q-analogs in finite expressions do not require q to approach 1 in a special way. Therefore, I removed from the "Introductory examples" the restriction on how q approaches 1. The text is
- For convenience, the limit as q → 1 inside the unit circle is written as the limit as q → 1− (which suggests the limit through real values tending up to 1; that is in fact more restricted, though the difference is not usually significant).
If someone adds examples of infinite series, where it does matter how q varies in the limit, then they may want to use this text. Zaslav 03:11, 7 August 2007 (UTC)
The definition of q-analog is unclear. The first sentence says "a q-analog is, roughly speaking, a theorem or identity in the variable q that gives back a known result in the limit, as q → 1 ". That sounds like it is saying that a q-analog of n is any expression whose limit is n. Is that true? Or is the q-analog of n only (1-q^n)/(1-q), and nothing else, by definition? That sentence suggests the former, but the examples suggest the latter. I would lean toward the former. But Mathworld says there can be multiple, different q-analogs for the same thing. So this is confusing.
One guess is that the q-analog of a number (or factorial or binomial) is a single expression, given by definition, and it's not all the other expressions with the same limit. But that different authors sometimes choose slightly different definitions, and so there are multiple, different q-analogs. Similar to how there are at multiple definitions of "Fourier transform", differing only in a leading constant, with different authors choosing different definitions. Is that the case for q-analog? If so, then the first sentence should be changed to something like:
- In mathematics, in the area of combinatorics and special functions, the q-analogs are expressions in the variable q that were defined to be useful generalizations of various theorems or identities, and which reduce to the common result in the limit, as q → 1 (from inside the complex unit circle in most situations). The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.
I won't make this change myself, because I'm not sure what a q-analog is. Could someone who knows what it is change the article? —Preceding unsigned comment added by 126.96.36.199 (talk) 07:30, 11 April 2011 (UTC)
Suggestions for rewriting
Can someone who knows something about the second, third and fourth paragraphs of the introduction verify if they are correct and meaningful and expand them into sections? I have a hard time seeing what business they have in the introduction, not being mentioned elsewhere in the article, and lacking any comprehensible explanation or example.
The article is way short on references to the combinatorial literature -- help?
Does anyone use either of the notions of q-addition, subtraction, etc., defined in the article for anything? (The first q-exponential defined is an example of something interesting, though you certainly can't tell that from the article.) Citation?
There appear to be zero examples of the relevance of special function theory -- help?
Can the section on Tsallis be replaced/expanded with content that explains why anyone should care?
q-Sperner or q-Ramsey theory: example?
Is there a meaningful distinction between "classical" q-theory and the stuff done currently by algebraic combinatorialists? This distinction is implicit but it's never made clear what "classical" q-theory actually consists of.
needs to be a separate article for q-series
Why is there a redirect from q-series to this? q-analogs only occupy a rather tiny portion of the actual theoretical work with most q-series: not every q-series can be thought of as a q-analog, and some require different proof strategies.