Who first gave the definition of quantum integers $$ [m]_q = \frac{1  q^m}{1  q} $$ and addition as $$ [m]_q \oplus_q [n]_q = [m]_q + q^m [n]_q $$ and multiplication as $$ [m]_q \otimes_q [n]_q = [m]_q [n]_{q^m} $$ and in which context?

$\begingroup$ In the first displayed equation, presumably $m$ is supposed to be $n$. $\endgroup$ – Joe Silverman Oct 20 '14 at 11:04

2$\begingroup$ By some reason physicists prefer $\frac{q^mq^{m}}{qq^{1}}$... $\endgroup$ – მამუკა ჯიბლაძე Oct 20 '14 at 12:00

3$\begingroup$ Gauss used Gaussian binomial coefficients, and quantum integers are special cases of these. $\endgroup$ – Chris Godsil Oct 20 '14 at 12:12

1$\begingroup$ I believe Gauss introduced Gaussian binomial coefficients in 1811 in Summatio quarumdam serierum singularium. I just flipped quickly through it, and I couldn't see any explicit discussion of quantum integers per se, but Chris Godsil is certainly right that they are implicit since they are the special case $m$ choose $1$ (and in any case I might have missed some broader discussion). $\endgroup$ – Henry Cohn Oct 20 '14 at 12:22

2$\begingroup$ See also "The history of qcalculus and a new method" by Ernst citeseerx.ist.psu.edu/viewdoc/… $\endgroup$ – Tom Copeland Sep 6 '16 at 1:28
I think qintegers were first introduced by F.H. Jackson in 1903 in the paper http://www.biodiversitylibrary.org/item/130137#page/15/mode/1up (On generalized functions of Legendre and Bessel).
Melvyn Nathanson, in Linear quantum addition rules claims the invention of the addition and multiplication rules of quantum integers, but notes that the polynomial representing a quantum integer itself has appeared previously in several contexts.

$\begingroup$ Thanks. Yes, I have seen a similar paper of his in Lecture Notes of Computer Science, but was wondering whether this was really the first appearance of these definitions. $\endgroup$ – Martin Peters Oct 20 '14 at 9:30