Weak quasiclassical asymptotics of polynomial solutions of three-term recurrence relations of high order

Capa

Citar

Texto integral

Acesso aberto Acesso aberto
Acesso é fechado Acesso está concedido
Acesso é fechado Somente assinantes

Resumo

For polynomials $Q_{n}(z):=z^n + \cdots$ defined by three-term recurrence relations
$Q_{n+1}=zQ_n-a_{n-p+1}Q_{n-p}$,
$p\ge {1}$, of order $p+1$ with the coefficient $a_{n}\equiv a_{n,N}$ (the variable recurrence coefficient) depending on the parameter $N$,
the weak asymptotics of $Q_n (z)$ are investigated in the quasi-classical regime as $n \to \infty$,
$n/N \to t$, and $a_{n,N} \to a(t)$.
The case $p=1$ (orthogonal polynomials) was studied earlier. The results obtained (for $p=2$) are applied to the problem of eigenvalues distributions of ensembles of normal random matrices.

Sobre autores

Alexander Aptekarev

Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow

Email: aptekaa@gmail.com
Scopus Author ID: 6603809965
Doctor of physico-mathematical sciences, Professor

Victor Novokshenov

Institute of Mathematics with Computing Centre, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa

Email: novik53@mail.ru
Doctor of physico-mathematical sciences, Professor

Bibliografia

  1. A. Aptekarev, V. Kaliaguine, J. Van Iseghem, “The genetic sums' representation for the moments of a system of Stieltjes functions and its application”, Constr. Approx., 16:4 (2000), 487–524
  2. A. I. Aptekarev. V. A. Kalyagin, E. B. Saff, “Higher-order three-term recurrences and asymptotics of multiple orthogonal polynomials”, Constr. Appox., 30:2 (2009), 175–223
  3. J. Nuttall, “Asymptotics of diagonal Hermite–Pade polynomials”, J. Approx. Theory, 42:4 (1984), 299–386
  4. A. I. Aptekarev, “Multiple orthogonal polynomials”, J. Comput. Appl. Math., 99:1-2 (1998), 423–447
  5. L. A. Pastur, “Spectral and probabilistic aspects of matrix models”, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996
  6. A. B. J. Kuijlaars, W. Van Assche, “The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients”, J. Approx. Theory, 99:1 (1999), 167–197
  7. P. Deift, K. T.-R. McLaughlin, A continuum limit of the Toda lettice, Mem. Amer. Math. Soc., 131, no. 624, Amer. Math. Soc., Providence, RI, 1998, x+216 pp.
  8. A. I. Aptekarev, W. Van Assche, “Asymptotics of discrete orthogonal polynomials and the continuum limit of the Toda lattice”, J. Phys. A, 34:48 (2001), 10627–10637
  9. A. I. Aptekarev, J. S. Geronimo, W. Van Assche, “Varying weights for orthgonal polynomials with monotonically varying recurrence coefficients”, J. Approx. Theory, 150:2 (2008), 214–238
  10. O. Costin, R. Costin, “Rigorous WKB for finite-order linear recurrence relations with smooth coefficients”, SIAM J. Math. Anal., 27:1 (1996), 110–134
  11. R. M. Kashaev, “A link invariant from quantum dilogarithm”, Modern Phys. Lett. A, 10:19 (1995), 1409–1418
  12. R. M. Kashaev, “The hyperbolic volume of knots from the quantum dilogarithm”, Lett. Math. Phys., 39:3 (1997), 269–275
  13. S. Garoufalidis, Thang T. Q. Lê, “The colored Jones function is $q$-holonomic”, Geom. Topol., 9 (2005), 1253–1293
  14. S. Garoufalidis, J. S. Geronimo, “Asymptotics of $q$-difference equations”, Primes and knots, Contemp. Math., 416, Amer. Math. Soc., Providence, RI, 2006, 83–114
  15. S. Garoufalidis, C. Koutschan, “Irreducibility of $q$-difference operators and the knot $7_4$”, Algebr. Geom. Topol., 13:6 (2013), 3261–3286
  16. A. I. Aptekarev, T. V. Dudnikova, D. N. Tulyakov, “Recurrence relations and asymptotics of colored Jones polynomials”, Lobachevskii J. Math., 42:11 (2021), 2580–2595
  17. A. I. Aptekarev, T. V. Dudnikova, D. N. Tulyakov, “Volume conjecture and WKB asymptotics”, Lobachevskii J. Math., 43:8 (2022), 2057–2079
  18. I. K. Kostov, I. Krichever, M. Mineev-Weinstein, P. B. Wiegmann, A. Zabrodin, “The $tau$-function for analytic curves”, Random matrix models and their applications, Math. Sci. Res. Inst. Publ., 40, Cambridge Univ. Press, Cambridge, 2001, 285–299
  19. P. Elbau, Random normal matrices and polynomial curves, Ph.D. thesis, ETH Zürich, 2006
  20. P. M. Bleher, A. B. J. Kuijlaars, “Orthogonal polynomials in the normal matrix model with a cubic potential”, Adv. Math., 230:3 (2012), 1272–1321
  21. P. Deift, Orthogonal polynomials and random matrices: a Riemann–Hilbert approach, Courant Lect. Notes Math., 3, Courant Inst. Math. Sci., New York; Amer. Math. Soc., Providence, RI, 1999, viii+273 pp.
  22. R. Teodorescu, E. Bettelheim, O. Agam, A. Zabrodin, P. Wiegmann, “Normal random matrix ensemble as a growth problem”, Nuclear Phys. B, 704:3 (2005), 407–444
  23. A. I. Aptekarev, “Spectral problems of high-order recurrences”, Spectral theory and differential equations, Amer. Math. Soc. Transl. Ser. 2, 233, Adv. Math. Sci., 66, Amer. Math. Soc., Providence, RI, 2014, 43–61

Arquivos suplementares

Arquivos suplementares
Ação
1. JATS XML
Baixar

Declaração de direitos autorais © Aptekarev A.I., Novokshenov V.Y., 2025

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).