On Polynomials Defined by the Discrete Rodrigues Formula
Tóm tắt
We study polynomials given by the discrete Rodrigues formula, which generalizes a similar formula for Meixner polynomials. Such polynomials are associated with the theory of Diophantine approximations. The saddle point method is used to find the limit distribution of zeros of scaled polynomials. An answer is received in terms of a meromorphic function on a compact Riemann surface and is interpreted using the vector equilibrium problem of the logarithmic potential theory.
Tài liệu tham khảo
J. Meixner, “Orthogonale Polynomsysteme mit Einer Besonderen Gestalt der Erzeugenden Funktion,” J. London Math. Soc. 9 (1), 6–13 (1934).
H. Bateman and A. Erdélyi Higher Transcendental Functions (McGraw-Hill, New York–Toronto–London, 1955), Vol. 2.
V. N. Sorokin, “On multiple orthogonal polynomials for discrete Meixner measures,” Sb. Math. 201 (10), 1539–1561 (2010).
V. N. Sorokin and E. N. Cherednikova, “Meixner polynomials with variable weight,” Sovrem. Problemy Mat. i Mekh. 6 (1), 118–125 (2011).
V. N. Sorokin, On Asymptotic Modes of Joint Meixner Polynomials (2016) [in Russian]. Preprint: Keldysh Inst. Appl. Math., no. 46
V. N. Sorokin, Angelesco–Meixner Polynomials (2017) [in Russian]. Preprint: Keldysh Inst. Appl. Math., no. 27
V. N. Sorokin, “On multiple orthogonal polynomials for three Meixner measures,” Proc. Steklov Inst. Math. 298, 294–316 (2017).
V. N. Sorokin, “Hermite–Padé approximants to the Weyl function and its derivative for discrete measures,” Sb. Math. 211 (10), 1486–1502 (2020).
V. N. Sorokin, “Asymptotics of Hermite–Padé approximants of the first type for discrete Meixner measures,” Lobachevskii J. Math. 42, 2654–2667 (2021).
A. V. D’yachenko and V. G. Lysov, On Multiple Discrete Orthogonal Polynomials on a Lattice with Shift (2018) [in Russian]. Preprint: Keldysh Inst. Appl. Math., no. 218
S. P. Suetin, “Two examples related to properties of discrete measures,” Math. Notes 110 (4), 578–582 (2021).
V. N. Sorokin, “Multipoint Padé approximation of the psi function,” Math. Notes 110 (4), 571–577 (2021).
A. A. Kandayan, “Multipoint Padé approximations of the beta function,” Math. Notes 85 (2), 176–189 (2009).
A. A. Kandayan and V. N. Sorokin, “Multipoint Hermite–Padé approximations for beta functions,” Math. Notes 87 (2), 204–217 (2010).
A. A. Kandayan and V. N. Sorokin, “Asymptotics of multipoint Hermite–Padé approximants of the first type for two beta functions,” Math. Notes 101 (6), 984–993 (2017).
J. Touchard, “Nombres exponentiels et nombres de Bernoulli,” Canadian J. Math. 8, 305–320 (1956).
M. Prevost, “A new proof of the irrationality of \(\zeta(2)\) and \(\zeta(3)\) using Padé approximants,” J. Comput. Appl. Math. 67 (2), 219–235 (1996).
R. Apery, “Irrationalité de \(\zeta(2)\) et \(\zeta(3)\),” Astérisque 61, 11–13 (1979).
V. N. Sorokin, “On the Zudilin–Rivoal theorem,” Math. Notes 81 (6), 817–826 (2007).
A. A. Gonchar, E. A. Rakhmanov, and V. N. Sorokin, “Hermite–Padé approximants for systems of Markov-type functions,” Sb. Math. 188 (5), 671–696 (1997).
E. A. Rakhmanov, “Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable,” Sb. Math. 187 (8), 1213–1228 (1996).
A. A. Gonchar and E. A. Rakhmanov, “Equilibrium distributions and rate of rational approximation of analytic functions,” Mat. Sb. 134(176) (3(11)), 306–352 (1987).
A. A. Gonchar, E. A. Rakhmanov, and S. P. Suetin, “Padé–Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the \(S\)-property of stationary compact sets,” Russian Math. Surv. 66 (6), 1015–1048 (2011).
E. A. Rakhmanov, “Orthogonal polynomials and \(S\)-curves,” in Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2012), Vol. 578, pp. 195–239.