Talk:Falling and rising factorials

message: There is an inconsistency in notation between this page and the `Hypergeometric Series' page.  One uses the downstairs index for the falling factorial and one for the rising factorial.  I don't know which is standard, so I'm not prepared to correct the appropriate page, but I thought they should be consistent, so I'm pointing it out here. —Preceding unsigned comment added by 152.2.6.241 (talk • contribs) 
: But this article explicitly says at the outset that there are TWO notational conventions: one that is standard in combinatorics and another that is standard in special functions.
: 
: You need to actually read it in order to understand what it says. Michael Hardy 23:16, 24 May 2007 (UTC)
replies: text: I tried to rephrase the sentence to make it clearer, but it is immediately obvious from the definitions. (a)(n) is defined as (a)(a+1)(...)(a+n-1). (a+n-1)n is defined as (a+n-1)(a+n-2)(...)(a). These are just the same product written in a different order. This is not obvious only to someone who doesn't know what the two symbols mean. 152.3.47.94 (talk) 05:45, 6 February 2008 (UTC)
thread_title: Inconsistency with Hypergeometric Series

message: I have a problem with this equation:
:(x)_{n}=x(x-1)(x-2)\cdots(x-n+1)=\frac{x!}{(x-n)!}.\,
The left side of the last equal symbol holds for all real x values but the right side only holds for x larger or equal to n since factorials are not defined for negative values.
So according to my convincement there should either not be a equal sign there or it should be mentioned that the equation only holds only for a limited range. I am though not so certain about this that I dare to edit the article myself. Any thoughts on this? --Orri Tómasson (talk) 22:16, 15 February 2010 (UTC)
replies: text: It really should use the Gamma function instead since factorials are only defined for integers. I unfortunately removed the gamma function and left behind the factorial because of misunderstanding the objections in a previous section, or perhaps that's what they meant. Anyway the gamma function should give the correct result. Well actually it has some problems for x a negative integer. I'll have to think about that. People have written it with the gamma function so it is verifiable but it isn't completely correct. I'll have to have a think about that. Dmcq (talk) 22:24, 15 February 2010 (UTC)

text: I would argue for putting this formula back in with the restriction that n has to be a positive integer. It would make the notation immediately clear for people that do not know that the Gamma function can be seen as a (shifted by one) factorial for positive reals. Andy (talk) 14:36, 6 November 2013 (UTC)
thread_title:  Negative values 

message: I'm attempting a cleanup for grammar, punctuation, and meaning.  Specifically:
*Use "Pochhammer symbol" only for the symbol, not for a function denoted by the symbol (which, as the article says, is ambiguous).
*Observe that rising/falling factorials are defined over any ring.  This might be better placed in articles on the rising/falling factorials.
*Rm "Furthermore, the product of four consecutive integers is a perfect square minus one."  This seems unrelated to the context (relationship between binomial coefficients and rising/falling factorials).  There must be an appropriate article for it and its proof.
*Rm "It follows from these expressions that the product of  consecutive integers is divisible by ."  If this is restored, it should be accompanied by a proof, including or citing an independent proof that the binomial coefficients are integers (not hard, but essential).
*New section "Generalizations".  This refers to generalized P. symbols, but also defines generalized rising and falling factorials, which might be better placed in articles on the rising/falling factorials.
Zaslav (talk) 06:47, 20 May 2011 (UTC)
replies: text: Sounds fair enough. I wouldn't get hung up on proofs though, they're normally only put in if very short or notable in their own right. Sometimes for where it looks like something like that should be in just a quick summary of the main details of how something is proved and a citation is put in. This is an encyclopaedia rather than a full blown textbook and we're better off pointing them at where they can get the stuff done more rigorously if they want. Dmcq (talk) 09:18, 20 May 2011 (UTC)
replies: text: Yes, very reasonable, but there was no reference to a proof, either.  Mainly, the statement seemed not relevant.  Zaslav (talk) 19:48, 20 May 2011 (UTC)
thread_title:  Cleanup 

message: Why not insert just a simple example!
71.139.161.9 (talk) 04:37, 28 August 2014 (UTC)
thread_title:  Example! 

message: :The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section. 
The result of the move request was: moved. Unopposed request. Number 57 17:25, 4 April 2016 (UTC)
replies: text: Pochhammer symbol →  – This article is not about about the symbol, which has 3 ambiguous meanings; it's about falling and rising factorials and their properties.  The title should be renamed to "Falling and rising factorials", the "Alternate notations" section changed into a "Notation" section, which describes the Pochhammer symbol, its ambiguity, its confusion with binomial coefficient, and the other, unambiguous notations for falling or rising factorials.   Wolfram MathWorld says: "The Pochhammer symbol for n>=0 is an unfortunate notation used in the theory of special functions for the rising factorial, also known as the rising factorial power or ascending Factorial."  24.188.204.252 (talk) 00:49, 27 March 2016 (UTC)

text: The above discussion is preserved as an archive of a requested move.  Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.
and I would recommend switching all the properties, etc to the unambiguous symbols for clarity, but that's up to other people.  — Preceding unsigned comment added by 71.167.72.62 (talk) 01:22, 14 June 2016 (UTC)
thread_title:  Requested move 27 March 2016 

message: Currently the article is of two minds as to whether x is or is not restricted to be an integer. The lead section doesn’t say which, and my addition of integer factorial notation was reverted on the grounds that x need not be an integer. The first equation in the Properties section uses n-choose-k notation and calls it a binomial coefficient, implying that x is an integer. Later the same section says “x can be taken to be, for example, a complex number”. Then the next paragraph says “The rising factorial can be extended to real values of n using the Gamma function“, and by calling it an extension it implies that the basic interpretation is as an integer. Only later does the section “Connection coefficients and identities“ define the x-choose-n notation for non-integer x.
So I think someone more familiar with this than me should go through the article and either start with defining x as an integer and then extend the analysis to non-integers, or start with defining x as a real number (with, I think, some non-permitted values mentioned) and then defining the n-choose-k notation for non-integers when it first appears. Loraof (talk) 17:13, 15 March 2018 (UTC)
thread_title:  Integer and non-integer notation 

message: In the portion
> Feller[7] describes (x)^n as "the number of ways to arrange n flags on x flagpoles".
It is not clear at this point in the article what the symbol (x)^n is meant to represent (the rising or falling factorial?)  — Preceding unsigned comment added by Dzackgarza (talk • contribs) 00:20, 7 June 2019 (UTC)
replies: text: Thanks for pointing this out; I have corrected it. --JBL (talk) 01:38, 7 June 2019 (UTC)
thread_title:  Incorrect/undefined symbol 

message: It currently says the falling factorial (x)_n gives you the number of ways to put n flags onto x flagpoles but shouldn't this be the rising factorial instead???
2607:9880:1A18:96:49E9:E6C0:8C93:DD9A (talk) 08:05, 29 March 2020 (UTC)
replies: text: Can't tell if this is just a pun (in which case, well done) or actually a mathematical question (in which case, it is correct as written). --JBL (talk) 13:20, 29 March 2020 (UTC)

text: If there are n > 1 flags and you put one of them on a pole, you have failed to place the n flags on the poles.  And indeed in this case the task cannot be done, and (1)_n = 0. What you are describing is "how to place x flags from a collection of n on x flagpoles". --JBL (talk) 12:07, 3 August 2020 (UTC)

text: OK, thanks for the explanation, I understand now. Sorry for my confusion. I have added a phrase to clarify that all flags must be used, but not all poles.
--Entropeneur (talk) 14:49, 4 August 2020 (UTC)
thread_title:  Flagpole mistake?