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?
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$\begingroup$ In the first displayed equation, presumably $m$ is supposed to be $n$. $\endgroup$– Joe SilvermanCommented Oct 20, 2014 at 11:04
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2$\begingroup$ By some reason physicists prefer $\frac{q^m-q^{-m}}{q-q^{-1}}$... $\endgroup$– მამუკა ჯიბლაძეCommented Oct 20, 2014 at 12:00
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3$\begingroup$ Gauss used Gaussian binomial coefficients, and quantum integers are special cases of these. $\endgroup$– Chris GodsilCommented Oct 20, 2014 at 12:12
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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 CohnCommented Oct 20, 2014 at 12:22
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2$\begingroup$ See also "The history of q-calculus and a new method" by Ernst citeseerx.ist.psu.edu/viewdoc/… $\endgroup$– Tom CopelandCommented Sep 6, 2016 at 1:28
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2 Answers
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I think q-integers 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).
answered Oct 20, 2014 at 13:19
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$\begingroup$ Many thanks! I did not know that one. $\endgroup$ Commented Oct 20, 2014 at 13:33
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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.
answered Oct 20, 2014 at 9:07
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$\begingroup$ 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. $\endgroup$ Commented Oct 20, 2014 at 9:30