TY - JOUR
T1 - Asymptotic expansions for second-order linear difference equations with a turning point
AU - Wang, Z.
AU - Wong, R.
PY - 2003/3
Y1 - 2003/3
N2 - A turning-point theory is developed for the second-order difference equation Pn+1 (x) - (Anx + Bn) Pn(x) + Pn-1(x) = 0, n = 1, 2, 3, ⋯, where the coefficients A n and Bn have asymptotic expansions of the form A n ∼ n-θ Σs=0
∞ αs/ns and Bn ∼ Σ s=0
∞ βs/ns, θ ≠ 0 being a real number. In particular, it is shown how the Airy functions arise in the uniform asymptotic expansions of the solutions to this three-term recurrence relation. As an illustration of the main result, a uniform asymptotic expansion is derived for the orthogonal polynomials associated with the Freud weight exp(-x4), x ∈ ℝ.
AB - A turning-point theory is developed for the second-order difference equation Pn+1 (x) - (Anx + Bn) Pn(x) + Pn-1(x) = 0, n = 1, 2, 3, ⋯, where the coefficients A n and Bn have asymptotic expansions of the form A n ∼ n-θ Σs=0
∞ αs/ns and Bn ∼ Σ s=0
∞ βs/ns, θ ≠ 0 being a real number. In particular, it is shown how the Airy functions arise in the uniform asymptotic expansions of the solutions to this three-term recurrence relation. As an illustration of the main result, a uniform asymptotic expansion is derived for the orthogonal polynomials associated with the Freud weight exp(-x4), x ∈ ℝ.
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U2 - 10.1007/s00211-002-0416-y
DO - 10.1007/s00211-002-0416-y
M3 - RGC 21 - Publication in refereed journal
VL - 94
SP - 147
EP - 194
JO - Numerische Mathematik
JF - Numerische Mathematik
SN - 0029-599X
IS - 1
ER -