1.      Iteration-series; further musings

 

Abstract

While in earlier articles I dealt with the alternating iteration series only I try now to also make some sense to the non-alternating iteration-series.

Note, that the iteration-series is in principle already known, because in a binomial-composition of the integer iterates it allows to compute fractional iterations as shown for instance by L. Comtet.

Here I try to get some more insight using the most basic aspects. While the properties of the series and especially of the differences of series for integer differences of heights are somehow obvious, I hope to find from these basics how to extend this to fractional height-differences as well, as it is –for instance- described for general series/sums with fractional summation-bounds in M. Müller [2005].

Gottfried Helms, 08'2009

1.1.Problem-exposition

            ρb        = 1 + b + b^b + b^b^b + …
                        =  0b + 1b + 2b + …

Generalization

            ρb(h) = 0+hb + 1+hb + 2+hb +…

In most explicite notation:

                 1 ≤ b ≤ e1/e

This sum is diverging for the given range of b's, since the terms approach a fixpoint >=1 in the limit and using the notation

            u € {0..1}      t=exp(u)      b = exp(u*exp(-u))

the limes

            lim h->infhb       = t = exp(u) ≥ 1

is the value of t.

1.2.What value do these divergent series have?

Before we can make further use of these series, we should first assign any meaningful value. But how to sum these divergent series to a value?

What we can try is sum the differences to the limiting value first. This sum should then converge (at least for u<1; for u=1 it seems as if we get a singularity)

The proposal is to use the following formula:

                 this is divergent

                        Subtract at each term the limiting value and correct its own sum

           

                        use zeta(0) for the infinite sum of a constant

           

This evaluation gives solutions; for instance using u=log(2), t=2, b=sqrt(2) we get the following table

h

hb

ρb(h)

0.0

1.00000000000

-3.69277834727

0.1

1.05656068331

-3.56363432866

0.2

1.10877179291

-3.44249658385

0.3

1.15709586665

-3.32866736214

0.4

1.20193093960

-3.22153353189

0.5

1.24362162767

-3.12055326747

0.6

1.28246796820

-3.02524528321

0.7

1.31873253593

-2.93518004693

0.8

1.35264621883

-2.84997254895

0.9

1.38441294251

-2.76927630506

1.0

1.41421356237

-2.69277834727

1.1

1.44220909146

-2.62019501196

1.2

1.46854339396

-2.55126837676

1.3

1.49334544588

-2.48576322880

1.4

1.51673124272

-2.42346447149

1.5

1.53880541744

-2.36417489514

1.6

1.55966261925

-2.30771325141

1.7

1.57938869393

-2.25391258286

1.8

1.59806169852

-2.20261876778

1.9

1.61575277708

-2.15368924757

2.0

1.63252691944

-2.10699190964

2.1

1.64844362100

-2.06240410342

2.2

1.66355745836

-2.01981177071

2.3

1.67791859324

-1.97910867467

2.4

1.69157321485

-1.94019571421

2.5

1.70456392961

-1.90298031258

2.6

1.71693010521

-1.86737587066

2.7

1.72870817531

-1.83330127679

2.8

1.73993191012

-1.80068046631

2.9

1.75063265712

-1.76944202465

3.0

1.76083955588

-1.73951882908

3.1

1.77057973012

-1.71084772442

3.2

1.77987845984

-1.68336922908

3.3

1.78875933581

-1.65702726791

3.4

1.79724439872

-1.63176892906

3.5

1.80535426450

-1.60754424219

3.6

1.81310823761

-1.58430597586

3.7

1.82052441349

-1.56200945210

3.8

1.82761977150

-1.54061237643

3.9

1.83441025924

-1.52007468177

4.0

1.84091086929

-1.50035838496

4.1

1.84713570912

-1.48142745455

4.2

1.85309806481

-1.46324768891

4.3

1.85881045937

-1.44578660372

4.4

1.86428470601

-1.42901332778

4.5

1.86953195702

-1.41289850669

4.6

1.87456274859

-1.39741421347

4.7

1.87938704197

-1.38253386559

4.8

1.88401426141

-1.36823214793

4.9

1.88845332898

-1.35448494101

5.0

1.89271269683

-1.34126925425

5.1

1.89680037684

-1.32856316366

5.2

1.90072396814

-1.31634575372

5.3

1.90449068248

-1.30459706308

5.4

1.90810736778

-1.29329803379

5.5

1.91158053000

-1.28243046371

5.6

1.91491635338

-1.27197696206

5.7

1.91812071927

-1.26192090757

5.8

1.92119922365

-1.25224640933

5.9

1.92415719348

-1.24293826999

6.0

1.92699970185

-1.23398195108

 

(See pictures for varying bases and constant initial height h=0 instead at appendix 2.)

1.3.Differences contain somehow slogs

The sum is in a sense "modular" wrt to its parameter h, because of the telescoping effect with integer differences of the height-parameter, say we expect

                        ρb(h) – ρb(h+1) = hb
                        ρb(h) – ρb(h+2) = hb + h+1b

                        ρb(h) – ρb(h+d) = hb + h+1b + … + h+d-1b

that all elements beyond the second height vanish, if the difference between two height parameters is integer. (The later interesting case will be the fractional difference...)

But this is not exactly the practical result. If we look at

            ρb(0) – ρb(1)           = –3.69277834727 + 2.69277834727
                                   = –1

we would expect that we get +1 instead.  We can try to make sense of it:

                                   =   1   –2
                                   =  0b – 1*t

If we proceed,

            ρb(1) – ρb(2)           = –2.69277834727 + 2.10699190964
                                   = – 0.58578643763
                                   =    1.414…   – 2 
                                   =  1b – 1*t

            ρb(2) – ρb(3)           = –2.10699190964 + 1.73951882908
                                   = – 0.367473080560
                                   =    1.63252691944...   – 2 
                                   =  2b – 1*t

we find, that we get a multiple of t as additional element. The same is true for the fractional heights, as long as we use integer differences between two height-parameters.

Checking this heuristically further and collecting results, we get indeed for integer differences d:

                        ρb(h) – ρb(h+d)      = ( hb + h1+1b + ... + h+d-1b ) – d*t

giving thus meaningful results. Moreover, we find the interesting fact that the d'th multiple of t occurs.

Table: ρ(0) – ρ(h)

h

ρ (0)- ρ (h)

comment

0.0

0.0

= 0 =(0b+1b+2b+…)–(0b+1b+2b+…)

0.1

0.0708559813874

 

0.2

0.149718236577

 

0.3

0.235889014873

 

0.4

0.328755184617

 

0.5

0.427774920205

= ? = (0b+1b+2b+…)–(0.5b+1.5b+2.5b+…)

0.6

0.532466935944

 

0.7

0.642401699661

 

0.8

0.757194201683

 

0.9

0.876497957794

 

1.0

1.00000000000

= 1 =(0b+1b+2b+…)–(1b+2b+…)

1.1

1.12741666469

 

1.2

1.25849002949

 

1.3

1.39298488153

 

1.4

1.53068612422

 

1.5

1.67139654787

= 0.4277.. + 1.24...  = (r(0)-r(0.5)) + 0.5b

1.6

1.81493490414

 

1.7

1.96113423559

 

1.8

2.10984042051

 

1.9

2.26091090030

 

2.0

2.41421356237

= 1 + b =(0b+1b+2b+…) – (2b+3b+…)

 

This suggests, that in some way in this formula an slog is present; however, I don't see it yet how to make use of this. With integer height-differences we can already feel some sense – not enough however to look at fractional height differences yet…

1.4.Computation

I've currently two ways for computation to crosscheck the results.

The first one is the actual summation according to the above formula, proceeding until the terms are smaller then a certain limit, say €=1e-40 using the zeta-correction-term:

F. 1.1:
            ρb(h) = t*ζ(0) + Σk=0inf( k+hb – t)                 where (k+hb –t)>1e-40

The difference-value

F. 1.2
            ρdb(h,d)        =  Σk=0inf( k+hb – t)– Σk=0inf( k+h+db – t) –d*t
F. 1.3
                                    =  Σk=0inf( k+hb – k+h+db) –d*t

The second one uses the formalism of schröder-function σ. We have (with the needed fixpoint-shift):

            hb = σ-1( uhσ(1-t)) +t

and the infinite sum is then

            ρb(h) =   σ-1( uh σ(1-t) )
                          + σ-1(uh+1 σ(1-t) )
                         + σ-1(uh+2 σ(1-t) )
                         + …
                         + σ-1( … σ(1-t) )
                         + t(1+1+1+1…)

Where we would like to apply some simplification, like the following general idea:

            ρb(h) = σ-1( (uh + uh+1 + uh+2 + …) σ(1-t) )  + t(1+1+1+1…)             // not a correct formula!
            ρb(h) = σ-1(      uh (1+ u + u2 + …)  σ(1-t) )  + t *ζ(0)
F. 2:

            ρb(h) = σ-1(      uh  * [1/(1–u)]           σ(1-t) )  + t*ζ(0)                    having |u|<1

Here in the powerseries-expansion of σ-1(x) we cannot simply use the powers (1/(1–u))k of the x-parameter; instead we must apply the running exponent k at u: 1/(1–uk) when the powerseries is written out. Thus the function must be computed by

           
                                   where ck is the kth coefficient of the powerseries for σ-1

If u is not too near to 1 then this converges pretty fast and needs not many coefficients.

 

The latter in matrix-notation:

            Ut = dV(u/t)* U *dV(t)                   // U is Bell-matrix for exp(x)–1
                        = W * D * WI                       // determine the eigenmatrices of Ut
then
            s1 = V(1-t)~ * W[,1]                      // the value of the schröder-function
            ρb(h) = V(s1 uh)~ *( I + D + D2 + D3 + …)         * WI [,1] + t*ζ(0)
                        = V(s1 uh)~ *dZ                               * WI [,1] + t*ζ(0)

where the diagonal-matrix dZ is in principle

            Z = (I – D)-1

but where the impossible reciprocal of zero in the first entry is replaced by zero, so

            Zr,r       = 0                             if r=0,
                        = (1 – ur)-1                if r>0

 

 

2.      Appendix

Pictures for varying bases; b=exp(u*exp(-u)), according to formula F1.1:

2.1.Pic 1: ρb(0) depending on u

2.2.Pic 2: exp(ρb(0)) depending on u

2.3.References/Links

[TetForum]Gottfried Helms
Iteration-series (msg 4)
msg 4 in "Tetration-forum"
http://math.eretrandre.org/tetrationforum/showthread.php?tid=241&pid=3596#pid3596

[LEuler]      Leonhard Euler:
Dilucidationes in capita postrema calculi mei differentialis de functionibus inexplicabilibus,
2nd ed. Commentatio 613 indicis enestroemiani,
M´emoires de l’acad´emie des sciences de St.-P´etersbourg 4 (1813), 88–119.
http://math.dartmouth.edu/~euler/pages/E613.html

[MMüller]   Markus Müller, Dierk Schleicher:
Fractional Sums and Euler-like Identities
http://arxiv.org/PS_cache/math/pdf/0502/0502109v3.pdf