I’ve said in several blog posts that multi-factorials come up often in practice, and given examples. This post will give a glimpse of why this is.
Rising powers
The kth rising power of a is
(a)k = a (a+1) (a + 2) … (a + k – 1).
So, for example, (5)3 = 5*6*7 and (1)k = n!.
Double factorials
Now let’s look at rising powers of fractions.
(1/2)k = (1/2) (3/2) (5/2) … (2k-1)/2 = (2k-1)!! / 2k.
This means double factorials of odd numbers can be written as rising powers.
(2k-1)!! = 2k (1/2)k .
Now for even numbers,
(2k)!! = 2k k! = 2k (1)k
and so even double factorials can be written in terms of rising powers too. This means that the term
(2k – 1)!! / (2k)!!
from the series in the previous post could be written as
(1/2)k / (1)k .
Hmm. Ratios of rising powers sound useful.
Generalizing
From the discussion above you can imagine how to create triple factorials out of rising powers of 1/3 and 2/3. In fact it’s easy to see that all multifactorials can be made out of rising powers of rational numbers.
Ratios of multifactorials come up often in coefficients of power series, and these can be rewritten as ratios of rising powers. Maybe we should have a name for functions whose coefficients are ratios of rising powers. And we do: hypergeometric functions. You can find the exact definition here.
At first the definition of a hypergeometric function seems arbitrary. I hope my previous posts, and especially this post, has been a sort of warm up to motivate why you might be interested in such functions.
Applications
In an old post I quoted W. W. Sawyer saying
There must be many universities today where 95 percent, if not 100 percent, of the functions studied by physics, engineering, and even mathematics students, are covered by the single symbol F(a, b; c; x).
The notation F(a, b; c; x) here denotes the hypergeometric function whose kth power series coefficient is
(a)k (b)k / (c)k k!
More general hypergeometric functions have two lists of parameters, separated by a semicolon. The first set corresponds to rising powers in the numerator, and the second set corresponds to rising powers in the denominator.
The exponential function is the simplest hypergeometric function: both parameter lists are empty, leaving only the k! term in the denominator.
The arcsine function mentioned above is an example of a function that is not hypergeometric per se, but can be written simply in terms of a hypergeometric function:
sin-1(x) = x F(1/2, 1/2; 3/2, x²).
This is more what Sawyer had in mind. Elementary functions like sine, cosine, and log note pure hypergeometric functions but are simply related to hypergeometric functions, similar to arcsine. The same is true of more advanced functions, like elliptic integrals and the CDFs of common probability distributions.
Hypergeometric is also the distribution for elections, where the sample s is the turnout, of N total registered voters, and where m is the total of N who would have voted candidate X. Of course, that requires sampling without replacement.
p(x) = m!(N −m)!s!(N −s)!/(x!(m −x)!N!(s −x)!(N −m −s + x)!)