Last modified on 5 September 2012, at 20:07

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.