See my question at the bottom of this post. The recurrence $P(n) x_{n+2} = Q(n)x_{n+1} - R(n)x_n$, where $P(n), Q(n), R(n)$ are polynomials of degree $1$, sometimes leads to interesting results. Probably the most basic cases are:
For $\log\alpha$:
$$P(n) = \alpha (n+2), Q(n) = (2\alpha-1)(n+1)+\alpha, R(n)=(\alpha-1)(n+1)$$ $$\mbox{with } x_1=\frac{\alpha-1}{\alpha}, x_2 = \frac{(\alpha-1) (3\alpha-1)}{2\alpha^2}$$
We have $\lim_{n\rightarrow\infty} x_n = \log\alpha$. The convergence is fastest when $\alpha$ is close to $1$. The related recurrence $$P(n) = 1, Q(n) = (2\alpha-1)(n+1)+\alpha, R(n)=(\alpha-1)\alpha(n+1)^2$$ $$\mbox{with } x_1=\alpha-1, x_2=(\alpha-1)(3\alpha-1)$$ yields $$\lim_{n\rightarrow\infty} \frac{x_n}{\alpha^n n!} = \log\alpha$$ and in addition $x_n$ is an integer if $\alpha>0$ is an integer.
For $\exp \alpha$:
$$P(n) = n+2, Q(n) = n+2+\alpha, R(n)=\alpha$$ $$\mbox{with } x_0=1, x_1 = 1+\alpha$$
We have $\lim_{n\rightarrow\infty} x_n = \exp\alpha$. The related recurrence $$P(n) = 1, Q(n) = n+2+\alpha, R(n)=\alpha(n+1)$$ $$\mbox{with } x_0=1, x_1=1+\alpha$$ yields $$\lim_{n\rightarrow\infty} \frac{x_n}{n!} = \exp\alpha$$ and in addition $x_n$ is an integer if $\alpha$ is an integer.
For $\sqrt{2}$:
$$P(n) = 4(n+2), Q(n) = 6n+11, R(n)=2n+3$$ $$\mbox{with } x_0=1, x_1 = \frac{5}{4}$$
We have $\lim_{n\rightarrow\infty} x_n = \sqrt{2}$. The related recurrence $$P(n) = 2(n+2), Q(n) = 4(6n+11), R(n)=32(2n+3)$$ $$\mbox{with } x_0=1, x_1=10$$ yields $$\lim_{n\rightarrow\infty} \frac{x_n}{8^n} = \sqrt{2}$$ and in addition $x_n$ is an integer.
Comment
These formulas (and tons of other similar formulas) are easy to obtain, yet I could not find any reference in the literature. It would be interesting to see if one is available for $\gamma$ (the Euler Mascheroni constant), but I don't think so. Also, what happens when you change the initial conditions? What if you replace the recurrence by its equivalent differential equation, for instance $$(x+2) f(x) - (x+2+\alpha) f'(x) + \alpha f''(x) =0$$ corresponding to the case $\exp\alpha$?
How to obtain these recursions?
The case $\sqrt{2}$ can be derived from this other question. To me, it is the most interesting case as it allows you to study the digits of $\sqrt{2}$ in base 2. Some of these recursions can be computed with WolframAlpha, see here for the exponential case, and here for $\sqrt{2}$. Numerous other recurrences, with much faster convergence, can be derived from combinatorial sums featured in this WA article.
My question
I am looking for some literature on these linear, non-homogeneous second order recurrences involving polynomials of degree $1$. Also, I will accept any answer for a recurrence that yields $\pi$. Should be easy, using formulas (37) or (38) in this article as a starting point.
from Hot Weekly Questions - Mathematics Stack Exchange
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