a lament for special functions

I was thinking about Felix Klein’s lament for theta functions:

When I was a student, abelian functions were, as an effect of the Jacobian tradition, considered the uncontested summit of mathematics, and each of us was ambitious to make progress in this field. And now? The younger generation hardly knows abelian function

And John D. Cook’s post:
The grand unified theory of 19th century math

And for some historical perspective, here’s Stephen Wolfram’s commentary on special functions: The History and Future of Special Functions

Mathematics curricula have moved on. People deal with groups and topologies and other fads these days. Few people doing analysis of dynamical systems these days even consider doing fractals of special functions (Jane Hawkins being a notable exception). It is entirely possible to get an undergraduate degree in mathematics (which I have to admit, I don’t have) without hearing about Bessel or theta or hypergeometric functions even once. A few people trickle in here or there, but for a large part, people are not so great at recognizing polylogarithms and whatnot. And as mathematics education slides onwards, the knowledge is becoming more and more esoteric. I think this is a bad thing. Educators don’t want to give out knowledge of the gamma function (which I remember reading about in high school) until sometime in a college analysis course. And elliptic functions? We should be getting people while their brains are young and fresh and show them domain colored plots of these things, not waiting until their cerebra have gelled!

another hard to evaluate product

$\prod_{m=2}^{\infty} \prod_{n=1}^{\infty} \left(1+\frac{1}{{}^{m}n}\right)$

That’s $n$, tetrated to the $m$th.

hilarious values…

There is a unique real number such that $\sqrt{x}=\zeta(x)$. Wolfram alpha says that it is around $\approx 2.207555582730262446854083026\ldots$. This is hilarious because, at that point, you are unable to tell if you just took the square root or the zeta function of the number.

“ethereal” functions

Say a function is ethereal iff:
1. all the coefficients in the power series representation of it are transcendental
2. there is no easy way of getting at those coefficients

2 is a crapshoot. Let me give a somewhat contrived example, take the Airy function $Bi(z)$, which has power series representation:
$\mathrm{Bi}(z) = \frac{1}{\pi\sqrt[6]{3}} \sum_{n=0}^{\infty} \frac{3^{n/3} \Gamma((1+n)/3) \sin(2(1+n)\pi/3)}{k!}$. I don’t consider this function ethereal because the coefficients are easy to get at, so to speak.

Now consider a function whose power series coefficients $a_{n}$ are such that $\mathrm{Bi}(a_{n})=n \forall n\in\mathbb{N}$. There is a sea of such functions, but they’re sort of slippery and hard to make

recipe for the construction of novel infinite products

1. Choose a special function that has limit zero. Airy functions work.
2. Compute $\prod_{n=1}^{\infty} \left(1+\mathrm{Ai}(n)\right) \approx 1.1839521034335512376\ldots$
3. See if you can eke any new identities out of it.

If a special function goes to infinity, its reciprocal goes to zero. Take the hyperfactorial $H(n) = \prod_{k=1}^{n} k^{k}$. You can then consider $\prod_{n=1}^{\infty} \left(1 + \frac{1}{H(n)}\right)\approx 2.52323943705157931311\ldots$