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I have tried many things, the relation is in the following picture, x' is a dummy variable.1

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    $\begingroup$ Welcome to Mathematica.SE! Please take a minute to read the tour. As written, this question is not answerable: it is not clear what task you want to accomplish (or what you mean by "write in Mathematica"). Please take a look at the existing highly-voted questions, and model yours on then: use a concise descriptive title. Clearly explain the task you want to solve, what you have done so far, and what difficulty you encountered. Please edit the question and make the necessary improvements (do not use comments to add critical information). $\endgroup$
    – Szabolcs
    Jun 29 at 14:36
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Something like this should work:

f[n_] := 
   f[n] = 
     Function[
       {x}, 
       Evaluate@Integrate[f[n - 1][x - x1] f[1][x1], {x1, 0, x}]
     ]

For example after specifying

 f[1] = Function[{x}, x]

f[10] gives

 x^19/121645100408832000

Of course, if f[1] is so complicated that Mathematica cannot do some of the convolutions explicitly, this will produce a mess.

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  • $\begingroup$ Thank you very much for your time !! It seems to work $\endgroup$
    – pedro
    Jun 29 at 14:59
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Another possibility is to use Convolve:

Clear[f]
f[n_] := f[n] = Function[
    y,
    Evaluate @ Convolve[f[n-1][x] UnitStep[x], f[1][x] UnitStep[x], x, y, Assumptions->y>0]
]
f[1] := Function[x, x]

Multiplication by UnitStep changes the convolution to be over a finite interval (by default, Convolve uses an infinite interval).

Then, we reproduce @mmeent's answer:

f[10][x]

x^19/121645100408832000

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  • $\begingroup$ thank you very much. Very helpful ! $\endgroup$
    – pedro
    Jun 29 at 23:14

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