Hi all,
I have already reported the unsatisfactory state of the binomial
function as implemented in Sage on 'Ask Sage' and had opened
a request for enhancement with ticket #17123.
Here I take up a suggestion of John Palmieri to discuss the
issue on this list. The request is:
Extend binomial(n,k) by the reflection formula, i. e. set
for n>=1 and k>=1
binomial(-k, -n) = (-1)^(n+k) binomial(n-1, k-1).
The reasons are:
(1) The current behavior introduces incompatibility with
Maple and Mathematica. This does not only question Sage as an
alternative for the other M-systems; because of the ubiquity
and importance of the binomial function it becomes difficult
to recommended Sage for implementing sequences on the OEIS.
The behaviour of Maple and Mathematica (which support the
reflection formula) seems to be the de facto standard. Many
mathematicians write their formulas in the OEIS without
explicitly mentioning when they assume the reflection formula.
And this extension was used by mathematicians prior to Maple
and Mathematica. (So it is certainly not a question of 'do we
have to mimic all the crap of other M-systems' like some
commentators seem to suggest.)
(2) The extension appears natural as the extended binomial
has the same behaviour as:
def limit_binomial(n,r):
return limit(gamma(n+x+1)/(gamma(r+x+1)*gamma(n+x-r+1)), x=0)
for n in (-5..5): print [limit_binomial(n,k) for k in (-5..5)]
Indeed I find it confusing that the current implementation
does not exhibit this behavior.
(3) Also the basic representation of Pascal's triangle in terms
of the matrix exponential (see the Wikipedia reference below)
makes the reflection extension appear natural as it is an extension
from the matrix with 1, 2, 3, ... subdiagonal and zero everywhere
else to the matrix with .., -3, -2, -1, 0, 1, 2, 3, .. as subdiagonal.
(4) For a comparison of different extensions run this:
def _binomial(n,k, pascal = true, reflection = true, uppernegation = true):
def b(n,k): return factorial(n)/(factorial(k)*factorial(n-k))
if pascal and n >= 0 and k >= 0 and k <= n: return b(n,k)
if reflection and n < 0 and k < 0 and k <= n: return
(-1)^(n-k)*b(-k-1,n-k)
if uppernegation and n < 0 and k >= 0: return (-1)^k*b(k-n-1,k)
return 0
R = [n for n in (-5..5)]
print "(1) Pascal's triangle, for the ultra-orthodox."
print "(cf. [BP])"
for n in R: print [_binomial(n,k,true,false,false) for k in R]
print "(2) Current behavior of Sage, also Magma, extends (1)."
for n in R: print [_binomial(n,k,true,false,true) for k in R]
print "(3) Suggested behavior."
print "Also Maple and Mathematica, extends (1) and (2)."
print "(cf. [RS], p.5071)"
for n in R: print [_binomial(n,k,true,true,true) for k in R]
print "(4) Extends (1) by reflection only."
print "(cf. [LMMS], p.637)"
for n in R: print [_binomial(n,k,true,true,false) for k in R]
(5) References
[BP] Blaise Pascal; Traité du triangle arithmétique; G. Desprez, 1665.
[LMMS] Ana Luzón, Donatella Merlini, Manuel A. Morón, Renzo Sprugnoli;
Identities induced by Riordan arrays; Linear Algebra and its Applications,
436 (2012) 631-647
[WP] Pascal's triangle
https://en.wikipedia.org/wiki/Pascal's_triangle#Extensions
[MJK] M. J. Kronenburg; The Binomial Coefficient for Negative Arguments;
arXiv:1105.3689v2
[RS] Renzo Sprugnoli; Negation of binomial coefficients; Discrete
Mathematics 308 (2008) 5070–5077
[PL] Peter Luschny; Extensions of the binomial; Blog post;
http://oeis.org/wiki/User:Peter_Luschny/ExtensionsOfTheBinomial
[trac] http://trac.sagemath.org/ticket/17123
I'd like to hear your opinions.
Peter
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