom, An elementary deriva­tion of the Wallis, Leibniz and Euler formulas for the number я, Uspekhi Matem. Nauk, 8 (1953), 181-187.); · Zbl 0053.03804
[10] http://functions.wolfram.com/ElementaryFunctions/Cot/27/01/0002/;
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.