MHB Looking for a recursion relation

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    Recursion Relation
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The discussion centers on a specific non-linear recursion relation, f(n + 1) = 2 - d(n)/f(n), where d(n) is arbitrary. The equation may lack a closed-form solution due to its non-linearity, leading to challenges in finding general conclusions. Alternative representations include a non-linear difference equation and a continued fraction approach. Participants agree that while a general solution is elusive, numerical methods could provide insights. The equation remains a complex and frustrating problem for those involved.
topsquark
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I don't know how to do a search for information on a specific equation. It's [math]f(n + 1) = 2 - \dfrac{d(n)}{f(n)}[/math], where d(n) is more or less arbitrary. It came up in some work I've been doing and I can't seem to get anywhere with it. Being non-linear it may not even have a closed form solution. There are two other ways to look at it. It's a non-linear difference equation: [math]f \Delta f + f(f - 2) = d[/math] and it can also be considered as a continued fraction. (I'm going to be looking up that idea tonight.)

Any thoughts?

-Dan
 
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topsquark said:
Any thoughts?
Yes, I think that it is impossible to make any general conclusions as the answer depends completey on ## d(n) ##.
 
Yes, thank you. I have found (but not proven) that this equation cannot be solved in general. I haven't even found a general way to approach it. It is a very annoying little equation!

-Dan

Addendum: Well, I should say "does not have closed form solutions in general." We can always do it numerically.
 
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