# Im trying to write this in Mathematica. It is a recurrence relation for convolution [duplicate]

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

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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.

• Thank you very much for your time !! It seems to work Jun 29 at 14:59

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).

f[10][x]


x^19/121645100408832000

• thank you very much. Very helpful ! Jun 29 at 23:14