How to solve this functional (recurrence) equation ?

In summary, the conversation revolved around solving the functional equation F(n)^2=n+F(n+1), specifically using the example of Ramanujan's nested radical. The conversation also mentioned the need for an initial condition to solve the problem, and suggested finding F(2) to reveal a pattern.
  • #1
jk22
729
24
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_radical#Square_roots
 
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  • #2
Don't know of a method. As stated the problem is incomplete - you need an initial condition (F(0) = ?).
 
  • #3
The problem is to find F(1), knowing that [tex]F(1)=\sqrt{1+\sqrt{2+\sqrt{3+\ldots}}}[/tex].
 
  • #4
jk22 said:
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.
 
  • #5
I would approach this problem by first understanding the nature of functional equations and their solutions. Functional equations are equations in which the unknown quantity is a function, rather than a variable. These types of equations are commonly used in mathematics and physics to describe relationships between variables that change over time or space.

To solve this specific functional equation, we can use a variety of methods such as substitution, iteration, or solving for specific values. One approach would be to first substitute n=0 and n=1 into the equation to see if any patterns emerge. This can give us a starting point for finding a general solution.

Another method would be to use iteration, where we plug in values for n and use the resulting value to find the next value until a pattern emerges. This method can be time-consuming, but it can also provide insights into the behavior of the function.

In addition, we can use algebraic manipulation techniques to transform the equation into a more manageable form. For example, we could try to isolate F(n) on one side of the equation and manipulate the other terms accordingly.

Moreover, we can also use techniques such as induction or proof by contradiction to prove the existence of a solution or to rule out certain possibilities.

In cases where the functional equation has a specific application, we can also use numerical methods such as computer simulations or graphing to approximate the solution.

It is also worth mentioning that this type of functional equation is closely related to the concept of nested radicals, as mentioned in the Ramanujan example. This can provide additional insights and techniques for solving the equation.

In conclusion, there are various methods and techniques that can be used to solve functional equations, and it may require a combination of these approaches to find a solution. It is important to carefully analyze the equation and use the appropriate tools to find a solution that satisfies the given conditions.
 

1. How do I approach solving a functional (recurrence) equation?

First, identify the type of recurrence equation it is (linear, quadratic, etc.) and determine the initial conditions. Then, use mathematical techniques such as substitution, iteration, and manipulation to find a general solution.

2. What is the difference between a functional and a non-functional recurrence equation?

A functional recurrence equation involves finding a function that satisfies the equation, while a non-functional recurrence equation involves finding a specific value or sequence that satisfies the equation.

3. Can I use a computer program or online calculator to solve a functional recurrence equation?

Yes, there are many programs and calculators available that can solve functional recurrence equations. However, it is still important to understand the mathematical concepts behind the solution.

4. What are some common techniques for solving functional recurrence equations?

Some common techniques include using generating functions, solving for the characteristic equation, and using the method of undetermined coefficients.

5. Are there any real-world applications for functional recurrence equations?

Yes, functional recurrence equations can be used to model various natural phenomena such as population growth, finance, and physics. They are also commonly used in computer science and programming to create efficient algorithms.

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