Implement geralized binomial coefficients

[rmathar]

Feature or enhancement

Proposal:

The Lib/fractions.py is missing the binomial function:

def binomial(n: fractions.Fraction, m :int) :
    """Computs n choose m.

    returns n(n-1)(n-2)...(n-m+1) / (1*2*3*...m)
    """
    val = fractions.Fraction(1)
    for i in range(1,m+1):
        val *= n+1-i
        val /= i
    return val 

Note that this is not the same as the implementation in the math module: here we want to compute the generalized binomial coefficients which allow numerators to be fractions!

Has this already been discussed elsewhere?

No response given

Links to previous discussion of this feature:

No response

[skirpichev]

See math.comb().

[rmathar]

This has been erroneously flagged completed. The math implementation does not allow fractional numerators, my implementation does. This should be re-opened and implemented here.

[terryjreedy]

In generalized binomial coefficients (EDIT: ref), the upper index (not 'numerator') can be complex. Thus, this would definitely not go in fractions, likely in the cmath module. However, I am leaving this closed as 'too specialized' for the CPython stdlib. New features like this have to be discussed elsewhere, usually on discuss.python.org.

[rmathar]

It does not help if the numerator parameter (or upper or whatever) is comples. I need (and others dealing with hypergeometric functions, pochhammer symbols etc) need an exact representation, and it does not help if one has implementations based on some floating-point implementations.

[terryjreedy]

The new issue template, which perhaps you used, begins

Proposing a feature to CPython?
You'll need to demonstrate widespread support for your idea among the community.
Major feature proposals should generally be discussed on Discourse before opening a GitHub issue. Wait until it's clear that most people support your idea before filling in this form.

It ends with boxes to link to evidence of that support, which are blank above. Don't argue here.

[skirpichev]

This has been erroneously flagged completed.

No, its not.

The math implementation does not allow fractional numerators

My point was, we already have an implementation of binomial coefficients for one domain. Having same function for multiple domains rather unusual for Python. (There is cmath module, but its a historical artifact.)

This should be re-opened and implemented here.

No. First you should argument for why we need to do this. The CPython "missing" a lot of functions, available in computer algebra systems (CAS). But its not a problem, because it's not a CAS.

I need (and others dealing with hypergeometric functions, pochhammer symbols etc) need an exact representation

You might want to use some Python-based CAS. E.g. SymPy, per default, does expansion of binomial coefficients, if the second argument is an Integer. (But not, if the first argument is a symbol.) Same for failing factorial (perm() in the stdlib):

In [1]: binomial(Rational(7, 3), 11)
Out[1]: 
  5474  
────────
14348907

In [2]: binomial(x, 7)
Out[2]: 
⎛x⎞
⎜ ⎟
⎝7⎠

In [3]: binomial(pi, 7)
Out[3]: 
     4    6       2         7        3      5
  7⋅π    π     7⋅π    π    π     29⋅π    5⋅π 
- ──── - ─── - ──── + ─ + ──── + ───── + ────
   48    240    20    7   5040    90     144 

In [4]: ff(Rational(7, 3), 4)
Out[4]: 
-56 
────
 81 

With PEP 791 we have a dedicated place for number-theoretic function. Perhaps, we could relax here requirements for arguments and allow sometimes Fractions. But I see following drawbacks:

• I agreed, its too much for the stdlib of general-purpose language;
• The Fraction type is not well integrated with the core language, it looks alien. I.e. true division of integers - returns a float, there is no dedicated dunder method to provide Fractions view for objects, and so on.