05/28/2008, 08:10 AM

Code:

`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