Mathematical Physics


Generalized Fibonacci Numbers and 4k+1-Fold Symmetric Quasicrystals

Authors: Carlos Castro

Given that the two-parameter $ p, q$ quantum-calculus deformations of the integers $ [ n ]_{ p, q} = (p^n - q^n)/ ( p - q) = F_n $ coincide precisely with the Fibonacci numbers (integers), as a result of Binet's formula when $ p = \tau = { 1 + \sqrt 5 \over 2}$, $ q = { \tilde \tau} = { 1 - \sqrt 5 \over 2 }$ (Galois-conjugate pairs), we extend this result to the $generalized$ Binet's formula (corresponding to generalized Fibonacci sequences) studied by Whitford. Consequently, the Galois-conjugate pairs $ (p, q = \tilde p ) = { 1\over 2} ( 1 \pm \sqrt m ) $, in the very special case when $ m = 4 k + 1$ and square-free, generalize Binet's formula $ [ n ]_{ p, q} = G_n$ generating integer-values for the generalized Fibonacci numbers $G_n$'s. For these reasons, we expect that the two-parameter $ (p, q = \tilde p)$ quantum calculus should play an important role in the physics of quasicrystals with $4k+1$-fold rotational symmetry.

Comments: 3 Pages.

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Submission history

[v1] 2019-01-28 22:50:29

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