I was curious about this as well. The paper you link is providing an extension of the binomial coefficients. Binomial[-1,-1] = 1 isn't the result of an "assumption" per se, but rather the choice of a particular approach to extending the coefficients (which are normally only defined for non-negative integers n, k). The paper does this by writing the coefficient formula in terms of gamma functions, and substituting a finite expression when the gamma function is infinite. Clearly, Knuth does not prefer this generalization.
The Knuth-coauthored text "Concrete Mathematics" takes a different approach, where instead of a substitution, the binomial formula takes its limiting value when the generalized gamma function form results in ∞/∞. So Knuth has
For any complex numbers z, w. It's not clear to me whether there is a further justification for this approach. I don't see a reference for this in the Concrete Mathematics text, but would be curious to learn more.
The Knuth-coauthored text "Concrete Mathematics" takes a different approach, where instead of a substitution, the binomial formula takes its limiting value when the generalized gamma function form results in ∞/∞. So Knuth has
For any complex numbers z, w. It's not clear to me whether there is a further justification for this approach. I don't see a reference for this in the Concrete Mathematics text, but would be curious to learn more.