# What is the origin of q-calculus notation?

It is kind of cute that q-analogues are used in physics (see this link for example), but it is also kind of confusing because the 'q' does not stand for 'quantum'. It predates that use!

So, where does the 'q' really come from? Is it only that McMahon (or someone else) first used that particular letter?

• You shouldn't expect the $q$ to stand for quantum because even classical problems have applications for functions such as $q$-exponentials. For example, $k=0$ in a Friedmann equation with an $\rho$-$a$ power law gives a $q$-exponential solution for $a$. – J.G. Mar 23 '18 at 16:01
• Perhaps it goes back to the use of $q=e^{i\pi\tau}$ or $q=e^{i2\pi\tau}$ in classical Jacobi elliptic function theory. – Gerald Edgar Mar 23 '18 at 18:14
• @JG That is exactly what I said, and why I asked the question... – Rodrigo A. Pérez Mar 25 '18 at 2:25

A good review of the history of the subject is Ch. 2 of Ernst's Comprehensive Treatment of q-Calculus, which names Bernoulli's numbers as an early appearance, and characterizes the work on elliptic and theta functions since 1750-s as "pre-$q$ mathematics".

Presumably, $q$ stands for "quotient" or rather its Latin counterpart "quotus", it traditionally appears in the geometric series notation. Euler in 1738 studied generating functions of the Bernoulli polynomials, and already introduced the $q$-series (a.k.a the Euler function) in the oft-cited chapter 16 of Introductio in Analysin Infinitorum (1740) called De Partitio Numerorum. Euler's work on partitions was prompted by questions about them raised to him by a French mathematician Naude, see Debnath's Brief History of Partitions of Numbers. Jacobi continued to use the letter in his work on theta functions.

Ernst divides the $q$-tradition into two major "schools", Watson's and Austrian, and makes interesting remarks about its notational history:

"The various Schools of $q$ to be outlined below have developed over a period of roughly 300 years since the Bernoullis and Euler. These Schools today use and communicate in such different languages, that they have problems understanding each other. The big School of Watson speaks—both literally and metaphorically— English. It has little exact consciousness of, indeed may not find it necessary, to study and know the roots of the vernacular... And we have in the q-area, in relation to other areas, a tendency to develop specific, new languages (e.g. notations), which surpasses this tendency in other fields of mathematics. This is either because practitioners find these more easy to speak, because they find them more beautiful or simply because they do not know and cannot pronounce the older forms.

[...] Both Schools or traditions recognize the early legacies of Gauß and Euler. Only the Austrian School, however, represents and incorporates the entire historical background which includes the pre-$q$ mathematics, namely the Bernoulli and Euler numbers, the theta functions and the elliptic functions. The vast area of theta functions and elliptic functions... is in fact $q$-analysis before $q$ was really introduced.

[...] The Austrian School is little known in the English speaking world, e.g. the USA and the Commonwealth, for two main reasons: Immediately after the first world war, ca. 1920 until 1925, the German and Austrian mathematicians were barred from participation in the big mathematical conferences... The second and more important reason has to do with languages. Most of the mathematicians of the Austrian School wrote in either German or French and, as regards the oldest, namely Euler, Gauß and Jacobi, in Latin. Only few English-speaking mathematicians today master these languages... The Watson School is today the most widespread and influential of the two principal Schools/traditions. But again this is mainly due to the language."

• “Jacobi continued to use the letter”: so, who started? – Francois Ziegler Mar 26 '18 at 6:47
• @FrancoisZiegler As far as I can tell, Euler. – Conifold Mar 26 '18 at 19:42
• I’m not seeing this... According to Roy (2011, p. 606), “Euler wrote $x$ for our $q$ and $z$ for our $x$. Note also that the term $q$-series came into use only in the latter half of the nineteenth century, appearing in the works of Cayley, Rogers, and others. Jacobi may possibly have been the first to use the symbol $q$ in this context”. I presume Roy means Jacobi (1829, p. 85). – Francois Ziegler Mar 26 '18 at 21:52
• @FrancoisZiegler You may be right, I did not check the sources and Ernst probably was not interested in tracing the $q$ itself. – Conifold Mar 26 '18 at 22:34