TY - JOUR
T1 - Uniform asymptotic formula for orthogonal polynomials with exponential weight
AU - Qiu, W. Y.
AU - Wong, R.
PY - 2000/4
Y1 - 2000/4
N2 - Let {pn(x)}n≥0 be the set of orthonormal polynomials with respect to the exponential weight w(x) = e-v(x), where v(x) = x2m + ⋯ is a monic polynomial of degree 2m with m ≥ 2 and is even. An asymptotic approximation is obtained for pn(x), as n → ∞, which holds uniformly for 0 ≤ x ≤ O(n1/2m. As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.
AB - Let {pn(x)}n≥0 be the set of orthonormal polynomials with respect to the exponential weight w(x) = e-v(x), where v(x) = x2m + ⋯ is a monic polynomial of degree 2m with m ≥ 2 and is even. An asymptotic approximation is obtained for pn(x), as n → ∞, which holds uniformly for 0 ≤ x ≤ O(n1/2m. As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.
KW - Exponential weight
KW - Orthogonal polynomials
KW - Turning point
KW - Uniform asymptotic approximation
KW - Zeros
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U2 - 10.1137/S0036141098344671
DO - 10.1137/S0036141098344671
M3 - RGC 21 - Publication in refereed journal
VL - 31
SP - 992
EP - 1029
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
SN - 0036-1410
IS - 5
ER -