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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)
Why does one do this? What motivated people to study these? — vivacissamamente ( talk) 23:10, 4 May 2008 (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 109.171.137.224 ( talk) 07:30, 11 April 2011 (UTC)
- Your version is a great deal more clear than what the article has currently; unfortunately, I also am not sure what a q-analog is exactly! It certainly would be nice for someone who really knows what this is about to clarify the lede. - Bryanrutherford0 ( talk) 13:59, 6 April 2015 (UTC)
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.
-- Joel B. Lewis ( talk) 22:31, 29 July 2011 (UTC)
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.
Bruce Berndt's What is a q-series. tryptographer ( talk) 17:54, 27 June 2013 (UTC)
The introductory paragraph currently concludes:
"There are two main groups of q-analogs, the "classical" q-analogs, with beginnings in the work of Leonhard Euler and extended by F. H. Jackson[2] and others.[3]"
This makes no sense as written. Could someone who knows what the intention was update the formulation? — Preceding unsigned comment added by 50.244.197.139 ( talk) 00:06, 11 March 2018 (UTC)
- The sentence is just missing a comma before "and others". But it has a number of other issues: it is not supported by the references and its assertion is not supported by the body of the article. I will remove it. -- JBL ( talk) 14:50, 16 March 2018 (UTC)
- Since the hypergeometric function is mentioned right off the bat in the lead, it would be good to have one or two paragraphs saying what it is, giving an example of an identity that it satisfies, and then giving q-versions. For example, it would be good to include the definition of 2F1, one or two nice identities that it satisfies (recover Vandermonde as a special case, for example, or a Pfaff transformation; even just recovering the binomial theorem from 1F0?). Then define the corresponding basic hypergeometric series and give the corresponding identities ( q-binomial theorem, q-Vandermonde, whatever). Right now links to these topics are missing or buried in this article.
- The current division of the first section makes no sense, and is introduced in a very pedagogical (rather than encyclopedic) way. Important functions like the q-exponential should be separated out with their own sub-headers; the q-binomial coefficient should be discussed once together, rather than twice separately.
- The field with one element section needs major work: the first sentences draws a distinction that is impossible to make sense of if one doesn't already understand it, and links to important related things like Tits building are absent. I own this book and some amount of this story could be glossed from it without too much trouble, I hope.
Some things that are not used as references but surely should be are the books of Gasper--Rahman and Stanley. Maybe one day I will get around to making some of these improvements (but anyone else is welcome to!). -- JBL ( talk) 15:24, 16 March 2018 (UTC)