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.