Let $x(1) = 1, x(k+1) = ax(k)+ bk, y(k) = -1 +a^{-k}x(k)$. This recursion seems quite intractable, but it has some interesting features and it is much more friendly than it looks at first glance. Here $a, b$ are two positive parameters, with $a>1$. Let $g(a,b) =\lim_{k\rightarrow\infty} y(k)$.
It is probably easy to establish that $g(a, b) = b f(a)$ where $f(a)$ is a smooth function. Values of $f(a)$ for various $a$'s are pictured below. It is possible that if $a$ is a rational number, then $f(a)$ is also a rational number? This happens frequently. More specifically, my questions are:
Questions
- Find a good approximation for $f(a)$.
- Or can you find an exact formula for $f(a)$?
- Is $f(8)$ a rational number?
The last question is very, very important to me as it has potential applications in finding a standard mathematical constant which is a normal number. My wish is that $f(8)$ is not a rational number, though it seems at first glance that the contrary should be true, unfortunately.
RSolve[{x[k+1]==a x[k]+b k,x[1]==1},x[k],k]
. $\endgroup$