---
tag: GFSDMJHHVA
kind: concept
name: q-analog
msc: [05A30, 11B65]
sources:
  - url: https://en.wikipedia.org/wiki/Q-analog
same_as:
  wikidata: Q3491352
  wikipedia: Q-analog
---

## Summary

A q-analog of a formula or a theorem is a version with an extra parameter $q$, see the [[definition|CK80HQNHEK]].
It gives back the original in the limit $q \to 1$.
The basic example is the q-integer $[n]_q = 1 + q + \cdots + q^{n-1}$, which tends to $n$.
From it one builds q-factorials, [[Gaussian binomial coefficients|YS09Z1YNEK]] and q-series.
Good q-analogs often have a meaning of their own, such as counts over [[finite fields|0ZXZ82FTX2]] or generating functions of statistics on [[permutations|05RWFKVTJ7]].

## Explanations

### L1 human

Take a familiar number, such as $3$, and replace it by the polynomial $1 + q + q^2$.
Setting $q = 1$ gives back $3$, so nothing is lost.
But the polynomial carries more information: for instance, it can count objects by size and by an extra feature at the same time.
A q-analog of a theorem is a statement about such polynomials that becomes the original theorem when $q = 1$.