Comparison of measures of totally positive polynomials

Cauchy biorthogonal polynomials

The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite–Padé approximation scheme. Associated with any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeros are simple and positive. We then specialize the kernel to the Cauchy kernel 1 x+y and show t...

Computing totally positive algebraic integers of small trace

We construct minimal polynomials of totally positive algebraic integers of small absolute trace by consideration of their reductions modulo auxiliary polynomials. Many new examples of such polynomials of minimal absolute trace (for given degree) are found. The computations are pushed to degrees that previously were unattainable, and one consequence is that the new examples form the majority of ...

Almost strict total positivity and a class of Hurwitz polynomials

We establish sufficient conditions for a matrix to be almost totally positive, thus extending a result of Craven and Csordas who proved that the corresponding conditions guarantee that a matrix is strictly totally positive. Then we apply our main result in order to obtain a new criteria for a real algebraic polynomial to be a Hurwitz one. The properties of the corresponding “extremal” Hurwitz p...

Oscillation Properties of Generalized Characteristic Polynomials for Totally Positive and Positive Definite Matrices

THE COMBINATORICS OF AL-SALAM-CHIHARA q-LAGUERRE POLYNOMIALS

We decribe various aspects of the Al-Salam-Chihara q-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Gr...