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'ĂȘtre 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 )



Possibly Related Threads…
Thread Author Replies Views Last Post
  [special] binary partition zeta function tommy1729 1 72 02/27/2023, 01:23 PM
Last Post: tommy1729
  bounded derivatives and semi-group iso ( repost ?? ) tommy1729 3 103 02/23/2023, 12:12 AM
Last Post: tommy1729
  The semi-group iso problem and bounded derivatives tommy1729 3 382 12/07/2022, 09:26 PM
Last Post: tommy1729
  1st iterated derivatives and the tetration of 0 Xorter 0 3,550 05/12/2018, 12:34 PM
Last Post: Xorter
  An intresting equation ? Taking squares by equation. tommy1729 0 3,699 05/08/2015, 11:37 PM
Last Post: tommy1729
  Binary partition at oo ? tommy1729 2 7,013 10/07/2014, 07:22 PM
Last Post: tommy1729
  Closed-form derivatives andydude 7 14,683 09/03/2009, 04:02 AM
Last Post: andydude
  q concerning derivatives Gottfried 0 4,294 03/19/2008, 08:28 AM
Last Post: Gottfried

Users browsing this thread: 1 Guest(s)