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There is no closed form except for the cases $a=0,1$. But you can find the asymptotic behavior. See, for example Fatou, Sur les equations fonctionnelles, Bull Soc. Math. France, 47 (1919), section 8 a …

Let $$f(z)=\sum_{n=0}^\infty a_nz^n$$ be the generating function of your sequence. Put $$\phi(z)=\sum_{n=0}^\infty b_nz^n=\sum_{n=0}^\infty z^{2^n}=z+z^2+z^4+\ldots,$$ where $b_n=1$ when $n$ is a pow …

Let us define the sequence $y_n$ by $y_0=|x_0|$, $y_{n+1}=ay_n+by_n^2$. Then we have $|x_n|\leq y_n$ for all $n$, since $ay+by^2$ is increasing for positive $a,b$, and it is enough to estimate the pos …

In the modern approach, this problem belongs to the dynamics of quadratic polynomial $f(z)=z-z^2/4$. You are interested in the particular orbit which begins at $z_0=1$. Polynomial $f$ has one (non-deg …

This type of equations can be solved in terms of "factorial series", as explained in the book: N. E. Nørlund, Leçons sur les équations linéaires aux differences finies, Paris, Gauthier-Villars, 192 …

Same method as for scalar equations works. Put matrices $M_{n+k-1},M_{n+k-2},...,M_{n}$ vertically into a big matrix $X_n$ of size $mk\times n$. Then your recurrence becomes a one-step recurrence $X_{ …

In general, it is not so. For example, orthogonal polynomials satisfy a second order linear ODE if and only if they are "classical orthogonal polynomials", MR0826863 Duistermaat, J. J.; Grünbaum, F. A …