Sometimes it’s useful to apply dimensional analysis where it doesn’t belong, to imagine things having physical dimension when they don’t. This post will look at artificially injecting dimensions into equations involving factorials and related functions.

Factorials

The factorial of n is defined as the product of n terms. If each of these terms had units of length, the factorial would have units of n-dimensional volume. It occurred to me this morning that a lot of identities involving factorials make sense dimensionally if you pretend the terms in the identities have units. The expressions on both sides of an equation have the same dimension, and only terms of the same dimension are added together. This isn’t always the case, so caveat emptor.

We could also think of an nth rising power or nth falling power as an n-dimensional volume. If we do, then the dimensions in a Newton series cancel out, for example.

Dimensional analysis for factorials and related functions could make it easier to remember identities, and easier to spot errors. And it could suggest the correct form of a result before the details of the result are filled in. In other words, artificial dimensional analysis can provide the same benefits of physically meaningful dimensional analysis.

Gamma function

For integers n, Γ(n) = (n − 1)!, and so we could assign dimension n − 1 to Γ(n), and more generally assign dimension z − 1 to Γ(z) for any complex z.

Now for some examples to show this isn’t as crazy as it sounds. For starters, take the identity Γ(z + 1) = z Γ(z). We imagine the left hand side is a z-dimensional volume, and the right hand side is the product of a term with unit length and a term that represents a (z − 1) dimensional volume, so the units check out.

Next let’s take something more complicated, the Legendre duplication formula:

Gamma (z)Gamma left(z+frac{1}{2}right)=2^{1-2z}sqrt{pi} ; Gamma (2z)
The left hand side has dimension (z − 1) + (z − ½) = 2z − ½. The left side has dimension 2z − 1, apparently contradicting our dimensional analysis scheme. But √π = Γ(½), and if we rewrite √π as Γ(½) in the equation above, both sides have dimension 2z − ½. The dimensions in other identities, like the reflection formula, also balance when you replace π with Γ(½)².

Hypergeometric functions

The hypergeometric function F(a, b; c; z) is defined by its power series representation. If we assign dimensions to the coefficients in the series as we’ve done here, then the numerator and denominator of each term have the same dimensions, and so the hypergeometric function should be dimensionless in our sense. This implies we should expect that identities for hypergeometric functions to be dimensionless as well. And indeed they are. I’ll give two examples.

First, let’s look at Gauss’ summation identity

F (a,b;c;1)= frac{Gamma(c)Gamma(c-a-b)}{Gamma(c-a)Gamma(c-b)}

provided the real part of c is greater than the real part of a + b. Notice that the numerator and denominator both have dimension 2c − a − b − 2.

Next, let’s look at Kummer’s identity

F (a,b;1+a-b;-1)= frac{Gamma(1+a-b)Gamma(1+tfrac12a)}{Gamma(1+a)Gamma(1+tfrac12a-b)}

Both the numerator and denominator have dimension 3a/2 − b.

Finally, let’s look at a more complicated formula.

begin{align*} F(a, b; c; z) &= frac{Gamma(c) Gamma(b-a)}{Gamma(b) Gamma(c-a)} ,(-z)^{-aphantom{b}} Fleft(a, 1-c+a; 1-b+a; frac{1}{z}right) \ &+ frac{Gamma(c) Gamma(a-b)}{Gamma(a) Gamma(c-b)} ,(-z)^{-bphantom{a}} Fleft(,b, 1-c+b; 1-a+b; ,frac{1}{z}right) \ end{align*}

In both the terms involving gamma functions, the dimensions of the numerator and denominator cancel out.

The post Dimensional analysis for gamma function values first appeared on John D. Cook.

Previous Post
Next Post