User:Daviddaved/Stieltjes continued fractions

Let $a_k$ be a set of n positive numbers. The Stieltjes continued fraction is an expression of the form

$\beta(z) = a_nz + \cfrac{1}{a_{n-1}z + \cfrac{1}{ \ddots + \cfrac{1}{a_1 z} }}.$

The function and its reciprocal define all rational n-to-1 maps of the right half of the complex plane onto itself,

$\beta,1/\beta:\mathbb{C^+}\xrightarrow[]{n\leftrightarrow 1}\mathbb{C^+},$

since

$\begin{cases} Re(z_1), Re(z_2) > 0 \implies Re(z_1+z_2) > 0, \\ Re(z) \implies Re(1/z) > 0, \\ Re(z) > 0, a > 0 \implies Re(az) > 0. \end{cases}$

The function $\beta$ is determined by the pre-image set $\Mu=\{\mu_k\}$ of the point {z = 1}, since

$\beta(z) = \frac{p(z^2)}{zq(z^2)}=1 \iff p(z^2)-zq(z^2) = 0,$

and a complex polynomial is determined by its roots up to a multiplicative constant by the fundamental theorem of algebra.

Let $\sigma_l$ be the elementary symmetric functions of the set $\Mu$. That is,

$\prod_k (z-\mu_k) = \sum_k \sigma_{n-k} z^k.$

Then the coefficients $a_k$ of the continued fraction are the pivots in the Gauss-Jordan elimination algorithm of the following n by n square Hurwitz matrix:

$S(\Mu) := \begin{bmatrix} \sigma_1 & \sigma_3 & \sigma_5 & \sigma_7 & \ldots & 0\\ 1 & \sigma_2 & \sigma_4 & \sigma_6& \ldots & 0\\ 0 & \sigma_1 & \sigma_3 & \sigma_5& \ldots & 0\\ 0 & 1 & \sigma_2 & \sigma_4& \ldots & 0\\ 0 & 0 & \sigma_1 & \sigma_3& \ldots & 0\\ \vdots & \vdots & \vdots & \vdots& \ddots& \vdots\\ 0 & 0 & 0 & 0& \ldots& \sigma_n\\ \end{bmatrix}$

and, therefore, can be expressed as the ratios of monomials of the determinants of its blocks.

Exercise 1 : Prove that

$a_1 = 1/ \sigma_1.$

Exercise 2 :

Let A be a diagonal matrix with the alternating in sign diagonal entries:

$A = \begin{bmatrix} 1/a_1 & 0 & 0 & \ldots & 0 \\ 0 & -1/a_2 & 0 & \ldots & 0 \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & -1/a_{n-1} & 0 \\ 0 & 0 & \ldots & 0 & 1/a_n \\ \end{bmatrix}$

and D the (0,1)-matrix

$D = \begin{bmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots & 0 \\ 0 & 1 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & 0 & 1 \\ 0 & 0 & \ldots & 1 & 1 \\ \end{bmatrix} .$

Prove that the continued fraction $\beta(z)$ evaluated at a point $z = \mu$ equals to 1 if and only if $\mu$ is an eigenvalue of the matrix AD.

Exercise 3 :

Use Exercise 1 to prove that

$\prod_k a_k = \frac{1}{\prod_k \mu_k} = 1/\sigma_n.$