# Formal power series arising from multiplication of quantum integers

@inproceedings{Nathanson2000FormalPS,
title={Formal power series arising from multiplication of quantum integers},
author={Melvyn B. Nathanson},
booktitle={Unusual Applications of Number Theory},
year={2000}
}
• M. Nathanson
• Published in
Unusual Applications of…
2000
• Mathematics, Computer Science
For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a sequence {f_n(q)} of polynomials such that f_n(0)=1 for all n. It is proved that if {f_n(q)} is a solution of this functional equation, then the sequence {f_n(q)} converges to a formal power series F(q). Quantum mulitplication also leads to the functional… Expand
8 Citations

#### Topics from this paper

• Mathematics
• 2005
Abstract For every positive integer n , the quantum integer [ n ] q is the polynomial [ n ] q = 1 + q + q 2 + ⋯ + q n - 1 . A quadratic addition rule for quantum integers consists of sequences ofExpand
The quantum integer [n]q is the polynomial 1 + q + q2 + ⋯ + qn-1, and the sequence of polynomials $\{[n]_q \}_{n = 1}^{\infty}$ is a solution of the functional equation fmn(q) = fm(q)fn(qm). In thisExpand
The quantum integer $[n]_q$ is the polynomial $1 + q + q^2 + ... + q^{n-1}.$ Two sequences of polynomials $\mathcal{U} = \{u_n(q)\}_{n=1}^{\infty}$ and $\mathcal{V} = \{v_n(q)\}_{n=1}^{\infty}$Expand
Quadratic addition rules for three $q$-integers
The $q$-integer is the polynomial $[n]_q = 1 + q + q^2 + \dots + q^{n-1}$. For every sequences of polynomials $\mathcal S = \{s_m(q)\}_{m=1}^\infty$, $\mathcal T = \{t_m(q)\}_{m=1}^\infty$, \$\mathcalExpand