The previous post looked at the probability that a random *n* by *n* matrix over a finite field of order *q* is invertible. This works out to be

This function of *q* and *n* comes up in other contexts as well and has a name that we will get to shortly.

## Pochhammer symbols

Leo August Pochhammer (1841–1920) defined the *k*th rising power of *x* by

Rising and falling powers come up naturally in combinatorics, finite difference analogs of calculus, and in the definition of hypergeometric functions.

*q*-Pochhammer symbols

The *q*-analog of the Pochhammer symbol is defined as

Like the Pochhammer symbol, the *q*-Pochhammer symbol also comes up in combinatorics.

In general, *q*-analogs of various concepts are generalizations that reduce to the original concept in some sort of limit as *q* goes to 1. The relationship between the *q*-Pochhammer symbol and the Pochhammer symbol is

For a simpler introduction to *q*-analogs, see this post on *q*-factorial and *q*-binomial coefficients.

## Back to our probability problem

The motivation for this post was to give a name to the function that gives probability a random *n* by *n* matrix over a finite field of order *q* is invertible. In the notation above, this function is (1/*q*; 1/*q*)_{n}.

There’s a confusing notational coincidence here. The number of elements in a finite field is usually denoted by *q*. The *q* in *q*-analogs such as the *q*-Pochhammer symbol has absolute value less than 1. It’s a coincidence that they both use the letter *q*, and the *q* in our application of *q*-Pochhammer symbols is the reciprocal of the *q* representing the order of a finite field.

I mentioned in the previous post that the probability of the matrix being invertible is a decreasing function of *n*. This probability decreases to a positive limit, varying with the value of *q*. This limit is (1/*q;* 1/*q*)_{∞}. Here the subscript ∞ denotes that we take the limit in (1/*q;* 1/*q*)_{n} as *n* goes to infinity. There’s no problem here because the infinite product converges.

## Mathematica and plotting

The *q*-Pochhammer symbol (*a*; *q*)_{n} is implemented in Mathematica as `QPochhammer[a, q, n]`

and the special case (*q*; *q*)_{∞} is implemented as `QPochhammer[q]`

. We can use the latter to make the following plot.

Plot[QPochhammer[q], {q, -1, 1}]

Recall that the *q* in our motivating application is the reciprocal of the *q* in the *q*-Pochhhammer symbol. This says for large fields, the limiting probability that an *n* by *n* matrix is invertible as *n* increases is near 1, but that for smaller fields the limiting probability is also smaller. For *q* = 2, the probability is 0.288788.

Plot[QPochhammer[1/q], {q, 2, 100}, PlotRange -> All]

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