Talk:Falling and rising factorials

Inconsistency with Hypergeometric Series

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)

Note that I am not linking to this page because the symbolic language is not useful to my purposes. The exclusive use of a minority symbolic representation is probably not as constructive as it might otherwise be.CarlWesolowski (talk) 21:24, 29 November 2016 (UTC)

It is, of course, obvious that (a)⁽ⁿ⁾ = (a + n − 1)_n. No, it is not obvious that this is the case. Instead of putting "it is obvious" or "it is trivial" could you please put how it is derived for people who do not understand like myself and many other students? —Preceding unsigned comment added by 129.81.123.46 (talk) 03:28, 13 December 2007 (UTC)

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)

Rightly or wrongly I wish to blame the pochhammer symbol for my not finishing my PhD thesis. Markjohndaley (talk) 07:27, 5 April 2009 (UTC)

Reversion to old leader

I've removed the reversion to the old leader because I think the old leader was far more confused. The Pochhammer symbol as a function referred to by name always refers to the rising factorial as far as I'm aware. As a symbol the Pochhhammer symbol refers to the form (x)_n wherever I've seen the symbolism used for something else. Dmcq (talk) 11:15, 26 December 2009 (UTC)

I disagree. The name Pochhammer symbol refers to a symbol, and that symbol is (x)_n with n an integer. Unfortunately as things go in mathematics, the meaning attached to a symbol can change over time. Pochhammer used the symbol to denote binomial coefficients, as noted in the article; if Knuth puts that in print, you can be sure he has spent many hours in the library studying 19-th century texts to find out the true history. So the current leader is at least misleading in that respect. Fortunately nobody uses the Pochhammer symbol any more for binomial coefficients (there remain plenty of options for those!), but to confuse things there are differences between the meanings attached to the symbol itself, and to the term "Pochhammer symbol". For my part, it would seem best to discontinue using either of them, and use the names rising/falling factorial power with the corresponding notation from Graham-Knuth-Patashnik, which are much better chosen from the point of suggestiveness, ambiguity and ease of use (and would incidentally reduce the risk of getting the number of h's in "Pochhammer" wrong;-). But to claim that "Pochhammer symbol" as a function always refers to the rising factorial power seems clearly wrong, as I just checked by looking at Wikipedia articles that link to this article; in fact I could detect no clear preference between rising and falling meaning (and also noticed that several pages were listed without apparently linking here at all, curiously). Note also that both "falling factorial" and "rising factorial" redirect it "Pochhammer symbol", with substantially more hits for the former. So I think the lede should explain the situation and be more neutral. Marc van Leeuwen (talk) 14:06, 27 December 2009 (UTC)

The article does say as a symbol the Pochhammer symbol means the form (x)_n. As a function without any qualifier without saying what the symbol stands for when people say Pochhammer symbol they mean the rising factorial as far as I'm aware. For instance in computer algebra systems like Mathematica one might say 'Pochhhammer' and be sure to get the rising factorial. Did you read what was there previously? I'm not saying what is there now is wonderful but the old thing they wanted to revert to could only be said to be okay in that it was so confused one could draw anything one wanted to out of it.

I just did a search for 'Pochhammer' which give a much smaller list and as far as I can see the only troublesome one is Binomial theorem, the other ones where the symbol is used for the falling factorial warn about it, otherwise it is used for the rising factorial without any reference to any problems.

Perhaps the article might be improved by removing the word symbol for the function as it is sometimes written that way? 14:42, 27 December 2009 (UTC)

I';ve tried moving all the problems and just referred to the 'Pochhammer function' for the function version to the start so they're easy to concentrater on, see what you think Dmcq (talk) 16:22, 27 December 2009 (UTC)

I approve of the changes made, although it is not quite clear to me which function if meant by "Pochhammer function"; presumably the function of the 'base' x obtained by taking a fixed 'exponent' n (therefore a polynomial function); this could be stated more clearly. I do plan to clean up the lede by making it concentrate more on interesting facts about the subject, less about confused notation/terminology surrounding it. However, I also think some stuff from "alternative notations" could be moved into the lede, and that the article would improve by using that notation throughout (even if it is admittedly not the "Pochhammer symbol"). Note that the end of the article already uses this notation.

One more thing. All mention of the Gamma function and factorials should be removed. Not only is it somewhat ridiculous to want to define a simple polynomial function in terms of a transcendental function like the Gamma function, or in terms of factorials (which are usually only used for natural numbers), but also this necessarily suggests points where the function would be undefined (for poles of the Gamma function for instance) while the factorial powers are always defined if the exponent in a natural number. Marc van Leeuwen (talk) 15:45, 3 January 2010 (UTC)

The Pochhammer function or Pochhammer polynomial is always the rising factorial I believe. It seems like original research to me to use symbolism that isn't in common use. I had been thinking of removing that use of the new notation at the end. And especially using non-standard symbolism in an article called Pochhammer symbol. If something like that was to be done I think it would be best to set up a separate article on the rising and falling factorials and move most of the content here to that. Or even rename this article and just have Pochhammer symbol link to it. Personally I do not find the alternate symbolism used at the end as intuitive or as readable as that used in the article.

Looking at that business about the Gamma function I can see a problem okay when one considers for example the Shaeffer sequence. The Gamma function definition defines the function as non-zero for negative values of n whereas it should really be zero if you don't want to restrict the sum to 0 to n. It isn't a big objection but I've a fondness for removing limits for sums like that. I'm not sure if anyone has considered negative or non-integer values of n in any context.

As to Shaeffer sequences that stuff should be amalgamated with the umbral calculus stuff as well somehow. Dmcq (talk) 17:58, 3 January 2010 (UTC)

I think I wasn't clear, so let me try again. I have absolutely no problem with the claim that Pochhammer function is always the rising factorial (given that I've never personally seen that term used, I'll accept that those who do use it that way). Only this sentence does not make clear whether one is talking about the function $x\mapsto x^{\overline {n}}$ for fixed n, or about $n\mapsto x^{\overline {n}}$ for fixed x, or about $(x,n)\mapsto x^{\overline {n}}$, which are three different functions. If you add the name "Pochhammer polynomial" it is clear the first one is meant, which is what I would have guessed; still the article could be clear about this.

My problem with the Gamma function is the equality in the lede

$x(x+1)(x+2)\cdots (x+n-1)={\frac {\Gamma (x+n)}{\Gamma (x)}}$

because the left hand side is always defined (assuming n∈N as is clearly done here), but the right hand side is not when x is a nonpositive integer. Another objection is that the right hand side does not shed any additional light on a perfectly clear left hand side. I'm not worrying about negative n, since the lede is not talking about that. I have less difficulty about talking about the Gamma function in a section about generalization beyond the case n∈N, although a discussion about when exactly things are defined would be in place, and more importantly one should verify that there is some interest in the literature in such cases in the first place. If the latter is not the case, then mention certainly does not meet Wikipedia:Notability standards and quite possibly constitutes original (though quite trivial) research. As for Sheffer sequences I don't see their pertinence to these matters, the article does not even seem to mention Pochhammer symbols or factorial powers at all, but in any case it always has n∈N.

Finally, I don't think you should call using a non-standard notation original research (supposing this was the case), since it doesn't present any facts (but nonetheless it would clearly be a very bad thing if individuals use Wikipedia to push their own favorite notation). However the notation by Graham, Knuth, Patashnik is in the published literature, it's used extensively in that book and I'm pretty sure elsewhere as well, so one cannot say that it is not in common use, just that it is introduced fairly recently. I am in no way affiliated to those authors, so my preference for this notation is certainly not my original whatever; it's just that I find they did a superior job of making up a good convention. Also note that they call them factorial powers, and they are indeed much more like powers than like factorials or products; yet even the section where they are discussed no longer mentions this but rather sequential products (somebody out there trying to wipe out terminology she doesn't like?) Marc van Leeuwen (talk) 10:15, 4 January 2010 (UTC)

I did actually do a quick search via google books and scholar for uses of the notation by references to the Knuth and Patashnik book when I said that and I only came up with one pdf file so I do think it would be at least synthesis in wikipedia terms to start presenting articles using them.

References to Pochhammer function occur in maths packages like Mathematica. I notice the Pochhammer polynomial is referred to only in the umbral calculus section and not actually defined. It is exactly the same as the Pochhammer function and should be in the leader. I can't see what the relevance of saying n is a constant is in what you were saying, I view functions as always depending on their parameters without any global constants. In a particular context one might have n a constant so as to simplify the expressions but that doesn't mean the Pochhhammer function doesn't depend on it.

As to the Gamma function in the way you say using the Gamma function in the way it is used in the article is quite common despite the problems you say, the problems can be dealt with by saying the value is the limit of the expression, basically the bottom is divided out of the top. I've absolutely no problems with it being removed though and just leaving the factorials instead.

For the Sheffer sequence I was just talking about the end of the properties section and the umbral calculus section which dealt with similar subjects. The Sheffer sequence article may not have mentioned rising and falling factorials but it is I believe basically because of that Sheffer sequence identity that rising and falling factorials appear in so many contexts. Dmcq (talk) 13:11, 4 January 2010 (UTC)

(Grmpf, how unclear can one get in explaining). The question was whether the "Pochhammer function" is a function of one or two arguments, and if it is one then whether that is x or n. Only as a function of x is it a polynomial (function), and if n is not an argument then it is fixed in the (nth) "Pochhammer function". Also, the problems I had with Gamma, I have even more so with factorials, because in addition to problems with negative integers, the factorial is not originally intended to be used with non-natural numbers (while Gamma clearly is). <rant> The mere fact that computer algebra packages and pocket calculators extend the definition (for pragmatic reasons; it avoids having to give an error message) does not change this basic difference in origins. Note also that there is more than one way to extend factorials analytically, although one can add some requirements to pick a unique one. Personally I would find it convenient if wherever I saw E! I could be sure that the expression E is meant to designate a natural number, which is true for 99.9% of the uses I've encountered, but alas this is not to be. In fact I must admit that Knuth, otherwise my hero, is one to promulgate this abuse of factorials. According to the factorial article, the (preferred) analytic extension of the factorial is called the Pi function (by Gauss).</rant> Marc van Leeuwen (talk) 09:40, 5 January 2010 (UTC)

I really do not understand the problem you have with n and whether it is a constant or not in the Pochhammer function. It definitely is not a constant as in e or π. If you look at the bit about the Sheffer sequence you'll see it is a sum with different values for n. It only varies over integers, is that what you mean? It all depends on how you want to think about it and context. In maths packages the Pochhammer function will be written as Pochhammer[x,n] or Pochhammer(x,n) so no distinction is made. For the Pochhammer polynomials I guess the n can be considered a considered a constant when one talks about the nth Pochhammer polynomial x⁽ⁿ⁾ the same as the Bernoulli polynomials B_n(x). It is about as useful I think as the difference between talking about the nth factorial and the factorial function.

The Pi function is exactly the same as the Gamma function just with the argument offset by one, it might make converting expressions easier but unfortunately time has raised Gamma as the main one, the other looks too much like multiplication I guess. As to using completely different analytic continuations, well I think this article has enough problems without dragging in something like that which isn't exactly mainstream. Anyway the Gamma or Pi function does have the right value when doing the divide which any others I've seen don't, that is with counting the value at negative integers as being the limit of the divide near the point.

So as I said before the only unhappiness I have is with what they are with negative n and that would be original research so doesn't count. Gamma function or factorial or both I haven't a great strength of feeling. As to removing them altogether that seems wrong, I'm sure they're both in the literature, in fact I think the Gamma one is the more common one. Dmcq (talk) 11:26, 5 January 2010 (UTC)

Negative values

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)

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)

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)

"Factorial" vs. "sequential product"

Someone changed most of the "rising factorial"s and "falling factorial"s to "rising sequential product"s and "falling sequential product"s and inserted the statement "(sometimes improperly called "factorials")". I am reversing this terminology in favor of the common names. Forty years of work in combinatorics have not acquainted me with the name "sequential product". It is certainly not generally used. There is nothing improper about the names "falling" and "rising factorials" – they are not unmodified "factorials", obviously, they are modified factorials. I would be happy to discuss the terminology if the "sequential product" person wishes to have a discussion. Zaslav (talk) 06:05, 20 May 2011 (UTC)

I'm not the "sequential product person", but I do agree that calling them factorials is improper and misleading (without any judgement about frequency of use). They are not modified factorials; the factorial is an operation of one argument, any form of the Pochhammer symbol is a notation with two arguments. I like Knuth's terminology "rising/falling factorial power", because it is exponentiation, not the factorial, that is modified (and these expressions are used much more like powers than like factorials, for instance as a linear basis of polynomial rings). If that is considered too lengthy, "rising/falling power" still properly conveys the essence. Maybe this makes me seem like a Knuth-adept, which I'm not (I disagree strongly with some of his notation); however in this case he clearly spent more thought than most people in combinatorics on proper notation and terminology. Marc van Leeuwen (talk) 16:47, 20 May 2011 (UTC)

It's best for an encyclopaedia just to stick to the sources unless there is a serious problem with them. Dmcq (talk) 18:26, 20 May 2011 (UTC)

I agree with Dmcq. Wikipedia is not the place for terminological innovation, unless there is serious confusion in the literature, in which case editors may have to choose a WP standard, or at least a standard for an article. (That's not so rare, but there has to be a very good reason.)

The most common usage is "falling/rising factorial". This is not a new coinage like Graham-Knuth-Patachnik's. It is widespread and thoroughly established. It follows that the term "factorial" does not mean a function of one argument, when it is modified by "rising/falling". I'm surprised that Marc van Leeuwen, who is sensible, is misled by one meaning of "factorial" to think it cannot have other meanings. Here is an example of what such thinking would lead to.

Consider the term "coefficient". The same kind of reasoning would insist that a "coefficient" has to be multiplying something. We would have to abandon the name "binomial coefficient". We would not be allowed to refer to "the value of the binomial coefficient ${\binom {p}{k}}$" or to say "this binomial coefficient is divisible by p when p is prime and k < p." Zaslav (talk) 19:45, 20 May 2011 (UTC)

I'm not saying Wikipedia should not use "falling factorial" if that is established, I just wanted to explain that there is some justification to consider the terminology less proper, and that more proper alternatives than the traditional ones can be found. In fact "falling factorial" is a neologism for "Pochhammer symbol" as well, just a bit less recent. There are few notations/terms that are in a more sorry state than the Pochhammer symbol with respect to universal agreement ("natural number" is another disaster that comes to mind). But I don't think the mission of WP is to enforce uniformity, either to some traditional use or to more rational new uses; it should merely establish existing differences clearly. There are more examples of competing notational schools (like always denoting polynomials as "f(X)", or never doing so) where I think policy is not to change existing articles only to change notational school.

To answer to the "binomial coefficient" example, they are of course originally truly coefficients used to multiply something, in the binomial formula; it would seem excessive to protest calling them coefficients when they are used on their own. By contrast I do take issue with "multiset coefficients", since the term refers to a recently introduced notation (many will simply write an equivalent binomial coefficient instead), which was not accompanied by that term (in fact not by any term at all), and moreover there is no "multiset formula" in which they are coefficients, so this seems to be terminology that should not be used, even less introduced, in Wikipedia. Marc van Leeuwen (talk) 06:05, 21 May 2011 (UTC)

Thanks, Marc. I agree that one can argue over what is the best terminology, and I'm glad you agree that WP isn't the place to change established terminology. (Though there are times I want to do so myself! I'm restrained by WP demons wearing white gloves....) Zaslav (talk) 12:20, 22 May 2011 (UTC)

Cleanup

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 n consecutive integers is divisible by n!." 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)

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)

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)

Example!

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

Also can't it be written as $\prod _{k=x-n+1}^{x}k$? — Preceding unsigned comment added by 24.188.204.252 (talk) 00:53, 27 March 2016 (UTC)

Requested move 27 March 2016

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)

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Pochhammer symbol → Falling and rising factorials – 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)

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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)

Integer and non-integer notation

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)

Incorrect/undefined symbol

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)

Thanks for pointing this out; I have corrected it. --JBL (talk) 01:38, 7 June 2019 (UTC)

Flagpole mistake?

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)

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)

The article says (x)_n = x (x-1) (x-2) ... n-factors, is the number of ways to arrange n flags on x flagpoles. This must be the other way round: (n)_x, or the roles of poles and flags must be exchanged. Consider the simple example of x=1 flagpole. If this flagpole must hold exactly one flag, then there are n states, not just x=1 of them. If there are x=2 poles, the first pole has n states and the second n-1 states, because one flag has been used up. In general, for x poles and n flags, the number of states is (n)_x. Or for n poles and x flags, the number of states is (x)_n. I corrected this in the article, but it was reverted. Maybe the flagpole example is not clear enough? How about changing this example to interpret (x)_n as the number of distinct n-letter words that can be written with an alphabet of x letters? --Entropeneur (talk) 07:40, 3 August 2020 (UTC)

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)

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)