The Binomial Coefficient for Negative Arguments — mentioned by tromp

Sat, 20 May 2023 16:59:39 GMT

If Knuth is correct then I wonder how this paper https://arxiv.org/pdf/1105.3689.pdf cited by 32, proves Theorem 2.1 that gives Binomial[-1,-1] = Binomial[0,0] = 1. What exactly is the assumption on which they differ? Whatever value you set this coeficient to, it's clear that not all of the familiar binomial properties 1. (n k) = (n-1 k-1) + (n-1 k) 2. (n 0) = 1 3. (n k) = (n n-k) can remain true, as 1. gives 1 = (0 0) = (-1 -1) + (-1 0), and 2. makes this sum 0 + 1, while 3. would make it 1/2 + 1/2.

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The Binomial Coefficient for Negative Arguments — mentioned by tromp