# User:DVD206/Stieltjes continued fractions

< User:DVD206(Redirected from User:Daviddaved/Stieltjes continued fractions)- 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,

since

- 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