TY - JOUR
T1 - Painlevé III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight
AU - Xu, Shuai-Xia
AU - Dai, Dan
AU - Zhao, Yu-Qiu
PY - 2015/4
Y1 - 2015/4
N2 - In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight w(x)=xαe-x-t/x, x∈. (0, ∞), t>. 0 and α. >. 0. When the matrix size n→. ∞, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for t∈. (0, d], d>. 0 fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large-. n asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift-Zhou nonlinear steepest descent method.
AB - In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight w(x)=xαe-x-t/x, x∈. (0, ∞), t>. 0 and α. >. 0. When the matrix size n→. ∞, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for t∈. (0, d], d>. 0 fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large-. n asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift-Zhou nonlinear steepest descent method.
KW - Asymptotics
KW - Hankel determinants
KW - Painlevé III equation
KW - Perturbed Laguerre weight
KW - Primary
KW - Riemann-Hilbert approach
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U2 - 10.1016/j.jat.2014.12.003
DO - 10.1016/j.jat.2014.12.003
M3 - 21_Publication in refereed journal
VL - 192
SP - 1
EP - 8
JO - Journal of Approximation Theory
JF - Journal of Approximation Theory
SN - 0021-9045
ER -