Abstract
A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation [Formula: see text] where An and Bn have asymptotic expansions of the form [Formula: see text] with θ ≠ 0 and α0 ≠ 0 being real numbers, and β0 = ±2. Our result holds uniformly for the scaled variable t in an infinite interval containing the transition point t1 = 0, where t = (n + τ0)-θx and τ0 is a small shift. In particular, it is shown how the Bessel functions Jν and Yν get involved in the uniform asymptotic expansions of the solutions to the above linear difference equation. As an illustration of the main result, we derive a uniform asymptotic expansion for the orthogonal polynomials associated with the Laguerre-type weight xα exp (-qmxm), x > 0, where m is a positive integer, α > -1 and qm > 0.
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