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Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation yn+1(x)− (Anx+ Bn)yn(x) + yn−1(x) = 0, where An and Bn have power series expansions of the form An ∼ ∞ ∑ s=0 αs ns , Bn ∼ ∞ ∑ s=0 βs ns with α0 = 0. Our results hold uniformly for x in an infinite interval containing the transition point x+ given by(More)
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dψ (x) = (k + α)k−1e−k k! at x = xk = ±(k + α)−1/2, k = 0, 1, 2, . . . . In this paper,we derive an asymptotic expansion for f (α) n (t/ √ ν) as n → ∞, valid uniformly for bounded real t , where ν = n + 2α − 1/2 and α is a positive parameter. The validity for bounded t can be extended to unbounded t by using a sequence of rational functions introduced by(More)
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