  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
$\mathcal{V} = \{v_n(q)\}_{n=1}^{\infty}$ define a {\em linear addition rule}
$\oplus$ on a sequence $\mathcal{F} = \{f_n(q)\}_{n=1}^{\infty}$ by
$f_m(q)\oplus f_n(q) = u_n(q)f_m(q) + v_m(q)f_n(q).$ This is called a {\em
quantum addition rule} if $[m]_q \oplus [n]_q = [m+n]_q$ 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
$f_m(q)\oplus f_n(q) = f_{m+n}(q)$ are computed.
