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

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Le lundi 20 octobre 2014 06:18:37 UTC-4, peter....@gmail.com a écrit :
> 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


You seem to have missed the bit in the documentation about the case that just n and n-r are negative integers.

    http://www.maplesoft.com/support/help/Maple/view.aspx?path=binomial

That says,

    In the case that exactly two of the expressions n, r, and n-r
    are negative integers, Maple also signals the invalid_operation
    numeric event, allowing the user to control this singular
    behavior by catching the event. See ?numeric_events for more
    information.

Using the first example from the help-page for the numeric event handler,

    http://www.maplesoft.com/support/help/Maple/view.aspx?path=NumericEventHandler

one can handle the cases you're considering.

restart:

formula:=(n,r)->limit(GAMMA(n+t+1)/(GAMMA(r+1)*GAMMA(n+t-r+1)),t=0):

seq( formula(-1,i), i=-5..5);
Error, (in GAMMA) numeric exception: division by zero

MyHandler := proc(operator,operands,default_value)
   NumericStatus( division_by_zero = false );
   return default_value;
end proc:

NumericEventHandler(division_by_zero=MyHandler):

seq( formula(-1,i), i=-5..5);

               0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1

The example in the help had an event handler that also issued a warning. You are, of course, free to make it do what you wish. The key is that it's been set up to provide programmatic control.

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Thread

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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