User:DVD206/Stieltjes continued fractions

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Let ${\displaystyle a_{k}}$ be a set of n positive numbers. The Stieltjes continued fraction is an expression of the form
${\displaystyle \beta (z)=a_{n}z+{\cfrac {1}{a_{n-1}z+{\cfrac {1}{\ddots +{\cfrac {1}{a_{1}z}}}}}}.}$

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

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

since

${\displaystyle {\begin{cases}Re(z_{1}),Re(z_{2})>0\implies Re(z_{1}+z_{2})>0,\\Re(z)\implies Re(1/z)>0,\\Re(z)>0,a>0\implies Re(az)>0.\end{cases}}}$
The function ${\displaystyle \beta }$ is determined by the pre-image set ${\displaystyle \mathrm {M} =\{\mu _{k}\}}$ of the point {z = 1}, since
${\displaystyle \beta (z)={\frac {p(z^{2})}{zq(z^{2})}}=1\iff p(z^{2})-zq(z^{2})=0,}$

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

Let ${\displaystyle \sigma _{l}}$ be the elementary symmetric functions of the set ${\displaystyle \mathrm {M} }$. That is,

${\displaystyle \prod _{k}(z-\mu _{k})=\sum _{k}\sigma _{n-k}z^{k}.}$
Then the coefficients ${\displaystyle a_{k}}$ of the continued fraction are the pivots in the Gauss-Jordan elimination algorithm of the following n by n square Hurwitz matrix:
${\displaystyle S(\mathrm {M} ):={\begin{bmatrix}\sigma _{1}&\sigma _{3}&\sigma _{5}&\sigma _{7}&\ldots &0\\1&\sigma _{2}&\sigma _{4}&\sigma _{6}&\ldots &0\\0&\sigma _{1}&\sigma _{3}&\sigma _{5}&\ldots &0\\0&1&\sigma _{2}&\sigma _{4}&\ldots &0\\0&0&\sigma _{1}&\sigma _{3}&\ldots &0\\\vdots &\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&0&\ldots &\sigma _{n}\\\end{bmatrix}}}$

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

Exercise 1 : Prove that

${\displaystyle a_{1}=1/\sigma _{1}.}$

Exercise 2 :

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

${\displaystyle A={\begin{bmatrix}1/a_{1}&0&0&\ldots &0\\0&-1/a_{2}&0&\ldots &0\\0&0&\ddots &\ddots &\vdots \\\vdots &\vdots &\ddots &-1/a_{n-1}&0\\0&0&\ldots &0&1/a_{n}\\\end{bmatrix}}}$

and D the (0,1)-matrix

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

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

Exercise 3 :

Use Exercise 1 to prove that

${\displaystyle \prod _{k}a_{k}={\frac {1}{\prod _{k}\mu _{k}}}=1/\sigma _{n}.}$