Is there a simplification for the following recursive fraction : (n/(W(n)))/(W(n/(W(n))))/(W(n/(W(n)))/(W(n/(W(n))))) The above formula uses a recursion 3 times. I'm looking for a simplification when we have such a finite recursion, for instance when this one appears i times. I would like to remove the recursion, i.e. obtain a single fraction. Thank you.

Karley Castillo

Karley Castillo

Answered question

2022-11-05

Simplifying a recursive fraction involving Lambert function
Is there a simplification for the following recursive fraction :
n W ( n ) W ( n W ( n ) ) W ( n W ( n ) W ( n W ( n ) ) )
The above formula uses a recursion 3 times. I'm looking for a simplification when we have such a finite recursion, for instance when this one appears i times. I would like to remove the recursion, i.e. obtain a single fraction.
Thank you.

Answer & Explanation

mignonechatte00f

mignonechatte00f

Beginner2022-11-06Added 13 answers

Since u = W ( u ) e W ( u ) , it follows that u W ( u ) = e W ( u ) and we get
n W ( n ) W ( n W ( n ) ) W ( n W ( n ) W ( n W ( n ) ) ) = e W ( e W ( e W ( n ) ) )
Which is the best simplification I can see.

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