# §13.27 Mathematical Applications

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

 13.27.1 $g=\begin{pmatrix}1&\alpha&\beta\\ 0&\gamma&\delta\\ 0&0&1\end{pmatrix},$ Permalink: http://dlmf.nist.gov/13.27.E1 Encodings: TeX, pMML, png

where $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers, and $\gamma>0$. Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. This identification can be used to obtain various properties of the Whittaker functions, including recurrence relations and derivatives.

For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).