Iteration by Ramanujan

05/27/2008, 08:12 AM
Post: #1




Iteration by Ramanujan
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 Gottfried Helms, Kassel 

05/27/2008, 08:23 AM
Post: #2




RE: Iteration by Ramanujan
galthaea Wrote: Wow thats amazing, can you give any references? @Gottfried: can you point galthaea to our forum? 

05/27/2008, 09:39 AM
Post: #3




RE: Iteration by Ramanujan
bo198214 Wrote:@Gottfried: can you point galthaea to our forum?Henryk  I'll try to send him a personal mail; hope his adress is valid. Gottfried Gottfried Helms, Kassel 

05/27/2008, 11:03 AM
(This post was last modified: 05/27/2008 03:37 PM by Ivars.)
Post: #4




RE: Iteration by Ramanujan
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^(x1)^(x2)..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: 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 convergenceif 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 

05/27/2008, 06:23 PM
Post: #5




RE: Iteration by Ramanujan
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 

05/27/2008, 08:38 PM
Post: #6




RE: Iteration by Ramanujan
andydude Wrote:Secondly, I found this on JSTOR, so I'm going to make a trip to the local library soon... One place Ramanujan considers infinite exponentials is Notebook 5: At Amazon.com The pages 490492 which speaks about convergence criteria for iterated exponentials can be read there by LookInside, but do not contain the formulas mentioned by Gottfried. Ivars 

05/27/2008, 09:35 PM
Post: #7




RE: Iteration by Ramanujan
Ivars Wrote:mentioned by Gottfried. ... just cited. I don't know anything about them (don't have JSTORaccess either). Source is the poster galaethea... Gottfried Helms, Kassel 

05/28/2008, 03:47 AM
Post: #8




RE: Iteration by Ramanujan
Ivars Wrote:One place Ramanujan considers infinite exponentials is Notebook 5: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 

05/28/2008, 06:52 AM
Post: #9




RE: Iteration by Ramanujan
andydude Wrote:...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. 

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