Thank you Fred! Seconds out indeed. Two conjectures: Kronenburg is a troll who wrote this to incite flamewars on mathematics fora, or Some other troll wrote this to embarrass Kronenburg. --rwg Captious graffiti: G,K&P, p156 "C'est une chose estrange combien il est fertile en proprietez." | B. Pascal [285] Then on the next page, I just hope I don't fall into this trap during the midterm. Google didn't find the Pascal quote except in a TeX source of GKP errata. Both Google and Google Translate wanted to respell the quote as C'est une chose *étrange* combien il est fertile en *propriété*,<https://www.google.com/search?client=firefox-a&hs=i7q&rls=org.mozilla:en-US:official&nfpr=1&biw=1052&bih=624&q=C%27est+une+chose+%C3%A9trange+combien+il+est+fertile+en+propri%C3%A9t%C3%A9.&spell=1&sa=X&ei=gY3OUazDM4SLjAL7m4C4Bw&ved=0CCsQBSgA> which Google then found in various Pascal collections. Wondering if the graffito could actually be mistyped, I found on http://www-cs-faculty.stanford.edu/~uno/gkp.html a warning: SPECIAL NOTE TO THE SPEAKERS OF FRENCH AND OTHER EXOTIC LANGUAGES: Numerous quotations and bibliographic citations found in this book have been copied verbatim from the original sources. If you believe you have found a typographic error, you must prove it by showing that the original was incorrectly transcribed; believe it or not, your language has changed over the years, just as English has. Digging deeper, sure enough, http://books.google.fr/books?hl=fr&id=UqgUAAAAQAAJ&q=fertile I wonder how many bogus bug reports this particular graffito collects. I also wonder why Google proper didn't find it in the various pirated GKPs online, nor in books.google.fr . Re turkeys and gem polishers, http://www.reviewjournal.com/news/nevada-and-west/protection-sought-rare-woo... Only you can prevent sanctimonious talking bears. Joerg wrote << By sheer coincidence I just stumbled upon http://arxiv.org/abs/1105.3689 Can anyone comment on validity of the statements given? The first error in the Kronenburg draft appears in paragraph 1 of section 2, where his (1.4) Gamma(s) Gamma(1-s) = pi/sin(pi s) is misapplied with s = 0 . He seems to have been seduced by the same siren call that has lured others (they know who they are) to their doom: assuming nice limiting behaviour at a singularity which --- as Bill eloquently observed --- actually resembles curled up lasagne. The mathworld plot I earlier managed to misinterpret at http://mathworld.wolfram.com/BetaFunction.html provides a passable identikit of the miscreant concerned. Defining the continuation of the binomial coefficient n_C_m for x,y real by z = z(x, y) = Gamma(x+1) / Gamma(y+1) Gamma(x-y+1) and examining the neighbourhood of one of these "lasagne points" in the third wedge region, it appears that for m > 0, n >= 0 integer and r^2 = x^2 + y^2 small z(-m + x, -m-n + y) = c (y/x) + O(r) (*) where c = {-m}_C_n = (-1)^n {n+m-1}_C_n . This expansion makes it obvious that approaching the singularity along direction y = 0 produces zeroes in this region; whereas approaching along y = x produces the deprecated nonzero third wedge. In contrast, an analogous analysis of z(-m, n) for m > 0, n >= 0 in the second wedge finds z(-m + x, n + y) = c (1 - y/x) + O(r) , showing that approaching along y = 0 [along the x-axis, 60 degrees below rightwards in the customary orientation] produces desired Taylor coefficients; whereas approaching along y = x produces only zeroes. Despite an earlier assertion to the contrary, it transpires that supporters of the nuclear option were guilty of wanting to have their cake and eat it! [Better go sweep up those crumbs now, I suppose ...] Fred Lunnon On 6/28/13, Fred lunnon <fred.lunnon@gmail.com> wrote: Bill Gosper: << In particular, he must have fudged to get the consistency he claims from his flawed premises >> But he is consistent. Restricted to integer arguments, the discussion we've been having boils down to a choice between: [R] Assuming the recursion n_C_m = {n-1}_C_{m-1} + {n-1}_C_m across the whole plane; and [S} Assuming the symmetry n_C_m = n_C_{n-m} across the plane, in which case recursion must be abandoned at (n,m) = (0,0) . He is adopting [S] , and you've plumped for [R] . While I've finally come round to accepting that you're probably correct, I'm astonished at how delicate the reasoning involved has proved to be. Fred Lunnon On 6/28/13, Bill Gosper < billgosper@gmail.com> wrote: JPropp>Carelessness or incompetence seems far more likely than fraud, which requires an intent to deceive. (Am I being over-literal?) Jim Propp On Thursday, June 27, 2013, Bill Gosper <billgosper@gmail.com> wrote: Jörg> By sheer coincidence I just stumbled upon http://arxiv.org/abs/1105.3689 Can anyone comment on validity of the statements given? Best, jj This paper says "The symmetry identity [3, 4] and the trinomial revision identity [3, 4] are valid for all complex x,y,z" where [3] is G,K,&P's Concrete Math, which the author does not admit to contradicting. But GKP says "So the equation (-1 choose k) = (-1 choose -1-k) is always false! 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 of its variables. Everyone who’s manipulated binomial coefficients much has fall into this trap at least three times." This paper is a fraud. --rwg "Never ascribe to malice that which can be explained by incompetence." I think there is deception here, even if Kronenburg imagines that Knuth et al agree with him. (But how much more explicit could they be?) In particular, he must have fudged to get the consistency he claims from his flawed premises, but I don't have time right now to catch him out. --rwg I haven't (re)read the part where GKP extend to noninteger lower index, where I hope they motivate (r choose negativeinteger) :=0. I recall Knuth in Vol 1 relying heavily on summing from -∞ to greatly simplify binomial sum manipulations, blissfully indifferent to the precise points of leftward termination. _______________________________________________