Confluent Hypergeometric Functions and the Heisenberg Group

The (a) Heisenberg Group consists of upper triangular matrices like this:

$latex \begin{pmatrix}

1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\

There are a number of papers on this, but my take is that I’m interested in places where theta and hypergeometric functions end up occuring in close proximity (as this has occured with the solutions to the quintic previously mentioned on this blog)

I just got the NIST book about mathematical functions

today, and was browsing it, and found something interesting in the section about confluent hypergeometric functions, when the beginning of section 13.27 caught my attention:

Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. The elements of this group are of the form:


1 & \alpha & \beta \\
0 & \gamma & \delta\\
0 & 0 & 1\\

Where \alpha, \beta, \gamma, and \delta are real numbers and $\gamma >0$”

Why the “special” in /special functions/ needs to go…

It’s more or less a consequence of applied to special functions. What are the opposites? General functions? There are generalized functions, like the Meijer-G function, but that seems kind of contrary to the point (they do have their applications, for example, Mathematica uses the Meijer-G function internally, but it’s not the sort of function which seems very user friendly). 

I’m going to kind of rally against some of the attitude espoused in this question:


Most real numbers are transcendental. Yes, they probably satisfy scads of strange identities. Do those identities conform to toolbox-fetishist-expectations of how the real or complex numbers work? Some of them do. But after a point, those are uninteresting and trivial. Most of the time I care about identities between functions and I live on the rainbow world of the complex plane, so mere identities between infinite series or products kind of fade. And, you know, you don’t have any of the tools of alien mathematics in that toolbox of yours. There are functions and relations which are undreamt of in that Primate sandbox of yours, so stop saber-rattling that toolbox.