A family of nonlinear difference equations: existence, uniqueness, and asymptotic behavior of positive solutions

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We study solutions (xₙ)ₙ ∈ ₙ of nonhomogeneous nonlinear second order difference equations of the type ℓₙ = xₙ (σₙ,₁ xₙ₊₁ + σₙ,₀ xₙ + σₙ,₋₁ xₙ₋₁) + κₙ xₙ, with given initial data x₀ ∈ R, x₁ ∈ R+ where (ℓₙ)n ∈ R+, (σₙ,₀)n ∈ R+ and (κₙ)n ∈ R and the left and right σ-coefficients satisfy either (σₙ,₁)n ∈ R+ and (σₙ,₋₁)ₙ∈ ₙ ∈ R+ or (σₙ,₁)n ∈ R+₀ and (σₙ,₋₁)n ∈ R+₀. Depending on one's standpoint, such equations originate either from orthogonal polynomials associated with certain Shohat-Freud-type exponential weight functions or from Painlevé's discrete equation #1.