2021/04/16 by Aleksandr Komlov, Александр Владимирович Комлов
Mathematics · Physics and Astronomy · #Combinatorics #Discrete mathematics #Geometry #Hermite polynomials #Holomorphic and Operator Theory #Mathematical analysis #Mathematics #Meromorphic and Entire Functions #Meromorphic function #Nonlinear Waves and Solitons #Partition (number theory) #Polynomial #Pure mathematics #Riemann hypothesis #Riemann surface #Surface (topology) #Tuple #math.CV
paper · pdf · doi:10.1070/sm9577
arxiv created 2021/04/16 · openalex publication_date 2021/10/14 · openalex created_date 2021/10/25 · arxiv updated 2022/03/09 · openalex updated_date 2026/05/21
For an arbitrary tuple of m+1 germs of analytic functions at a fixed point, we introduce the so-called polynomial Hermite-Padé m-system (of order n, n∈\mathbb N), which consists of m tuples of polynomials; these tuples, which are indexed by a natural number k∈[1,…,m], are called the kth polynomials of the Hermite-Padé m-system. We study the weak asymptotics of the polynomials of the Hermite-Padé m-system constructed at the point ∞ from the tuple of germs [1, f1,∞,\dotsc, fm,∞] of the functions 1, f1,…,fm that are meromorphic on some (m+1)-sheeted branched covering π\colon \mathfrak R→\widehat\mathbb C of the Riemann sphere \widehat\mathbb C of a compact Riemann surface \mathfrak R. In particular, under some additional condition on π, we find the limit distribution of the zeros and the asymptotics of the ratios of the kth polynomials for all k∈[1,…, m]. It turns out that in the case, where fj = fj for some meromorphic function f on \mathfrak R, the ratios of some kth polynomials of such Hermite-Padé m-system converge to the sum of the values of the function f on the first k sheets of the Nuttall partition of the Riemann surface \mathfrak R into sheets.