Binomial[-1,-1]

Question

According to various sources e.g.

http://www.proofwiki.org/wiki/Definition:Binomial_Coefficient

and Wolfram themselves

http://functions.wolfram.com/GammaBetaErf/Binomial/02/ , the binomial coefficient ${n\choose k}$ is is defined as 0 whenever $k$ and $n$ are negative integers and $k\le n$. But when I type

Binomial[-1,-1]

Mathematica returns

1

I looked up the documentation for the definition of Binomial and it says

In general, ${n\choose m}$ is defined by $\Gamma(n+1)/\Gamma(m+1)\Gamma(n-m+1)$ or suitable limits of this.

Apparently, when $n=-1$ or $m=-1$ since $\Gamma(0)$ is not defined the suitable limit case is applied.

So, why does Mathematica return 1 for ${-1\choose -1}$? What precisely is the "suitable limits"?

Answer 1

Looked up the rules on this. This is how it works. If $n$ and $r$ are negative integers, there is a symmetry relation $\binom{n}{r}=\binom{n}{n-r}$ and now the limit is used.

But now $\binom{n}{r}=\binom{-1}{0}$ from above. Hence the above limit is, where $n=-1$ and $r=0$ is

    n = -1; r = 0;
    Limit[ Gamma[n + t + 1]/(Gamma[r + 1] Gamma[n + t - r + 1]), t -> 0]
    (* 1 *)

Maple also agrees with Mathematica

Reference: Maple help on Binomial

Answer 2

Without expanding Binomial into Gamma functions, you can also see that the result is correct based on the following true statement:

SeriesCoefficient[(1 + x)^n, {x, 0, k}, Assumptions -> k >= 0]

(* ==> Binomial[n, k] *)

This is the binomial expansion, valid in particular for n = -1. But that case leads to the alternative expression

SeriesCoefficient[(1 + x)^-1, {x, 0, k}, 
 Assumptions -> k >= 0]

(* ==> (-1)^k *)

which by comparison with the previous line implies that the following is the correct result:

Binomial[-1, 0]

(* ==> 1 *)

Now we have the symmetry relation

FullSimplify[Binomial[n, k] == Binomial[n, n - k]]

(* ==> True *)

Using this with n = -1 and k = -1 then confirms what Mathematica says:

Binomial[-1, 0] == Binomial[-1, -1]

(* ==> True *)

Combined with the previous equation the result Binomial[-1, -1] == 1 follows.

Answer 3

Version 14.1 introduced PascalBinomial, which preserves Pascal's identity for all integer values, and uses a different definition for negative integer $n$ than Binomial. You can read more about this in the v14.1 release blog post.

Binomial[-1, -1]
(* 1 *)

PascalBinomial[-1, -1]
(* 0 *)

The accompanying limits, differing in the order of the limiting:

Limit[Limit[Limit[Gamma[n + ϵ + 1]/(Gamma[k + ϵ + 1] Gamma[n - k + ϵ + 1]), 
   k -> -1], n -> -1], ϵ -> 0]
(* 1 *) 

Limit[Limit[Limit[Gamma[n + ϵ + 1]/(Gamma[k + ϵ + 1] Gamma[n - k + ϵ + 1]), 
   ϵ -> 0], k -> -1], n -> -1]
(* 0 *)