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bruce berndt's release of ramanujan's notebooks
  is one of the best resources for aspiring combinalgebraicists

ramanujan really was one of the greats
  and it is a shame that books like "the man who knew infinity"
  really do not shine light on ramanujan's methods

if you really want to know what fascinated ramanujan
  bruce berndt does a much better job of describing it

          ^..^

i was recently asked to join a tetration discussion group at http://math.eretrandre.org
and it was mentioned that some of the ramanujan references i have posted may be new to the group

i just wanted to describe
  to the best of my knowledge
  after a long love affair with ramanujan's notebooks
ramanujan's interest and approach to what is now called tetration

ramanujan has a trait that i strongly admire

he likes to break things
  by looking at them bigger than they are

if it applies in one place
  and he doesn't immediately see why it can't apply everywhere
then he assumes it applies everywhere
and sees where it takes him

he gets a new tool and starts using it everywhere

ramanujan regularly took discrete relations and looked for continuous generalisations
that was one of his big contributions to number theory

in his quarterly notes
  he explores the whole nature of continuous iteration
but elsewhere he shows fascination with the exponential in particular

this is natural since the exponential is fundamental here

in chapter 3 of his notebooks
  just after having found out about lagrange inversion
(you know
  that point in the budding combinalgebraicists education
  where they learn how to invert x = y e^y
just as lambert did so many years ago)
ramanujan suddenly focuses in on

       x
x = a e        (|a| <= e)

and shows

      oo
     ---       j-1
x   \    (j+1)       j
e  = /    --------   a
     ---    (1)
     j=0       j

not much
kinda one of those exercises after learning lagrange inversion
  where you substitute ln y

but then he sees the secret to taking another iterate
looks in on

x
e  = a + x       (a >= 1)

and shows

        oo
  x    ---       j-1
e     \    (j+1)      -aj
e   =  /    --------  e
       ---    (1)
       j=0       j

that type simple and beautiful creation is always very signature of ramanujan

and of course
  the next step is the easy for him

     .
    .
   x      oo
  x      ---       j-1
x       \    (j+1)            j
x     =  /    --------  (ln(x))                |ln(x)| < 1/e
         ---    (1)
         j=0       j

at this point in the notebook
  berndt lists a huge list of resources on the history of these types of forms
including papers on the convergence of

          .
        .
      .
    x3
  x2
x1

ramanujan goes on to generalise the whole apparatus in the rest of chapter 3

by the time he gets to chapter 4
  he is ready to return to iterated exponentiation
and after defining

F (x) = x
0

F   (x) = exp{F (x)} - 1
r+1           r

he decomposes the iteration in two different ways

         oo                 oo
        ---                ---
        \             j    \           j
F (x) = /    phi (r) x   = /    f (x) r
r      ---     j          ---   j
        j=0                j=0

notice that the second sum is a series in terms of r
  which can be taken as a real number

he proves some properties about the f_j like

n f (x) = f (x) f'  (x)
   n       1     n-1

and
             oo
            ---
            \
f'(x) = x + /    B  f (x)
1          ---   n  n
            j=0

(where B_n is the nth bernoulli)

among all sorts of other beautiful theorems and evaluations of f

this expansion of iterated exponentials in continuous r
  and the subsequent discovery of many properties of the coefficient functions
is in my opinion ramanujan's largest contribution to tetration
or iterated exponentiation
or whatever you want to call it

although

http://en.wikipedia.org/wiki/Tetration

says that a complex extension of tetration has not been shown to exist
it seems straightforward
(using the expansion of f as

                  oo
         / x \n  ---
f (x) = | - |   \        j-1          j
  n      \ 2 /   /    (-1)    psi (n) x
                 ---             j
                 j=1

by ramanujan)
that the expansion in f has a positive radius of convergence in r
  for a given x < 1

berndt concludes the section with this little bibliographical note
  that i have not seen on pages from the tetration community
"I. N. Baker [1][2] has made a thorough study of iterates of entire functions
with particular attention paid to the exponential function in his second
paper.  These papers also contain references to work on iterates of
_arbitrary_complex_order_.  But we emphasise that no one but Ramanujan
seems to have made study of the coefficients phi_j(r) and f_j(x).  A
continued development of this theory appears desirable."

how is that for motivating a student?

now
  i personally don't think there is much of a mystery to tetration

i think it's actually a field with quite a history and literature
  and i've followed up on many of the leads from berndt
    (bell, carlitz, becker, riordan, ginsburg, stanley, ...)
  so i've seen some great work in the area

but i suspect there may be a community with interest in these things
who may not be aware of some of this other work

i've tried to point this out on several occasions in the past
but the recent questions i received
  from several sources
and interest by the community mentioned above
shows i may need to be more explicit

i will post this to the community forums
and can try to answer questions if any additional references are needed
  but i have not pursued tetration in my own studies

instead
  except for an early fascination with x^x, x^(-x), and inversion of y^y=x
when i've looked at iterates
i've mostly looked at iterates of other entire functions

                / |0   x \
like  G   (x) = | |   e  | - 1
     n r+1      \ |n     /

and the related lagrange inversion problems on the generalised coshinusi

where you can take ramanujan's work and immediately reapply it...

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galathaea: prankster, fablist, magician, liar

sorry for formatting and spaces eaten by forum
galathaea Wrote:but elsewhere he shows fascination with the exponential in particular

this is natural since the exponential is fundamental here

in chapter 3 of his notebooks


Hi galathaea,

Which of the notebooks chapter 3? Chapter 4? Berndt has released 5 Ramanujan notebooks. Or is it in Quaterly reports?

Ivars
Ivars Wrote:
galathaea Wrote:but elsewhere he shows fascination with the exponential in particular

this is natural since the exponential is fundamental here

in chapter 3 of his notebooks


Hi galathaea,

Which of the notebooks chapter 3? Chapter 4? Berndt has released 5 Ramanujan notebooks. Or is it in Quaterly reports?
Code:
berndt keeps the chapters progressive across the books
so the second book
  for instance
starts with chapter 10
and by the very end of book 5
  he's on chapter 39

chapters 3 and 4 are in book 1

also
  the quarterly reports
  are copied at the end of book 1

the biggest part of concentrated development here
  is in book 1 in the chapters on combinatorial sums
  (chapter 4 is half dedicated to iterated exponents)
and although the relevant forms reappear in a few other places
most of the main work is found in book 1

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
galathaea Wrote:
Code:
bruce berndt's release of ramanujan's notebooks
  is one of the best resources for aspiring combinalgebraicists

(...)

Hi galathaea -

I have seen similar formulae for the fractional indexed infinite sum sometimes, but never got into it.
I try to translate this to the current case; correct me if I'm completely false yet.
Assume, I have a function f(n) which gives

f(n)= b^^n + b^^(n+1) + b^^(n+2) +...

where thus

f(n)-f(n+1) = b^^n

is it then possible to get from this the half-iterate

f(n+1/2) - f(n+3/2) = b^^(n+1/2) ?

Is that the idea?

If this would be right, then could I use the same idea with the function

g(n) = (-1)^n*( b^^n - b^^(n+1) + b^^(n+2) -... )

which I actually have (seem to have...)?

My problem is the understanding of the general concept and technique of such an idea. I think, it is somehow similar to what Euler did, but he uses also an integral as remainder, where I had no idea how to apply this in our context.

Gottfried
Yep, found it.

The function defined as series of n times iterated exponential (with base e) terms divided by corresponding n-th hyperfactorials (2^3^4^...n....) is on p. 326-327, Berndt, Ramanujan Notebooks, Vol 1., in excerpts from Second Quaterly report. It would be interesting to see if the original contains something more.

Berndt states that it converges for every x and for every n>0 .

He does not mention entire function there.

Ivars

P.S. by adhering to Berndt's endnote of giving way to formal math, we can define by simple analogy and make conjectures:

tetra e = 1+1/2^3+1/2^3^4+ .......... = 1+1/8 + rest = probably around 1,125...because of extreme slowness.

and use it as basis for taking tetra-logarithms of iterated exponential on iteration parameter ( if we have e[4] n = f(e,4,n) then basis tetra e would be used to take tetra-log f(e,4,n) = tetra-log (n). For other bases a the value would have to be adjusted. For other n-tations we need different base for penta log etc.

Further analogies may include tetra Taylor expansion, etc.
Well I wanted to extend this while I still have the ideas in my mind.

Probably, we have to start with tetra-integers n(4) , meaning n in a[4]n(4). These integers will have :

-Non associative, non-commutative partitions (since 2(4)+3(4) = 5(4)? will not be the same as 3(4)+2(4) in terms of the result of operation a[4]5 = a[4](2(4)+3(4)) is not the same as a[4](3(4)+2(4))
-correspondingly different combinatorics, ordered
-different notion of primes , etc.

- once these rules are established, it should be possible to create :

tetra rationals Q(4)
tetra reals R(4)
tetra negative numbers and Z(4)

now, with this, it would be possible to extend the notion of equation to e.g.:

x[4]n+x[4]m = C

Which would allow to define tetra imaginary unit perhaps as solution to :

x[4]2=-1 (x^x=-1) and lead to properties of tetra complex numbers C(4).

if we have function a[4]n = f(a,4,n) obviously taking n-> infinity ends up with principially unreachable scale of infinity (tetra-infinity) if we take f(a,4,n) to infinity via taking a->infinity.

This allows to look for another notion of tetra - infinitesimal which would allow to establish tetra-differentiation and tetra-integration.

Now we can also answer the question about h(z) moving things to complex numbers from reals.

The result h(z) = complex means that from the math point of view we use, the next scale(s) of infinity ( tetra and above) shows up like imaginary for C(3) .They are however real in R(4). However, from equation x^x=-1 and generally x[n]2=-1 it is obvios that there is a structure behind this. However, what the speed of tetration does, is to push the result over the limits of infinity scale we work in with normal, exponential functions and normal integers.

Obviously, the number of discrete scales of infinity thus corresponds to the number of operations- tetra, penta, etc. We may again form operation integers based on this and ask what are their partitions ,combinatorics etc and extensions to rational, real, etc values and calculus.

So we should end up with 3 types of integers

1) Integers for numbers
2) Integers for operations
3) integers for application times of operations to numbers

Each of them will have different partition laws and combinatorics; integers for NUMBERS is clear and well known, properties of integers for operations is not known, properties for integres for number of application of operations will depend on the number of operations. If operation number is fixed, e.g tetra= 4 these combinatorics/partitions of INTEGER n for number of applications of operations will be a function of operation so, n(4).

Now we can define pentation as exponentiation in scale based on tetration and move on with finding out the laws of arithmetics of n(5), n(6) etc. and corresponding Q(5), R(5), Z(5), C(5) and look for patterns how they develop, as we have N(n), Q(n), R(n), Z(n), C(n) etc. and corresponding calculus.

If we find them, we can extend these PATTERNS of laws of arithmetics/combinatorics to rational, real, imaginary operation numbers and see what happens.

Then we look back and try to understand the whole thing together.

The first instructive thing would be to develop artithmetics and combinatorics of tetration integers n(4) = application times of tetration to some number.

I got these ideas from Ramanujans f(x) mentioned above and Hardy's "Orders of infinity", combined. I have these in djvu.

Ivars

P.S. The question about the physical essence of these things are what really interests me, so please excuse for non-rigid notions. The fine structure of imaginary unit appearing as a result of cumulative action of all scales of infinities defined above could be related to alpha, which I have tried to catch earlier, but without understanding why and what it would be shooting in the dark. "Jumping" over infinity, as tetration does , is also a very useful idea for phase changes in general (for me things always happen in general, somehow...Wink).
Based on previous, we can define tetra-logarithm as:

log[4](1+x)=1-x[4]2/2+x[4]3/3-x[4]5/4+....

and take x=(1/e)^2=0.13533528323661269189399949497248

then log[4](1+(1/e)^2)) = e^(1/e)-1=0.44466786101 (well I do not know for sure as series converge too slow for my methods).

Since log(1+(1/exp(1))^2)) = 0.126928011043

log[4] has smaller base than e.

To find it, e(4)^log[4](1+(1/e)^2))= e(4)^0.44466786101

0.44466786101 *ln(e(4)) = 0.126928011043

ln(e(4)) = 0.126928011043/0.44466786101 =0.285444535512

e(4)=exp^0.285444535512 = 1.33035328597

That should be the sum of 1+1/2^3+1/2^3^4 ..............?

Of course, it can not be true. I have driven to far into wrong direction, but idea was appealing.

Ivars
Ivars Wrote:Yep, found it.

The function defined as series of n times iterated exponential (with base e) terms divided by corresponding n-th hyperfactorials (2^3^4^...n....) is on p. 326-327, Berndt, Ramanujan Notebooks, Vol 1., in excerpts from Second Quaterly report. It would be interesting to see if the original contains something more.

Berndt states that it converges for every x and for every n>0 .

He does not mention entire function there.

a function that has a power series representation
that converges for every x
is entire

Quote:P.S. by adhering to Berndt's endnote of giving way to formal math, we can define by simple analogy and make conjectures:

tetra e = 1+1/2^3+1/2^3^4+ .......... = 1+1/8 + rest = probably around 1,125...because of extreme slowness.

and use it as basis for taking tetra-logarithms of iterated exponential on iteration parameter ( if we have e[4] n = f(e,4,n) then basis tetra e would be used to take tetra-log f(e,4,n) = tetra-log (n). For other bases a the value would have to be adjusted. For other n-tations we need different base for penta log etc.

Further analogies may include tetra Taylor expansion, etc.
Code:
ramanujan was fascinated with the orders of growth
and played with many types of iteration

he has nested radical theorems
  continued fractions
  iterated trigonometric
and many much more complicated functional forms iterated

when combining iteration and summing
orders of growth are what dictate global convergence

there are quite a vast array of different forms one may sum

d'alembert's ratio test is the key algebraic tool for entire functions

if one were to formalise the theory to build natural generalisations
one needs to apply the classical hierarchies of orders
  to build a map of what interesting forms appear

it is clear that ramanujan does this in many places

these basic entire forms all look like

oo
---
\     f(x, j)
/     -------
---     g(j)
j=0

where forAll x,

| f(x, j+1) g(j) |
| -------------- |  --> 0  as j --> oo
| f(x, j) g(j+1) |

the exponential is the ur-entire function
and is very clever about the way it does this

for f(x, j)
  x^j is the taylor basis
and just multiplies x together j times

to get the right type of term growth
g(j) needs to grow like some j things multiplied together or faster

with the factorial
  the n things multiplied together grow as n->oo
  which compared to the "constant" x
always wins out
i.e. eventually for some n

n          n
---        ---
| |        | |
| |  x  <  | |  j
j=1        j=1


ramanujan took

/  n         \
|  O  exp(.) |(x)
\ j=1        /

as his f(x, n)
and

n+1
^
/|\
|   (j+1)
j=1

as his g(n)
(with n=0 creating the special term 1 instead of 2 expected)

although they are both exponential towers
the first has constants e for all but one entry
  which is the "constant" x
(in all this discussion - i mean constant as n->oo)
and the latter expression has terms that grow

possibly more symmetric with taylor may have been

n
^
/|\
|  x
j=1

as f(x, n)
and

n
^
/|\
|  (j+1)
j=1

as g(n)

e.g.

               x
         x    x
    x   x    x
1 + - + -- + --- + ...
    2    3     4
        2     3
             2

but there are trade-offs in properties

the e^e^..^x have pretty good derivative properties
e.g. d/dx (e^e^x) = e^x e^e^x

but

d/dx (x^x^x) = d/dx (e^(ln(x^x^x)) = d/dx (e^(x^x ln(x)))
= d/dx(x^x ln(x)) x^x^x = (d/dx (x ln(x)) x^x ln(x) + x^(x-1)) x^x^x
= ((ln(x) + 1) x^x ln(x) + x^(x-1)) x^x^x

(or something like that)
which is much more complicated to work with

..

in the formal setting
  jumping to iterating exponentiation misses a whole lot of other growth orders

in fact
  we can start with iterating the original functions
    the numerator over x
    and the denominator over factorial

this gives two different possible directions

for iterating the numerator
f(x,n) = x^n

                    2
                   n
so f(f(x,n), n) = x

this is multiplying n^2 copies of x
so an appropriate denominator might be

g(n) = (n^2)!
or (n!)^n

repeated iteration of f gives the various series of entire functions

     oo     k
(   ---    j     )oo
/   \     x      \
\   /    -----   /
(   ---    k     )k=1
    j=0  (j )!

     oo     k
(   ---    j     )oo
/   \     x      \
\   /    -----   /
(   ---      k   )k=1
    j=0  (j!)

and since k here plays the part of constant
there is the "limit" types with terms

   j             j
  j             j
x             x
-----   and   -----
  j               j
(j )!         (j!)

this can continue anew with iterations
  producing constants
    that can be eaten by some limit form
    (which play much the same role as limit ordinals
     in the order theory of function asymptotics)

[[notice that the second form is not just a projection of the exponential series]]

alternatively
  the factorial may be iterated
giving terms with j! numbers in the denominator
so a natural term would be

  j!
x
-----
(j!)!

this
  like the first term type above
is just a projection
but substituting j^j for j!
in any single place
  produces new forms that have more interesting structure

and this is key to the generalisation needed
because there are many ways to ensure the correct asymptotic order conditions
  using a variety of iterative techniques to build function orders

all of these lie between the realm of the exponential and ramanujan's beast
  and there is an infinite hierarchy even beyond
each waiting for a theory to develop and interesting relations to find

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galathaea: prankster, fablist, magician, liar
Dear galathaea,

your explanations really made appetite to read the Ramanujan notebooks.
So I will go today to the library and fetch them Smile
Thanks for pointing out these rather unreferenced works about iteration/tetration.
This is an interesting site for Ramanujan, with handwritten Originals of Notebooks 1,2 and some 37 publications in 2 versions of "Collected Papers".

Ramanujan Site

Ivars
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