sequence of functions

• To: mathgroup at smc.vnet.net
• Subject: [mg120110] sequence of functions
• From: rych <rychphd at gmail.com>
• Date: Fri, 8 Jul 2011 04:55:33 -0400 (EDT)

```I would like to build a function sequence inductively, for example
f[n][x]=f[n-1][2x]+f[n-1][2x-1], f[0][x] = x(1-x)

This is what I tried in Mathematica,

ClearAll["Global`*"]
f[n_] = Function[x, Simplify[f[n - 1][2 x] + f[n - 1][2 x - 1]]];
f[0] = Function[x, x (1 - x)];

f[2][x]
Out:
-20 + 64 x - 64 x^2

?f
Out:
Global`f
f[0]=Function[x,x (1-x)]
f[n_]=Function[x,Simplify[f[n-1][2 x]+f[n-1][2 x-1]]]

My questions: it doesn't seem to cache the previous f[n-1] etc? If I
f[n_] := f[n] = Function[...

Then it does cache the f[n-1] etc. but still in the unevaluated form,
Global`f
f[0]=Function[x,x (1-x)]
f[1]=Function[x\$,Simplify[f[1-1][2 x\$]+f[1-1][2 x\$-1]]]
f[2]=Function[x\$,Simplify[f[2-1][2 x\$]+f[2-1][2 x\$-1]]]
f[n_]:=f[n]=Function[x,Simplify[f[n-1][2 x]+f[n-1][2 x-1]]]

How do I force the reduction so that I see f[1] = -2 (1 - 2 x)^2, etc.
in the definitions?

Thanks,
Igor

```

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