User:Daviddaved/Stieltjes continued fractions

< User:Daviddaved
Let be a set of n positive numbers. The Stieltjes continued fraction is an expression of the form

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


The function is determined by the pre-image set of the point {z = 1}, since

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

Let be the elementary symmetric functions of the set . That is,

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

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

Exercise 1 : Prove that

Exercise 2 :

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

and D the (0,1)-matrix

Prove that the continued fraction evaluated at a point equals to 1 if and only if is an eigenvalue of the matrix AD.

Exercise 3 :

Use Exercise 1 to prove that