ramanujan and tetration - Printable Version

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...



-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-

galathaea: prankster, fablist, magician, liar

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



-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-

galathaea: prankster, fablist, magician, liar