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On the binary partition and taking derivatives
Recently we talked about the binary partition function

f(n) - f(n-1) = f(n/2).

And Jay's asymptotic

J ' (x) = J(x/2).

This leads to a general question :
" how to take a derivative of an unsolved equation ? "

I will clarify with the binary partion function as example :

a >= h

where h is the positive infinitesimal.
( I will use 0 for h later , use lim interpretation )

(f_a(x) - f_a(x-a)) / a = f_a(x/2)

The questions are , without solving for the f_a(x) first with respect to x ( asymptoticly ) ,

df/da f_a(x) = ??

f_a(x)/f_(a-1)(x) = ??

df/da f_a(x)/f_(a-1)(x) = ??

and similar ones.

A good techniques for such problems should exist.

Notice Jay conjectured

f_1(x)/f_0(x) ~ C

And with good approximations of both f_0,f_1 that should be easy to prove.

But to show it directly is the goal.

Hence the reason d'être of this thread and its questions.

It is my philosophy of math , that math Always tries to shortcut everything.

Multiply and divide are shortcuts to addition.
Finding shortcuts to addition and multiplication is a motivation for linear algebra and dynamical systems.

Shortcuts to matrix powers led to diagonalization and Jordan forms etc etc.
Asymptotic Shortcuts to counting primes led to PNT.

Series acceleration is another example.

I think you get the idea.
( not having a known " shortcut " ( for computation ) leads to difficult problems in math , for instance collatz conjecture.
I like to count the difficulty of a math problem in terms of unknown shortcuts vs known shortcuts related to the question )



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