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Re: The value of binomial(-1, r)

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On 10/20/2014 5:18 AM, peter.luschny@gmail.com wrote:
> I am interested in the value of binomial(-1, r).
>
> The docs say: In the case that n is a negative integer,
> binomial(n,r) is defined by this limit:
>
> binomial(n,r) = limit(GAMMA(n+t+1)/(GAMMA(r+1)*GAMMA(n+t-r+1), t=0)
>
> OK, so let's try the definition:
>
> n := -1;
> limit(GAMMA(n+t+1)/(GAMMA(r+1)*GAMMA(n+t-r+1)), t=0);
>
> [1] Maple out: undefined
>
> Now let's execute
>
> seq(binomial(-1, r), r=-5..5);
>
> [2] Maple out: 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1
>
> [1] contradicts [2]. Are the docs or is the value in error?
>
> Or better asked:
>
> What is a good definition of binomial(-1, r) (r real, r >= 0) in terms
> of the Gamma function?
>
> Peter
>

The doc is wrong.

It should be like this (notice the extra 't' in the first
term in the denominator)

---------------------------
myBinomial:=(n,r)-> limit(GAMMA(n+t+1)/(GAMMA(r+t+1)*GAMMA(n+t-r+1)), t=0);
seq(myBinomial(-1, r), r=-5..5);
       1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1
--------------------------

which now matches

------------------------------
seq(binomial(-1, r), r=-5..5);
       1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1
------------------------------

Maple 18.01
ps. Same with M:

-----------------------------
f[n_, r_] := Limit[Gamma[n + t + 1]/(Gamma[r + t + 1]*Gamma[n + t - r + 1]), t -> 0]
Table[f[-1, r], {r, -5, 5}]
        {1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1}
------------------------------


--Nasser

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The value of binomial(-1, r) peter.luschny@gmail.com - 2014-10-20 03:18 -0700
  Re: The value of binomial(-1, r) "Nasser M. Abbasi" <nma@12000.org> - 2014-10-20 06:00 -0500
    Re: The value of binomial(-1, r) peter.luschny@gmail.com - 2014-10-21 02:44 -0700
  Re: The value of binomial(-1, r) acer <maple@rogers.com> - 2014-10-22 05:43 -0700

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