2006/03/27 by Melvyn B. Nathanson, Nathanson, Melvyn B.
Mathematics · #11B13 #11B37 #11P81 #65Q05 #81R50 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Quantum Algebra (math.QA) #math.CO #math.NT #math.QA #msc:11B13 #msc:11B37 #msc:11P81 #msc:65Q05 #msc:81R50
paper · pdf · doi:10.48550/arxiv.math/0603623
8 pages; to appear in Integers: The Electronic Journal of Combinatorial Number Theory
arxiv created 2006/03/27 · arxiv updated 2009/12/01
The quantum integer [n]q is the polynomial 1 + q + q2 + ... + qn-1. Two sequences of polynomials U = \un(q)\n=1∞ and V = \vn(q)\n=1∞ define a \em linear addition rule ⊕ on a sequence F = \fn(q)\n=1∞ by fm(q)⊕ fn(q) = un(q)fm(q) + vm(q)fn(q). This is called a \em quantum addition rule if [m]q ⊕ [n]q = [m+n]q for all positive integers m and n. In this paper all linear quantum addition rules are determined, and all solutions of the corresponding functional equations fm(q)⊕ fn(q) = fm+n(q) are computed.