08/09/2014, 09:16 PM

I realised that solving the equation

f ' (2x) = 3 f(x)

can be done in the same way as the equation f ' (2x) = f(x) or equivalent f ' (x) = f(x/2).

Maybe that leads to a closed form solution too.

( closed form here means expressible as standard functions , sums and integrals )

---

As for the equations

f(x) - f(x-1) = f(x/2)

or

f(x+1) - f(x) = f(x/2)

and Jay's comments ; Im aware of the weaknesses of the equations.

I was just trying to find a better approximation then the equation f ' (x) = f(x/2) gives.

Let A(x) be one of the Original sequences and let T(x) be the nonconstant analytic solution to T(x) - T(x-1) = T(x/2).

Then I think one of the A(x) satisfies

A(x) - T(x) = O(T(x)^C)

where 0<C<1

and also

lim x-> +oo A(x)/T(x) = 1.

---

I think I have seen 2d cellular automaton that give the Original sequences or approximated the functional equations given here.

I believe they came from Wolfram and Cook but Im not sure.

Anyone else who might recall this ?

---

As for J(x) <=> J ' (x) = J(x/2)

I think I know a way to prove lim x-> +oo A(x)/J(x) = 1.083...

But what really intrests me is a closed form for 1.083...

Havent I seen this number before ??

regards

tommy1729

" Choice is an illusion " tommy1729

f ' (2x) = 3 f(x)

can be done in the same way as the equation f ' (2x) = f(x) or equivalent f ' (x) = f(x/2).

Maybe that leads to a closed form solution too.

( closed form here means expressible as standard functions , sums and integrals )

---

As for the equations

f(x) - f(x-1) = f(x/2)

or

f(x+1) - f(x) = f(x/2)

and Jay's comments ; Im aware of the weaknesses of the equations.

I was just trying to find a better approximation then the equation f ' (x) = f(x/2) gives.

Let A(x) be one of the Original sequences and let T(x) be the nonconstant analytic solution to T(x) - T(x-1) = T(x/2).

Then I think one of the A(x) satisfies

A(x) - T(x) = O(T(x)^C)

where 0<C<1

and also

lim x-> +oo A(x)/T(x) = 1.

---

I think I have seen 2d cellular automaton that give the Original sequences or approximated the functional equations given here.

I believe they came from Wolfram and Cook but Im not sure.

Anyone else who might recall this ?

---

As for J(x) <=> J ' (x) = J(x/2)

I think I know a way to prove lim x-> +oo A(x)/J(x) = 1.083...

But what really intrests me is a closed form for 1.083...

Havent I seen this number before ??

regards

tommy1729

" Choice is an illusion " tommy1729