Mathematica 7.0.1: As a bivariate function, Binomial is not continuous in both variables at negative integers: In[1]:= Binomial[-3, -5] Out[1]= 0 In[2]:= Binomial[-3.0000001, -5.0000001] Out[2]= 6.00000035000001 Mathematica 11.1: The value of Binomial at negative integers is determined via Binomial[n,m]==Binomial[n,n-m]: In[2]:= Binomial[-3, -5] Out[2]= 6 GKP: The restriction that n and k be integers in identity (5.4) is obvious, since each lower index must be an integer. But why can't n be negative? Suppose, for example, that n = −1. Is −1 ? −1 ( ) = ( ) k −1−k a valid equation? No. For instance, when k = 0 we get 1 on the left and 0 on the right. In fact, for any integer k ≥ 0 the left side is −1 (−1)(−2) . . . (−k) k ( ) = ------------------- = (−1) k k! which is either 1 or −1; but the right side is 0, because the lower index is negative. And for negative k the left side is 0 but the right side is −1 −1−k ( ) = (−1) , −1−k which is either 1 or −1. So the equation −1 −1 ( ) = ( ) is always false! k −1−k The symmetry identity fails for all other negative integers n, too. But unfortunately it's all too easy to forget this restriction, since the expression in the upper index is sometimes negative only for obscure (but legal) values [ I just hope I don't fall into this trap during the midterm.] of its variables. Everyone who's manipulated binomial coefficients much has fallen into this trap at least three times. ---end GKP --- As I hope Fred's opus makes clear, this asymmetry is not a special case kludge. Remarkably, the real-valued surface Binomial[y,x] is symmetrical, but the discretized surface Binomial[n,k] = Limit[Limit[Binomial[y,x],x->k],y->n] is not! In order to maintain their symmetry fiction, WRI broke the addition formula GKP(5.8) Out[5]= Binomial[r, k] == Binomial[-1 + r, -1 + k] + Binomial[-1 + r, k] In[6]:= % /. r -> 0 /. k -> 0 Out[6]= False But they forgot to break QBinomial! In[9]:= QBinomial[-3, -5, q] Out[9]= 0 --rwg On 2017-07-22 22:00, Henry Baker wrote:
Dropbox is a pain, unless you add "?dl=1" to the end, as in:
https://www.dropbox.com/s/9vvl6l6ym1zkret/binomial.pdf?dl=1
The "?" indicates that you're passing parameters, and the "dl=1" says -- in essence -- "download equals 1 -- i.e., *yes*".
It took me quite a while to figure this out about dropbox.
At 06:16 PM 7/22/2017, Fred Lunnon wrote:
@#$%^&?! ... DropBox is getting steadily less user-friendly --- why give me a link and immediately disable the infernal thing? Anyway, I have copied the file, deleted it and re-up-loaded it: try instead
https://www.dropbox.com/s/9vvl6l6ym1zkret/binomial.pdf
WFL
On 7/23/17, Neil Sloane <njasloane@gmail.com> wrote:
Fred, when I tried to look at your discussion, https://www.dropbox.com/s/anykne0pd55ehjg/binomial.pdf I got a rude "go away" message
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sat, Jul 22, 2017 at 7:20 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
See the note I wrote up following a discussion on this list circa 2014 --- https://www.dropbox.com/s/anykne0pd55ehjg/binomial.pdf
[ I did contemplate trying to get this published in the Math. Intelligencer, but retreated in the face of an editor's mysterious antipathy to computer- generated illustrations. ]
WFL
On 7/22/17, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
I'm guessing that a similar lapse of conviction ruined Mathematica's Binomial function: Some moron complained about the asymmetry so long after it was implemented that WRI forgot why it's asymmetrical, and accommodated the moron.
For those of us who aren't familiar with all Mathematica's quirks, would you care to say what erroneous symmetry Mma gives to the binomial coefficients?
(My apologies if this was already being discussed in some other thread and I just missed it.)
-- g