2015/02/04 by Andrei Martínez-Finkelshtein, Martínez-Finkelshtein, Andrei, Evgenii A. Rakhmanov +3
Mathematics · #30C85 #30E10 #41A20 #41A21 #41A45 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:30C85 #msc:30E10 #msc:41A20 #msc:41A21 #msc:41A45 #msc:42C05
paper · pdf · doi:10.48550/arxiv.1502.01202
40 pages, 1 figure. Minor modifications and error corrections
arxiv created 2015/05/19 · arxiv updated 2015/05/21
Type I Hermite--Padé polynomials for a set of functions f0, f1, ..., fs at infinity, Qn,0, Qn,1, ..., Qn,s, is defined by the asymptotic condition Rn(z):=(Qn,0f0+Qn,1f1+Qn,2f2+...+Qn,sfs)(z) =\mathcal O (\frac1zs n+s), z→∞, with the degree of all Qn,k≤ n. We describe an approach for finding the asymptotic zero distribution of these polynomials as n→ ∞ under the assumption that all fj's are semiclassical, i.e. their logarithmic derivatives are rational functions. In this situation Rn and Qn,kfk satisfy the same differential equation with polynomials coefficients. We discuss in more detail the case when fk's are powers of the same function f (fk=fk); for illustration, the simplest non trivial situation of s=2 and f having two branch points is analyzed in depth. Under these conditions, the ratio or comparative asymptotics of these polynomials is also discussed. From methodological considerations and in order to make the situation clearer, we start our exposition with the better known case of Padé approximants (when s=1).