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The current extension to complex numbers is misleading. It correctly reproduces the double factorial for odd integers, but does not reproduce the double factorial for even integers. Interestingly, Mathematica can evaluate fractional arguments for Factorial2, which reproduces both, but I can't find the definition they use to compute it.
-- Kaba3 ( talk) 10:53, 15 June 2015 (UTC)
I think this article is not bad, but it seems to me, it risks to be a bit misleading as it misses the main point of the double factorial, which is, first of all, just giving a useful notation for a common expression, rather than defining some new function to be extended to negative or complex values. For a non-negative integer n we give a special notation to the product of all positive integers not larger than n and with the same parity of n, namely because it arises frequently in mathematical expressions --such as coefficients of certain series expansions, enumerative formulas, repeated integrals etc, disregarding to the parity of the number n (even if within formulas that may depend on the parity of n). For this reason, I don't see any reason for treating differently odd and even numbers, and I do not see much interest in making emphasis on x!! for negative or complex numbers, nor I see other reason of dropping the alternative definition than indulging oneself in mathematical mysticism or amazing naive readers. -- pm a 10:27, 29 November 2013 (UTC)
The traditional name of this function is semifactorial. I am moving the page. It is not "double", it is half of a factorial as it omits half the factors of a factorial. Zaslav ( talk) 02:59, 17 July 2014 (UTC)
The double factorial in the sense of semifactorial, is defined for all nonnegative integers. That is the core meaning, as far as I can see from other WP articles and my own experience. I've revised the article to make this "the" definition. I kept the parts about extending odd double factorials to negatives or complexes. Is this acceptable?
I also removed the term "odd factorial" as, in reviewing various WP articles, I never found an "odd factorial" of an even number (or hardly ever and then I forgot it). I have no problem with "odd factorial" = 1⋅3⋅5⋅... up to n if odd or n−1 if even, but I don't know that it is existing terminology. I did, however, find this incorrect definition of double factorial (possibly of an odd number only, but not so stated) in a few places: n!! = 1⋅3⋅5⋅..., which needs correction.
I hope the interested persons will review this and either improve it or decide it's another mistake. Zaslav ( talk) 17:48, 18 July 2014 (UTC)
In the Relation to section, I'm wondering why it isn't stated that n! = n!! x (n-1)!! for all n>0. It is clearly true.?? 98.21.70.161 ( talk) 19:54, 25 October 2017 (UTC)
Well, it's actually given in the generalizations section of the factorial function page in more generality as
Maxie ( talk) 22:01, 25 October 2017 (UTC)
@ Quantling: One result of this edit is that in the sentence beginning "From this", the referent of "this" is now the fact that something is log convex. I suspect that this is not right. Can you please look over the section and check for global coherence? ( This version before you moved things around perhaps does not have the same problem.) Thanks, -- JBL ( talk) 20:38, 1 February 2023 (UTC)
The last equation in this section reads:
![]() | This article has not yet been rated on Wikipedia's content assessment scale. |
The current extension to complex numbers is misleading. It correctly reproduces the double factorial for odd integers, but does not reproduce the double factorial for even integers. Interestingly, Mathematica can evaluate fractional arguments for Factorial2, which reproduces both, but I can't find the definition they use to compute it.
-- Kaba3 ( talk) 10:53, 15 June 2015 (UTC)
I think this article is not bad, but it seems to me, it risks to be a bit misleading as it misses the main point of the double factorial, which is, first of all, just giving a useful notation for a common expression, rather than defining some new function to be extended to negative or complex values. For a non-negative integer n we give a special notation to the product of all positive integers not larger than n and with the same parity of n, namely because it arises frequently in mathematical expressions --such as coefficients of certain series expansions, enumerative formulas, repeated integrals etc, disregarding to the parity of the number n (even if within formulas that may depend on the parity of n). For this reason, I don't see any reason for treating differently odd and even numbers, and I do not see much interest in making emphasis on x!! for negative or complex numbers, nor I see other reason of dropping the alternative definition than indulging oneself in mathematical mysticism or amazing naive readers. -- pm a 10:27, 29 November 2013 (UTC)
The traditional name of this function is semifactorial. I am moving the page. It is not "double", it is half of a factorial as it omits half the factors of a factorial. Zaslav ( talk) 02:59, 17 July 2014 (UTC)
The double factorial in the sense of semifactorial, is defined for all nonnegative integers. That is the core meaning, as far as I can see from other WP articles and my own experience. I've revised the article to make this "the" definition. I kept the parts about extending odd double factorials to negatives or complexes. Is this acceptable?
I also removed the term "odd factorial" as, in reviewing various WP articles, I never found an "odd factorial" of an even number (or hardly ever and then I forgot it). I have no problem with "odd factorial" = 1⋅3⋅5⋅... up to n if odd or n−1 if even, but I don't know that it is existing terminology. I did, however, find this incorrect definition of double factorial (possibly of an odd number only, but not so stated) in a few places: n!! = 1⋅3⋅5⋅..., which needs correction.
I hope the interested persons will review this and either improve it or decide it's another mistake. Zaslav ( talk) 17:48, 18 July 2014 (UTC)
In the Relation to section, I'm wondering why it isn't stated that n! = n!! x (n-1)!! for all n>0. It is clearly true.?? 98.21.70.161 ( talk) 19:54, 25 October 2017 (UTC)
Well, it's actually given in the generalizations section of the factorial function page in more generality as
Maxie ( talk) 22:01, 25 October 2017 (UTC)
@ Quantling: One result of this edit is that in the sentence beginning "From this", the referent of "this" is now the fact that something is log convex. I suspect that this is not right. Can you please look over the section and check for global coherence? ( This version before you moved things around perhaps does not have the same problem.) Thanks, -- JBL ( talk) 20:38, 1 February 2023 (UTC)
The last equation in this section reads: