What is the general method for finding the aymptotics of large $n$ of the sequence $(a\_n)\_{n=0}^\infty$ defined by the recursion
$$a\_{n} = (\alpha\_1n+\alpha\_2) a\_{n-1} + (\alpha\_3n+\alpha\_4) a\_{n-2}+\delta \tag1$$
where $\alpha\_i$'s are constant real numbers and $\delta\in\{0,1\}$ is constant.


Here is [an example](https://math.stackexchange.com/q/3386637/64809) of the above recursion and my frustrated attempt at using the generating function for the un-simplified version. That particular example is solvable with a generating function once [it is transformed](https://math.stackexchange.com/a/3389236/64809). However, not every recursion of the form $(1)$ can be simplified through a simple transformation.



Asymptotics of the general second order affine recursion
I do not know about *the* general method, but here's a recipe for $\delta = 0$. 


One can view this as a 3-term recurrence relation, and consider the corresponding orthogonal polynomials. [Favard's Theorem](https://en.wikipedia.org/wiki/Favard%27s_theorem) guarantees such polynomials exist, and in fact [Riesz Representation Theorem](https://en.wikipedia.org/wiki/Riesz_representation_theorem) upgrades the orthogonality to the usual one if you ensure $\alpha\_3 n + \alpha\_4 < 0$. This transfers the problem to a problem to a problem about asymptotics of orthogonal polynomials, which can be done using multiple methods (WKB, Riemann-Hilbert, amongst many others).


**Example:** 


Consider $\alpha\_1 = 0, \ \alpha\_2 = 2, \ \alpha\_3 = -2, $ and $\alpha\_4 = 2$. Then, assuming no algebra mistakes, we have (I re-index so I can use Wikipedia)
$$
a\_{n + 1} = 2a\_n - 2na\_{n - 1}, \ a\_0 = 1, \ a\_2 = 2
$$
This recurrence is satisfied by Hermite polynomials $H\_n(x)$ (evaluated at $x = 1$), which satisfy 
$$
\int\_{-\infty}^{\infty} x^k H\_n(x) \ e^{x^2} = 0 \quad \text{ for } k = 0, 1, ..., n-1.
$$
and 
$$
H\_{n + 1}(x) = 2xH\_n(x) - 2nH\_{n - 1}(x)
$$
Now, again the wikipidia page tells me that (this I did not check too carefully, but I know asymptotic expansions of Hermite Polynomials exist)
$$
a\_n = H\_n(1) \sim e^{1/2} \dfrac{2^n}{\sqrt{\pi}} \Gamma\left( \dfrac{n + 1}{2} \right) \cos \left( \sqrt{2n} - \dfrac{n \pi}{2} \right).
$$


Now, in the general situation, a glance at Chihara's book "[Introduction to Orthogonal Polynomials](https://rads.stackoverflow.com/amzn/click/com/0486479293)" (Pages 215 - 217) tells me that for recurrences of the form you give will often be handled using Hermite and Charlier polynomials (maybe others if you get funny about what $x$ value you consider). 


P.S. I am new to this, so if this needs downgraded to a comment or is somehow not what you wanted, my bad!


