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Re: [Axiom-developer] Issue 336

From: Waldek Hebisch
Subject: Re: [Axiom-developer] Issue 336
Date: Sat, 17 Mar 2007 19:41:39 +0100 (CET)

Martin Rubey wrote:
> If the second argument k is negative, we could return zero. However, I vaguely
> remember that the definition in terms of Gamma functions does not yield this 
> as
> a limit, at least not for any n. Do you happen to remember?
> In fact, if n=k, the definition in terms of Gamma functions always gives one,
> no matter whether n is negative of not...
> Axiom gives 0 currently, due to the definition of ibinom. This looks all quite
> inconsistent...

Hmm, really inconsistent:

(4) ->  binomial(n, n)
   (4)  1
                                                     Type: Expression Integer
(5) -> binomial(-3, -3)
   (5)  0
                                                     Type: NonNegativeInteger

I would tend to say that binomial(n, k) with k beeing a negative
integeriis an error.  Definition in term of Gamma functions has
discontinuity there: if n is non-integer and k tends to negative integer
then we get 0 as a limit.  Now, if we keep k fixed and let n go to
any value we get 0.  But if we allow n and k to change simultaneously
we may get nonzero limit (so limit as a function of two variables does
not exist).  We have discontinuity only for integer n and k such that
k <= n < 0 (otherwise limit is 0).

                              Waldek Hebisch

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