Linear difference equations with a transition point at the origin

A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation Pn+1(x) ? (Anx + Bn)Pn(x) + Pn?1(x) = 0, where A_n and B_n have asymptotic expansions of the form An ∼ n-o ∞ θ s=o αs/ns, Bn ∼ ∼ n-o ∞ θ s=o βs/ns with θ ≠ 0 and A₀ ≠ 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 J_ν and Y_ν 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 x ^Aexp(-q_mx^m), x > 0, where m is a positive integer, A > -1 and q_m > 0.