2015/06/26 by Nikolay Ikonomov, Nikolay R. Ikonomov, Ralitza K. Kovacheva +6 · 1 citation
Mathematics · #30E10 #41A21 #Advanced Mathematical Identities #Complex Variables (math.CV) #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical functions and polynomials #math.CV #msc:30E10 #msc:41A21
paper · pdf · doi:10.48550/arxiv.1506.08031
Bibliography: 79 items; figures: 72 items
arxiv created 2015/06/26 · openalex publication_date 2015/06/26 · arxiv updated 2015/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper are discussed the results of new numerical experiments on zero distribution of type I Hermite-Padé polynomials of order n=200 for three different collections of three functions [1,f1,f2]. These results are obtained by the authors numerically and do not match any of the theoretical results that were proven so far. We consider three simple cases of multivalued analytic functions f1 and f2, with separated pairs of branch points belonging to the real line. In the first case both functions have two logarithmic branch points, in the second case they both have branch points of second order, and finally, in the third case they both have branch points of third order. All three cases may be considered as representative of the asymptotic theory of Hermite-Padé polynomials. In the first two cases the numerical zero distribution of type I Hermite-Padé polynomials are similar to each other, despite the different kind of branching. But neither the logarithmic case, nor the square root case can be explained from the asymptotic point of view of the theory of type I Hermite-Padé polynomials. The numerical results of the current paper might be considered as a challenge for the community of all experts on Hermite-Padé polynomials theory.