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

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

OEIS
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### References:

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