Linear quantum addition rules
Abstract
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 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 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.
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