Functional Python Programming

Chapter 16

Both of these functions will require a factorial calculation, n!. We've seen several
variations on the fractions theme. We'll use the following one:

@lru_cache(128)

def fact(k):

if k < 2: return 1

return reduce(operator.mul, range(2, int(k)+1))

This is k!=Π 2 ≤≤i k i: a product of numbers from 2 to k (inclusive). We've omitted the
unit test cases.

Computing the partial gamma value

The partial gamma function has a simple series expansion. This means that we're
going to compute a sequence of values and then do a sum on those values. For more
information, visit http://dlmf.nist.gov/8.

()

()

0

1

,

!

k sk

k

z

sz

ks k

γ

+

≤≤∞

−

=

+

∑

This series will have a sequence of terms that—eventually—become too small to be
relevant. The calculation ()− 1 k will yield alternating signs:

−=1 1^0 , 1−=^1 −−1, 1 1^2 =−, 1^3 =− 1

The sequence of terms looks like this with s=1 and z=2:

2/1, -2/1, 4/3, -2/3, 4/15, -4/45, ..., -2/638512875

At some point, each additional term won't have any significant impact on the result.

When we look back at the cumulative distribution function, Fx();k, we can consider
working with fractions.Fraction values. The degrees of freedom, k, will be an
integer divided by 2. The X^2 value, x, may be either a Fraction or a float value; it
will rarely be a simple integer value.

When evaluating the terms of γ, the value of

()^1

!

k

k

−

will involve integers and can be

represented as a proper Fraction value. The value of
zsk+
could be a Fraction or
float value; it will lead to irrational values when sk+ is not an integer value. The
value of sk+ will be a proper Fraction value, sometimes it will have the integer
values, and sometimes it will have values that involve 1/2.