Iteration by Ramanujan

[Gottfried]

Hi, just today I found this msg in the sci.math newsgroup, which may be of interest here. Especially the second function of Ramanujan, which combines (a somehow inverse to) Andrew's E()-function and the ("Tetra-")series of increasing heigths. Maybe, you like this

Gottfried

subject: yeah sure!!
author: galathaea@gmail.com

Code:

i've already pointed out in this thread
that ramanujan worked on continuous iteration in his quarterly reports

these were written between august 5th, 1913
  and march 9th, 1914

ramanujan actually expands the notion of iteration
  into a power series

            oo            j
           ---  psi (x)  n
           \       j
(f)^n(x) = /    -----------
           ---      (1)
           j=0         j

where
  because n could be any value in the convergence radius
there is a potential continuous definition

but ramanujan was by no means the first either

i've also mentioned comtet's book
  which even berndt's coverage recommends
to put this on a rigorous foundation

this has been around since before euler

fractional differentiation
  for instance
was developed from several different transform approaches
from the very early transform studies

it's natural that if
  (-ik)^n corresponds to n-th differentiation
  in the transform language
  then there is a clear generalisation of differentiation
    that allows real orders

..

to show some of the other cool things in ramanujan's reports
and to connect to the tetration threads
  ramanujan studies

                         x
                  x     e
            x    e     e
           e    e     e
f(x) = 1 + -- + --- + ---- + ...
            3     4      5
           2     3      4
                2      3
                      2

ramanujan shows that this function is enitre
and yet grows faster than any
      x
     .
    .
   e
  e
finitely iterated exponential


(...)
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar

[bo198214]

Code:

i've already pointed out in this thread
that ramanujan worked on continuous iteration in his quarterly reports

these were written between august 5th, 1913
  and march 9th, 1914

ramanujan actually expands the notion of iteration
  into a power series

            oo            j
           ---  psi (x)  n
           \       j
(f)^n(x) = /    -----------
           ---      (1)
           j=0         j

where
  because n could be any value in the convergence radius
there is a potential continuous definition

but ramanujan was by no means the first either

i've also mentioned comtet's book
  which even berndt's coverage recommends
to put this on a rigorous foundation

this has been around since before euler

fractional differentiation
  for instance
was developed from several different transform approaches
from the very early transform studies

it's natural that if
  (-ik)^n corresponds to n-th differentiation
  in the transform language
  then there is a clear generalisation of differentiation
    that allows real orders

..

to show some of the other cool things in ramanujan's reports
and to connect to the tetration threads
  ramanujan studies

                         x
                  x     e
            x    e     e
           e    e     e
f(x) = 1 + -- + --- + ---- + ...
            3     4      5
           2     3      4
                2      3
                      2

ramanujan shows that this function is enitre
and yet grows faster than any
      x
     .
    .
   e
  e
finitely iterated exponential

Wow thats amazing, can you give any references?

@Gottfried: can you point galthaea to our forum?

[Gottfried]

@Gottfried: can you point galthaea to our forum?

Henryk - I'll try to send him a personal mail; hope his adress is valid.

Gottfried

[Ivars]

Are those 2^3,2^3^4, 2^3^4^5... = 2^3^4^...n what Andrew calls E factorial as E(n) ? Does it have generalization to x? Obviously they can not start at 1. The smallest integer is 2, unlike ordinary factorial.

What if its turned around, so 2^1, 3^2^1, 4^3^2^1, 5^4^3^2^1.. n^..3^2^1. It is also a fast growing number. x^(x-1)^(x-2)..1, but much slower then the other.

This slower one has been called exponential factorial:

Exponential Factorial Wolfram MathWorld

It is given by recurence relation:

\( a_n=n^{a_{n-1}} \)
\( a_1=1 \)

Ramanujan's factorial would be bigger.


Do I understand right that by applying some transformation involving such factorials the summation of many divergent series can be brought to some sort of convergence-if their speed of growth is slower than these factorials?

Then these perhaps can be applied to power series of extremely slow functions directly, like e.g. 1/h(z).

Ivars

[andydude]

First of all, that is not the exponential factorial. The EF is (5^4^3^2) whereas this is (2^3^4^5). Secondly, I found this on JSTOR, so I'm going to make a trip to the local library soon...

Andrew Robbins

[Ivars]

Secondly, I found this on JSTOR, so I'm going to make a trip to the local library soon...

Andrew Robbins

One place Ramanujan considers infinite exponentials is Notebook 5:
At Amazon.com

The pages 490-492 which speaks about convergence criteria for iterated exponentials can be read there by LookInside, but do not contain the formulas mentioned by Gottfried.

Ivars

[Gottfried]

mentioned by Gottfried.

... just cited. I don't know anything about them (don't have JSTOR-access either). Source is the poster galaethea...

[andydude]

One place Ramanujan considers infinite exponentials is Notebook 5:
At Amazon.com

Yes indeed, this book can also be found on http://scholar.google.com/ but sadly they require money or something for pages 410 and 490 which are where Ramanujan's iterated exponential formulas are... but luckily the surrounding pages talk about how Bachman recently proved this formula to be true. Bachman's article can be found here.

Andrew Robbins

[bo198214]

...but luckily the surrounding pages talk about how Bachman recently proved this formula to be true.

Haha, ya Ramanujan was the guy who wildly wrote down a lot a formulas, which mostly could be shown to be true.