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4
questions
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votes
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answers
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Asymptotics for this limit iteration with $f(x)= 2x + x^5 ,g(x) = x + x^3$
Consider $x>0$
Let
$$f(x)= 2x + x^5$$
$$g(x) = x + x^3$$
$$f(r(x))=r(f(x))=id(x)$$
$$g(u(x)) = u(g(x))=id(x)$$
Where $id(x)$ is the identity function mapping all values to itself.
Let $*^{[y]}$ ...
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1
vote
0
answers
72
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Fibonacci like sequence $f(n) = f(n-1) + f(n-2) + f(n/2)$ and closed form limits?
Consider
$$f(1) = g(1) = 1$$
$$f(2) = A,g(2) = B$$
$$f(3) = 1 + A,g(3) = 1+B$$
And for $n>3$ :
$$f(n) = f(n-1) + f(n-2) + f(n/2)$$
$$g(n) = g(n-1) + g(n-2)$$
where we take the integer part of the ...
- 16.4k
1
vote
0
answers
77
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About $\sqrt[k]{l + \sqrt[k]{l + \sqrt[k]{l + ...}}} $ asymptotics
Consider simple nested radicals
More precisely Let
$$ K > 1 , 1 \leq l $$
$$X(j,K) = X_\infty(j,K)$$
$$X_0(j,K) = a(j,K)$$
$$X_n(j,K) = \sqrt[k]{j + X_{n-1}(j,K)}$$
$$Y(j,K) = \frac{j + X_{\...
- 16.4k
9
votes
2
answers
653
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Integral $S_\ell(r) = \int_0^{\pi}\int_{\phi}^{\pi}\frac{(1+ r \cos \psi)^{\ell+1}}{(1+ r \cos \phi)^\ell} \rm d\psi \ \rm d\phi $
Is there a closed form for $|r|<1$ and $\ell>0$ integer?
The solution for the special cases $\ell=2$ and $4$ would also be interesting if the general case is not available.
Integrating ...
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