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 $\gamma >0$”


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