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How to solve this functional (recurrence) equation ?

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jk22
#1
Nov16-13, 10:18 AM
P: 159
I'm in a problem where I have to solve the following functional equation :

[tex]F(n)^2=n+F(n+1)[/tex]

Does anyone know some methods to solve this kind of problems ?

A similar equation happens in Ramanujan example of root denesting : http://en.wikipedia.org/wiki/Nested_...l#Square_roots
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mathman
#2
Nov16-13, 03:56 PM
Sci Advisor
P: 6,057
Don't know of a method. As stated the problem is incomplete - you need an initial condition (F(0) = ?).
jk22
#3
Nov17-13, 02:32 AM
P: 159
The problem is to find F(1), knowing that [tex]F(1)=\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}}[/tex].

Mentallic
#4
Nov17-13, 05:42 AM
HW Helper
P: 3,531
How to solve this functional (recurrence) equation ?

Quote Quote by jk22 View Post
The problem is to find F(1), knowing that [tex]F(1)=\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}}[/tex].
Should that be to find F(n) given that [itex]F(1)=\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}}[/itex] ?

Find F(2) and the pattern becomes clear.


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