These were first considered in the case \(m=2,n=1\) by Wallis in 1655, and later by Euler and more systematically by Gauss. The case \(m=3,n=3\) was considered by Clausen in 1828; the general case was introduced by Thomae [121] in 1870.

Definition26.1.2.

For parameters \(\alpha_1,\dots,\alpha_m, \beta_1,\dots,\beta_n\text{,}\) the basic hypergeometric series (or \(q\)-hypergeometric series)

where \((x; q)_k\) denotes the \(q\)-Pochhammer symbol

\begin{equation*}
(x; q)_k = \prod_{i=0}^{k-1} (1 - x q^k).
\end{equation*}

Note that the term \((q; q)_k\) corresponds (up to signs and factors of \(q\)) to the factor \(k!\) in the ordinary hypergeometric series. For the Gaussian case (i.e., \(m=2, n=1\)) this was first introduced by Heine [64] in 1846.

Remark26.1.3.

The process of \(q\)-deformation has a long rich history, of which we give a scandalously brief summary here.

Products of \(q\)-Pochhammer symbols appear naturally in the study of generating functions connected to partitions, which first appear in the work of Euler, were systematically studied by Jacobi, occur prominently in the notebooks of Ramanujan, and have a continuing history far beyond the scope of this remark. See [48] for a comprehensive development (circa 1988).

A quantum group is a certain noncommutative algebra which can be viewed as a \(q\)-deformation of the universal enveloping algebra of a Lie algebra (or an affine Lie algebra). These were introduced by Drinfeld and Jimbo in the 1980s with a view towards quantum statistical mechanics.

One way to make sense of (some) statements about the putative field with one element is to consider statements about the finite field of \(q\) elements and then specialize at \(q=1\text{.}\) As an elementary example, taking the formula

and setting \(q=1\) gives \(n!\text{,}\) the order of the group \(S_n\text{,}\) which by chance happens to be the Weyl group of \(\GL_n\text{.}\) One is thus led to treat Weyl groups as “algebraic groups over the field with one element”.

Going in the opposite direction, it is quite common in combinatorics to look for statements about finite sets that can be promoted to statements about finite-dimensional vector spaces. One recent development in this direction is the \(q\)-analogue of a matroid ([76]).

Remark26.1.4.

Note that Heine uses the letter \(q\) as we do nowadays for the deformation parameter. However, this was presumably following the model of Jacobi, who used the letter \(q\) in the notation for the Jacobi theta function. This comes from the usage where \(q\) stands for the value \(e^{2 \pi i\tau}\) where \(\tau\) is a value in the upper half-plane, as in the theory of elliptic functions.

The point of all this is that \(q\) does not stand for “quantum”, as it was entrenched in the notation a full 50 years before the first inklings of quantum mechanics!

Subsection26.2Jackson's \(q\)-calculus

We focus here on a specific instance of \(q\)-deformation introduced by Jackson [71], [72] in 1908.

Definition26.2.1.

Given a function \(f(x)\) and a parameter \(q\text{,}\) the \(q\)-derivative of \(f(x)\) is defined as

provided that we are working in some context where the infinite sum makes sense. For example, if this is taking place in a \(q\)-adically complete ring and \(f\) is itself \(q\)-adically continuous, then one may check that performing this operation and then taking the \(q\)-derivative recovers \(f(x)\text{.}\)

Remark26.2.3.

Without the \(q\)-deformation, hypergeometric series have long been viewed as solutions of the hypergeometric differential equation

starting with the work of Gauss in the case \(m=2,n=1\text{.}\) In modern language, hypergeometric differential equations (particularly with \(n = m-1\)) provide important examples of Picard-Fuchs equations which describe the variation of algebraic periods on some family of algebraic varieties; in particular, there is a natural construction via which hypergeometric equations emerge from de Rham cohomology.

The \(q\)-analogue of de Rham cohomology was first considered by Aomoto [7], [8] in 1990 in order to provide a similar geometric description of the Jackson integrals that appear when one tries to transport the previous construction to the setting of \(q\)-hypergeometric series.

Subsection26.3The \(q\)-de Rham complex of Aomoto

Definition26.3.1.

For \(R \in \Ring\text{,}\) define the framed \(q\)-de Rham complex associated to \(R[x]\) as

For \(R \in \Ring_{\QQ}\text{,}\) we can identify \(q\Omega^\bullet_{R[x_1,\dots,x_r]/R,\square}\) with the usual de Rham complex by a Taylor series construction. In particular, in this case the construction is independent of the choice of coordinates. (Compare [109], Lemma 4.1.)

Definition26.3.3.

To promote \(q\Omega^\bullet_{R[x_1,\dots,x_r]/R,\square}\) to a \(\ZZ \llbracket q-1 \rrbracket\)-dga, we must account for the asymmetry in the product rule (26.1). To this end, we equip \(q\Omega^1_{R[x_1,\dots,x_r]/R,\square}\) with a \(q\Omega^0_{R[x_1,\dots,x_r]/R,\square}\)-bimodule structure using the standard action on the left and the action on the right via \(f(x) \mapsto f(qx)\text{.}\)

With this, we may view \(q\Omega^\bullet_{R[x_1,\dots,x_r]/R,\square}\) as a \(\ZZ \llbracket q-1 \rrbracket\)-dga; however, the asymmetry we just introduced means that this dga is not commutative.

Remark26.3.4.

We wish to emphasize that by contrast with the ordinary de Rham complex, the definition of the \(q\)-de Rham complex exhibits a genuine dependence on the choice of coordinates (so in particular it is not functorial enough to admit a left Kan extension). For example, there is no way to promote the automorphism \(x \mapsto x+1\) on \(R[x]\) to an \(R\llbracket q-1 \rrbracket\)-linear morphism of complexes as illustrated in Figure 26.3.5.

Namely, such a map would have to send \(x^{n-1}\) to

but the coefficients in the latter expression are not necessarily contained in \(R \llbracket q-1 \rrbracket\) unless \(R\) is a \(\QQ\)-algebra.

However, we can instead hope to prove that \(q \Omega^\bullet_{R[x_1,\dots,x_r]/R,\square}\) is indeed functorial in \(R[x_1,\dots,x_r]\) as an object (and even a commutative algebra object) in \(D(R \llbracket q-1 \rrbracket)\text{;}\) this was conjectured by Scholze in [109]. To make this more explicit in the previous example, consider the isomorphisms

becomes a quasi-isomorphism once we take the universal quotient that kills \(x-y+1\) in degree \(0\) (and similarly with the variables reversed).

Using Remark 26.3.2, one can reduce the claim to the corresponding statement after derived \(p\)-completion. In this case, one can provide a coordinate-free interpretation of \(q\)-de Rham cohomology using prismatic cohomology; see Theorem 27.3.8.