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integral of t^Z exp(-t) dt, and as (2.) a PRODUCT operator, that multiples a series of terms index by a dummy variable k. The person who wrote this article was talking to themselves and made about 100 hidden assumptions. If he or she would share those hidden assumption, as they are made, and stop using the same symbol for multiple meaning, the article might actually help readers, instead of just confusing them.
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2698:. When you generalize from integers to reals, it may be the case that part of a formula that vanish for integers becomes visible for reals. So although I agree that the usual Γ is usually the correct interpolation, I don't see the rationale for insisting that only functions obeying the integer version of the recurrence can be of any interest. —
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the "correct" extension to non-integer values. But that's not the case (and what I wanted to make clearer to readers). The Gamma function is the most commonly used interpolation because it has useful properties, but using it to extend the factorials to non-integer values, in general, is convention rather than correct.
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Do you even read the paper that I' m making reference? Look the name of the paper is Beta SM project in the introduction say the objective of the project is literally this "This project want to resolve all the problems or functions that are calculations with a difficult solution or they are limits of
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include the not in Z as stated in other sources? I think it might be possible also to state the line before slightly more clearly took me a short second to understand. Perhaps something like "This definition can be extended to the rest of the complex plane by solving Euler's reflection formula (EQN).
905:
Nope. I see no evidence that this is a significant enough connection to factorials to mention anywhere in the article, let alone to put in the table in the lead. I can find some sources (chemistry texts mostly) noting that the factorial of
Avogadro's number is huge; they don't tend to give the value.
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Reading it gives one no clue of what (-1/2)! is or how to calculate it. If an illustration of the expansion of (-1/2)! were added, it would make the article 10 times easier to understand. The author blithely writes (-1/2)! and expects people to know what it means. Furthermore, the author asserts
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The article also states Euler's original Gamma function as capital Pi function = to a Limit, as n goes to infinity, of a ratio with n^z n! as the numerator. Most readers will have no clue what infinity raised to a power is, or what infinity factorial is. Most will not not know if such terms
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correct interpolation. Had there been infinitely many interpolations but only one satisfying the recurrence relation for non-integer values, it may very well have been the case that many interpolations are interesting for one reason or another but the only one with that property might be considered
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The recurrence relation is a fundamental and defining property of factorials—it's what makes factorials factorials, so to speak. When I first heard of interpolating factorials many years ago, I took for granted that the recurrence relation would hold for the non-integer relations, because I thought
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The Gamma and Pi functions Main article: Gamma function The Gamma function, as plotted here along the real axis, extends the factorial to a smooth function defined for all non-integer values. The factorial function, generalized to all complex numbers except negative integers. For example, 0! = 1! =
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The reason given for the first revert was "not an improvement, broken citation, technically not make reader understand". The reason given for the second revert was "This was recently reverted. Please do not reinsert it without first discussing it on the talk page". Another reason would be that the
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The point is that you have to constrain things somehow to be able to say that Gamma is the canonical interpolation. Your edit adds half the constraint, turning that thought from "you have to constrain things" to "we have already constrained things but you have to constrain more things". What is so
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Besides nonnegative integers, the factorial function can also be defined for non-integer values, but this requires more advanced tools from mathematical analysis. One function that "fills in" the values of the factorial (but with a shift of 1 in the argument) is called the Gamma function, denoted
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In the Wiki
Article, that is quoted in part below the dashed line, on the binomial theorem and its extension to negative non integer, we find the Gamma Function. It uses capital Pi in two completely different ways that seems designed to confuse readers. It uses it as: (1.) a function Pi(z) =
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specifically. It is true more generally that factorials naturally appear in formulae from quantum and statistical physics, because one considers all the possible permutations of a set of particles. There might be something worth saying about that in the article; I'll have to think about it.
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The page says: "Peter
Luschny. The Homepage of Factorial Algorithms (no longer existent)." This is no longer true. The page does exist again. However, I was not able to eliminate the "(no longer existent)" from the text. If you can change it please remove this misleading comment. Thanks.
1025:
Here's a draft of a sentence we could use for the statistical side of this, but I'm not sure of the best source for it. Also because this is material I'm unfamiliar with I'm likely to have made a mistake in summarizing it or in choosing the level of detail appropriate for this topic:
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Article listed as GA the day before DYK nomination and is long enough, cited, copyvio-free + neutral. Hook facts cited inline and in the article. QPQ done. Personally my preference is for ALT1, or a variant of ALT2 that gives a brief explanation of what
Benford's law is, such as:
2090:
The code in "Computation" section, to my mind, could be formatted better - currently it's in a bi template, I see no reason for it not to be a code block. Granted - it's pseudo code - but to my eyes it's strange to have it in text style like that. Makes article flow less well
2638:, and only consider log-convex functions? Or why not leave the constraints out of it until they are needed for uniqueness? I'm not embarrassed to say trivial things when they're relevant. Not every statement in our mathematics articles has to have deep reasoning behind it. —
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Support ALT2b. 99% of main page readers will not know
Benford's law so ALT2 just becomes "factorials obey something" (or "something obeys something" if they don't know factorials). People like me who actually know Benford's law would be surprised if factorials did
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Overall this article seems to pass the GA criteria to me without too much bother. Have left some suggestions below for mild improvements but I don't think any preclude this from GA status. This is significantly improved from the last nomination - well done all!
2522:. That, to me, misses the point. It also excludes interpolating the points linearly or indeed arbitrarily. I suppose it would be possible to rephrase it to make both statements at once (e.g. "There are infinitely many ways to extend the factorials to a
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and that function diverges to infinity rather than having a finite value at all negative integers. Additionally, the link you give cannot be used as a reference (Knowledge cannot be used as a reference for itself).
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that (if the secret computation were revealed), (-1/2)! = square root of pi, with not one shred of evidence as to why. Not one reader concerned about the topic, in 10,000 will have any idea what is going on here.
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And what you say of original research if you read a little bit more you can see "material—such as facts, allegations, and ideas" is some of this the paper I making reference no so it not a original research.
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I remember you and import limit of the gamma function is that he can' t represent negative factorial that' s why the limits are infinity to 0 if can resolve also negative factorials will be to infinity to
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This is of course true of any set of discrete points. The interesting part is that there are infinite ways to do it within certain confines, most importantly while still satisfying the recurrence relation.
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This is ridiculously easy to solve. Merely take a pencil and draw some curve—any curve will do—which passes through the points. Such a curve automatically defines a function which solves the interpolation
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It is worth mentioning that there is an alternative notation that was originally introduced by Gauss which is sometimes used. The Pi function, denoted Π(z) for real numbers z no less than 0, is defined by
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Please read the lead carefully and you will see that the factorial function and the gamma function are not defined at all for negative integers. Thus the equations you inserted make no sense at all.—
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a positive integer)? They are both expressions for the factorial function at negative integers. Besides being incorrect (for the standard extension of factorial to gamma) this material appears to be
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However this fails to assign a value to the gamma function: the reflection formula only holds for non-integer values of z to avoid division by 0" though I recognise that's getting very clunky
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Something like that could definitely go in the applications section (probably in the paragraph about applications to fields beyond mathematics) if it can be adequately sourced. —
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890:= 6.02214076×10) to the table of factorials. It has some practical significance in that the factorial is the number of possible arrangements of molecules in one mole of gas.--
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848:) is e, so I wonder, is the sum of the reciprocals of the product of the first n Factorials (the Superfactorials), which is 1.5868056, or the sum of the reciprocals of the
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only permits monospaced typewriter text. That would make the variables in the code inconsistent in appearance with the same variables in the article text, undesirable. —
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Yeah - just feel a computer science user would recognise that better, and absolutely right you don't wanna get into omega/big o/theta etc... will continue review today
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that without that property it wouldn't be much of an interpolating function. The non-uniqueness of the Gamma function in this regard is, I think, very important.
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That is not true the gamma function is for $ (n -1)!$ and the equation that I make reference find value for $ (-n)!$ and the reference is to this paper
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Will some one figure out what the author was trying to say, determine if it is correct, and then rewrite it so it is correct and can be understood.
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Is it worth mention the basic commands for the factorial in some programming languages or math environments like matlab or maple? 9 April 2007
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Hadamard's Γ obeys a form of the recurrence relation. But it is a form with an extra term that happens to be zero on the positive integers. —
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Well set out, appropriate sections, lead is very clear covering main topics at an introductory level. Layout is good, guiding user well
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has any special meaning as a parameter of the Gamma function, making it significant enough to report its value in that article? —
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Article seems stable - good discussion about improvements and article development in talk page does not seem overly contentious.
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You probably could find a section titled "Continuous interpolation and non-integer generalization" and headed by "Main article:
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You are right figshare isn' t a very good reference I will comment the problem that you say to the mail of the author
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Source: See six separately sourced bullet points for different explanations of this claim in the article
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Before we continue talking please read the paper that I' m making reference and please also read this
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standard extension of the factorial to numbers other than the non-negative integers is given by the
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for non-integer values."), but I don't think the bare statements that we have now is satisfactory.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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are well defined or exist. Some explanation is obviously require to make this readable.
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has put in a lot of work to bring this up to standard (thanks!), article does indeed seem
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That sounds right, and it could be sourced to pages 107–110 of the textbook that I added.
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And what Anita say is just the equation you have to isolate $ (-n)!$ having the solution
1493:: it represents viewpoints fairly and without editorial bias, giving due weight to each.
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The following is an archived discussion of the DYK nomination of the article below.
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Properties section is enhanced with use of graphs - appropriate and well captioned
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That's not what I'm saying. What I'm saying is way closer to the Gamma function
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though really all of the hooks are fine, it's just a matter of interestingness.
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These equations are mathematically incorrect. Please do not insert them again.—
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https://es.wikipedia.org/Factorial#Soluci%C3%B3n_n%C3%BAmero_negativo_factorial
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Subsequent comments should be made on the appropriate discussion page (such as
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on this page. Adding the factorial of the imaginary unit can be quite useful.
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natural about that choice? Why not instead start with the other half of the
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The statement "There are infinitely many ways to extend the factorials to a
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and cannot therefore be used here. Otherwise, I fully agree with Anita192.
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https://figshare.com/articles/journal_contribution/beta_SM_project/24901614
1465:. it stays focused on the topic without going into unnecessary detail (see
1180:. The edit link for this section can be used to add comments to the review.
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In
Applicatios (computer science) might it be worth explicitly mentioning
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I appreciate your efforts to remove cruft, bit if some textbooks mention N
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1514:: it does not change significantly from day to day because of an ongoing
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of the first n
Factorials which is 1.47608642, expressible in terms of e?
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being the "correct" interpolation—or perhaps even more to the point not
2434:." is true, but trivial and rather useless. As the cited source states,
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The sum of the reciprocals of the product of the first n integers (the
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Is the reference ok it' s all ok even in the version in
Spanish it is
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but I don't want to take the space here to explain Omega and anyway
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so yes is solving a problem of the maths is the point of the paper.
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The sum of the reciprocals of the sum of the first n integers (the
789:\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\prod_{k=0}^n (z+k)}. \!
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Its relation to the factorials is that for any natural number n
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Taking this on as part of the
January Drive. Previous reviewer
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No, I do not think so. In Maple, for example, you can write 5!
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Good level of detail - I think pretty well pitched throughout
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I must admit that I don't understand what you're getting at.
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on the basis that it would exclude an interpolation based on
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by dividing by the factorials of the numbers of each type of
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way of calculating factorial may be more understandable. In
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I am trying to clean the article of cruft, not add more. —
3032:, forbidden on both the Spanish and English Wikipedias. —
777:\Gamma(z)=\int_0^\infty t^{z-1} e^{-t}\, \mathrm{d}t. \!
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Moved from the article. Is this really worth mention? --
2921:. And do you really think there is a difference between
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Knowledge Did you know articles that are good articles
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Euler's original formula for the Gamma function was
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2677:07:35, 20 August 2023 (UTC)
2663:16:07, 19 August 2023 (UTC)
2648:13:02, 19 August 2023 (UTC)
2628:11:36, 19 August 2023 (UTC)
2415:08:40, 6 January 2022 (UTC)
2388:05:18, 6 January 2022 (UTC)
2335:01:11, 6 January 2022 (UTC)
2271:larger than 20! will cause
2136:Knowledge talk:Did you know
2128:this nomination's talk page
2105:18:54, 5 January 2022 (UTC)
2084:18:54, 5 January 2022 (UTC)
2057:18:54, 5 January 2022 (UTC)
1938:10:12, 5 January 2022 (UTC)
1920:18:29, 4 January 2022 (UTC)
1203:17:48, 4 January 2022 (UTC)
1038:Boltzmann's entropy formula
872:20:01, 12 August 2011 (UTC)
154:The text of the entry was:
3460:
2300:was introduced in 1808 by
1946:Euler's reflection formula
1814:{\displaystyle O(n\log n)}
1339:the layout style guideline
1050:indistinguishable particle
1044:must correct the count of
659:21:38, 25 April 2003 (UTC)
3399:Mathematics good articles
3329:Your above cites are not
2520:Hadamard's gamma function
2514:for non-integer values",
1525:
1422:
1309:
1234:
1224:
823:) 22:56, 23 October 2010
802:\Pi(z) = \Gamma(z+1) \,.
312:
245:
224:
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33:Mathematics good articles
2290:: ... that the notation
2094:<syntaxhighlight: -->
1944:Should the equation for
1221:
532:
452:
319:project's priority scale
133:appeared on Knowledge's
3409:GA-Class vital articles
3394:Knowledge good articles
2132:the article's talk page
2113:Did you know nomination
2065:greatest common divisor
1601:to the topic, and have
1278:. it complies with the
1263:Well written and clear
1042:Sackur–Tetrode equation
276:WikiProject Mathematics
3211:before it is defined.—
3205:
3157:
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2970:
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2949:{\displaystyle (n-1)!}
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990:
989:{\displaystyle N_{A}!}
877:
719:Editing the references
687:) 19:38, 29 April 2007
503:# multiply result by x
428:Lowercase sigmabot III
156:Did you know ... that
131:fact from this article
3206:
3204:{\displaystyle (-n)!}
3158:
3023:
3003:
3001:{\displaystyle (-n)!}
2971:
2951:
2834:
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2636:Bohr–Mollerup theorem
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2333:). Self-nominated at
2265:: ... that computing
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1905:
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1778:
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1427:Broad in its coverage
1030:statistical mechanics
991:
743:) 06:48, 30 June 2007
712:) 06:48, 30 June 2007
535:// function factorial
203:level-4 vital article
39:good article criteria
3183:
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2069:primitive polynomial
1956:
1910:is more accurate). —
1863:
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1714:
1645:
1518:or content dispute.
1410:copyright violations
1389:no original research
1320:no original research
970:
299:mathematics articles
106:Good article nominee
87:Good article nominee
3331:WP:reliable sources
2917:That is also not a
2832:{\displaystyle i+1}
2524:continuous function
2432:continuous function
2067:of the values of a
1444:. it addresses the
1366:. All content that
839:Tetrahedral numbers
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2199:Anuyogadvāra-sūtra
2032:
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1773:
1697:
1625:Overall assessment
1569:copyright statuses
1532:, if possible, by
1300:list incorporation
1178:Talk:Factorial/GA2
1032:, calculations of
986:
835:Triangular numbers
783:n!=\Gamma(n+1).\,
438:Moved from article
268:Mathematics portal
212:content assessment
66:Article milestones
3108:
3030:original research
3021:{\displaystyle n}
2969:{\displaystyle n}
2540:{\displaystyle f}
2442:I therefore added
2338:
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2273:64-bit arithmetic
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2011:
1635:
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1603:suitable captions
1575:are provided for
1408:. it contains no
1168:
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884:Avogadro's number
878:Avogadro's number
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2244:: ... that the
2229:: ... that the
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2753:imaginary unit
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1467:summary style
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1284:lead sections
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1055:
1054:Gibbs paradox
1052:to avoid the
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315:High-priority
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240:High‑priority
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141:Did you know?
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82:March 3, 2019
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1597:. media are
1594:
1563:. media are
1560:
1533:
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1488:
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1462:
1446:main aspects
1441:
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1384:
1364:cited inline
1355:
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1239:Well-written
1238:
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1184:
1171:
1160:Instructions
1060:— Preceding
881:
854:— Preceding
849:
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677:63.150.207.3
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441:
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388:
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314:
274:
218:WikiProjects
201:
176:Good article
175:
157:
155:
146:
138:
104:
85:
49:
47:
43:please do so
31:
30:
26:
2407:PrimeHunter
2354:: ... that
2074:Reworded. —
2047:Reworded. —
1567:with their
1530:Illustrated
1174:transcluded
1046:microstates
807:—Preceding
727:—Preceding
696:—Preceding
671:—Preceding
360:speedy keep
290:Mathematics
281:mathematics
237:Mathematics
3388:Categories
3226:-infinity.
2723:TompaDompa
2683:TompaDompa
2655:TompaDompa
2620:TompaDompa
2357:factorials
2297:factorials
2268:factorials
2247:factorials
2194:factorials
1414:plagiarism
1315:Verifiable
1216:improved!
1127:Authorship
1113:GA toolbox
1082:XOR'easter
999:XOR'easter
846:Factorials
841:) is 1.5.
444:imperative
159:factorials
145:column on
92:Not listed
37:under the
3349:Arrobaman
3317:Arrobaman
3293:Anita5192
3279:Arrobaman
3265:Anita5192
3251:Arrobaman
3237:Arrobaman
3213:Anita5192
2905:Arrobaman
2867:Arrobaman
2232:factorial
1226:Attribute
1186:Reviewer:
1150:Templates
1141:Reviewing
1106:GA Review
206:is rated
135:Main Page
27:Factorial
3335:D.Lazard
2437:problem.
2421:T:DYK/P4
2313:Reviewed
2306:Source:
2281:Source:
2277:overflow
2256:Source:
2149:promoted
1930:Vogon101
1599:relevant
1536:such as
1516:edit war
1199:contribs
1189:Vogon101
1155:Criteria
1074:contribs
1062:unsigned
1036:such as
868:contribs
856:unsigned
829:Hmmm....
821:contribs
813:Jaimster
809:unsigned
741:contribs
729:unsigned
710:contribs
698:unsigned
685:contribs
673:unsigned
423:365 days
381:Archives
350:deletion
208:GA-class
51:reassess
2757:Bera678
2173:Comment
1490:Neutral
1296:fiction
1040:or the
1034:entropy
944:WP:CALC
526:and in
317:on the
137:in the
74:Process
2384:(talk)
2379:iolite
2294:! for
1571:, and
1565:tagged
1538:images
1512:Stable
1298:, and
1288:layout
860:Robo37
637:return
521:result
518:return
494:result
470:result
450:code:
448:Python
214:scale.
111:Listed
77:Result
3179:uses
3008:(for
2956:(for
2401:" to
2352:ALT2b
2250:obey
1546:audio
1544:, or
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1318:with
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1176:from
601:<=
485:: -->
479:while
195:This
3353:talk
3339:talk
3321:talk
3297:talk
3283:talk
3269:talk
3255:talk
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3217:talk
3038:talk
2909:talk
2892:talk
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2727:talk
2704:talk
2687:talk
2673:talk
2659:talk
2644:talk
2624:talk
2411:talk
2360:are
2331:talk
2288:ALT4
2263:ALT3
2242:ALT2
2227:ALT1
2177:view
2157:talk
2101:talk
2080:talk
2053:talk
1934:talk
1916:talk
1362:are
1214:much
1193:talk
1086:talk
1070:talk
1017:talk
1003:talk
958:talk
949:and
934:talk
912:talk
896:talk
864:talk
817:talk
737:talk
706:talk
681:talk
656:Taku
458:fact
358:was
309:High
162:are
71:Date
2809:in
2718:the
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2419:To
2399:one
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2275:to
2211:of
2201:of
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1999:sin
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1469:).
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952:. —
930:agr
892:agr
850:sum
643:ret
622:ret
580:for
568:ret
559:int
547:int
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538:int
455:def
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