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- W2912288090 abstract "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 ) = { 1over 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." @default.
- W2912288090 created "2019-02-21" @default.
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- W2912288090 date "2019-01-01" @default.
- W2912288090 modified "2023-09-26" @default.
- W2912288090 title "Generalized Fibonacci Numbers and 4k+1-Fold Symmetric Quasicrystals" @default.
- W2912288090 hasPublicationYear "2019" @default.
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