negative-binom 0.1.4

A binomial function that extends the domain of combinations to include negative integers.

Negative Binom

A binomial function that extends the domain of combinations to include negative integers.

Installation

You can install the package using: $$bash pip install negative-binom $$

Usage

Import the function and use it as follows: $$python from negative_binom import binom

Standard case:

print(binom(5, 2)) # Output: 10

Extended cases:

print(binom(-5, 2)) # Output: 15 print(binom(5, -2)) # Output: 0 print(binom(-5, -2)) # Output: 0 $$

Mathematical Explanation

The binomial coefficient is traditionally defined as: $$ \binom{m}{n} = \frac{m!}{n!(m-n)!}, \quad \text{for }m \geq n \geq 0 $$ The binom function extends the definition to handle negative values of m and n using "The Pascal Hexagon" provided by Hilton and Pedersen in Extending the Binomial Coefficients to Preserve Symmetry and Pattern:

The hexagon is broken into different parts depending on if they contain zeros or non-zeros. Formulas below then can be used to determine the values of the binomial coefficients:

1. Standard case $ (m \geq 0) $

• If $ m \geq n \geq 0 $: $$ \binom{m}{n}=\frac{m!}{n!(m-n)!} $$

• Otherwise: $$ \binom{m}{n}=0 $$

2. Extended case $ (0 > m) $

• If $ n \geq 0 $: $$ \binom{m}{n}=(-1)^n \binom{-m+n-1}{n} $$

• If $ m \geq n $: $$ \binom{m}{n}=(-1)^{m+n} \binom{-n-1}{-m-1} $$

• Otherwise: $$ \binom{m}{n}=0 $$

This extended definition of binomial coefficients makes it possible to work with negative values of $ m $ and $ n $.

Personal Notes

I made this package because I needed a binomial function that included negative arguments as well. math.comb results in error and scipy.stats.binom sets the result to 0 for negative arguments, so they didn't work for me. Hope this helps anyone else.

License

MIT License