Notes on U-tetration ( iteration of
x-> exp(x)-1 )

 

(Last update at13.Feb.2011     remark: I changed the use of the symbol U of the previous version to get a clearer distinction between symbols for the function and that for the matrices)

 

The basic notation, the formal powerseries

Here I consider the formal powerseries for iterates of the function d(x) = exp(x)-1 .

I'll use the notations

                d(x, 1)                   = d(x) = exp(x) – 1
                d(x, h+1)             = d( d(x,1),h)
                d(x, 0)                   = x

I call the number-of-iterations "height" (of the powertower) and use the letter h for that parameter.

For negative integer h we have the iterates of log(1+x):

                d(x,–1)                 = log(1+x)
                d(x,–2)                 = log(1+log(1+x))
                ...

It is possible to define formal powerseries for fractional iterations consistent with the addition of (iteration-) heights:

                d( d(x, h1), h2) = d(x, h1+h2)

 

Using a "matrix-operator" (a conveniently scaled "Bell"-matrix)

A procedure for the computation can be found for instance in L. Comtet. Comtet introduces the self-composition using the Newton-binomial-formula for the formal powerseries.

In this article I simply use the matrix-logarithm of a small modification of the Bell-matrix for the d(x)

 

                Let U1 the column-vector of coefficients of the formal powerseries for d(x)
                U1 = column(0,1,1/2!,1/3!,…)

 

                Then let U0 the column-vector for d(x)0  = 1
                U0 = column(1,0,0,0,…)

                Let Uc the column-vector for the c'th power of d(x)c  (not the iterate!)
                Uc = column( )

Then let U the matrix of concatenation of all Uc-vectors. U is then the factorially scaled matrix of the Stirling-numbers 2nd kind, as, for instance defined in [A&S]

 

 

Matrixoperator U:

Matrix of Stirlingnumbers 2nd kind
(the Bell-matrix for d(x))

 

Further, let's define a general type of Vandermondevector

                V(x) = row(1, x, x2, x3, …)

With this we can construct the (iterable) matrix-product:

                V(x) * U = V(d(x)) = V(d(x,1))
                V(x) * U2 = V(d(x,1))*U  =  V(d(x,2))

                V(x) * Uh = V(d(x,h))

This expression is so far only meant as expression for the formal composition of the powerseries, which we find in the second column of V(d(x,h)). This means, to get that powerseries we need only consider the second column of the h'th power of U, let's write it this way:

                d(x,h) = V(x) * Uh [,1]

 

 

Formal powerseries for the fractional iterates d(x,h) , h noninteger

We can get formal powerseries for fractional iterates analoguosly if we can define fractional powers of the matrix U.

Because the matrix has a unit-diagonal the best method to compute a fractional power is to use the matrix-logarithm defined by the mercator-series using U–I as parameter.

                UL = Log( U ) = (U–I) – (U–I)2/2 – (U–I)3/3 + … – …

The argument U–I is nilpotent to any selected truncation size, so that series gives finite expressions for the entries of UL at any finite row/column-index and these evaluate then to exact entries in rational numbers. This allows a meaningful interpretation of that matrix-logarithm.

To compute fractional powers of U we can then similarly use the matrix-exponential

                Uh = Exp( h * Log(U))

And then, as above we get the formal powerseries for the fractional iterate by the second column of the h'th power of U:

                d(x,h) = V(x) * Uh [,1]                   // also valid for all h

For instance, for h=1/2 we get the matrix U1/2

 

 

Matrixoperator U1/2:

 

 

giving the formal powerseries

               

whose coefficients increase strongly such that the radius of convergence of this series is zero.

The first 20 coefficients of that formal powerseries are

                0, 1, 2, 2, 0, 8, -56, 32, 10176, -215808, -78784, 150990912, -3405688576,
                 -139041794560, 10385778676736, 130003936220160, -43016304236761088,
                 526545841919713280, 266085261164348628992, -12347306589339686547456

and it seems, that the above scaling ensures that those coefficients are all integer.

 

 

General expression for the formal powerseries depending on the variable h-parameter

Because of the convenient properties of the logarithm of the matrix U we can express the coefficients of the powerseries depending on a variable h in terms of polynomials of that h:

                d(x,h)   =  1x
                                + (1/2*h)x2
                               + ( 1/4*h2 - 1/12*h) x3
                               + ( 1/8*h3 - 5/48*h2 + 1/48*h) x4
                               + ( 1/16*h4 - 13/144*h3 + 1/24*h2 - 1/180*h) x5
                               + …

or, attempting to get integer coefficients

               

 

From this we can write the two-parameteric function d(x,h) formally as matrix-product introducing the matrix POLY of coefficients of that polynomials

                d(x,h) = V(x) * POLY * V(h) ~

where the top-left truncation of  POLY looks like

 

Matrix POLY of coefficients for the two-parametric function d(x,h):

 

 

For a given height h this means, that we have to multiply POLY with the transpose V(h)~ to get the column-vector U1 corresponding to Uh[,1] which contains the coefficients for the formal powerseries for the h'th iterate:

                U1 = Uh[,1] = POLY * V(h)~

and then

                d(x,h) = V(x) * U1

An eyeball-check gives for h=1 : POLY * V(1)~ for the first few rows:

                1 = 1                                                     row 1 for coefficient at x
                1/2*h = 1/2                                      row 2 for coefficient at x2
                -1/12*h+1/4*h2 = 1/6                 row 3 for coefficient at x3
                1/48–5/48+1/8 = 1/24              row 4 for coefficient at x4

                … giving coefficients for exp(x)-1

For h=-1 : POLY * V(-1)~ for the first few rows:

                1 = 1                                                     row 1 for coefficient at x
                1/2*h = -1/2                                    row 2 for coefficient at x2
                -1/12*h+1/4*h2 = 1/3                 row 3 for coefficient at x3
                -1/48–5/48-1/8 =– 1/4             row 4 for coefficient at x4

                … giving coefficients for log(1+x)

 

Table POLY and change of order of summation

(truncation to 32 x 12)

 

0

0

0

0

0

0

0

0

0

0

0

0

1.00000000000

0

0

0

0

0

0

0

0

0

0

0

0

0.500000000000

0

0

0

0

0

0

0

0

0

0

0

-0.0833333333333

0.250000000000

0

0

0

0

0

0

0

0

0

0

0.0208333333333

-0.104166666667

0.125000000000

0

0

0

0

0

0

0

0

0

-0.00555555555556

0.0416666666667

-0.0902777777778

0.0625000000000

0

0

0

0

0

0

0

0

0.00127314814815

-0.0157986111111

0.0515046296296

-0.0668402777778

0.0312500000000

0

0

0

0

0

0

0

-0.000148809523810

0.00526620370370

-0.0258680555556

0.0506365740741

-0.0453125000000

0.0156250000000

0

0

0

0

0

0

-0.0000454695767196

-0.00133308531746

0.0113136574074

-0.0321180555556

0.0434317129630

-0.0290364583333

0.00781250000000

0

0

0

0

0

0.0000199790564374

0.000158110119048

-0.00405953758818

0.0174816743827

-0.0336202739198

0.0340108989198

-0.0178943452381

0.00390625000000

0

0

0

0

0.0000113214653145

0.0000198297876249

0.00106265615814

-0.00798717321612

0.0220179880401

-0.0313159079218

0.0249550471230

-0.0107166108631

0.00195312500000

0

0

0

-0.0000113193776388

0.0000281142067166

-0.000188055739271

0.00292184945207

-0.0121852839322

0.0240907720872

-0.0267906803443

0.0174372597346

-0.00627919353505

0.000976562500000

0

0

-0.00000172664698128

-0.0000398044512447

0.0000929097888191

-0.000857463809011

0.00558563546987

-0.0156722021953

0.0237718056857

-0.0214808783063

0.0117297621631

-0.00361631686095

0.000488281250000

0

0.00000705612171179

0.00000110606798251

-0.0000973900278552

0.000319743843236

-0.00214332382319

0.00857213099601

-0.0178075901564

0.0216702768669

-0.0163663738175

0.00765414014275

-0.00205391667794

0

-0.00000101301777217

0.0000255407498408

0.0000193146034564

-0.000221106056740

0.000853415265246

-0.00399194850843

0.0112958472089

-0.0184076755112

0.0185526492239

-0.0119658680773

0.00487200212379

0

-0.00000559541003662

-0.00000844104671673

0.0000565866905508

0.0000736013774889

-0.000480293863401

0.00179037366013

-0.00615026857187

0.0132851855803

-0.0176563735180

0.0150939818079

-0.00845607452096

0

0.00000278500897643

-0.0000218764183589

-0.0000308783186148

0.0000918274988497

0.000200451899171

-0.000956523820017

0.00311688150890

-0.00826461703283

0.0142971284796

-0.0159372946748

0.0117720730416

0

0.00000556362245182

0.0000161788895796

-0.0000534680155645

-0.0000764323476234

0.000107980977407

0.000450686111708

-0.00170513688559

0.00469959885089

-0.0100009496866

0.0143229295042

-0.0136786842136

0

-0.00000548903126386

0.0000232632702353

0.0000552525079060

-0.0000999706949845

-0.000145443330382

0.0000656096485154

0.000876233786845

-0.00272141966761

0.00632621838743

-0.0111319804655

0.0135249843031

0

-0.00000669291680506

-0.0000312146749719

0.0000608496312550

0.000140289074251

-0.000154302742138

-0.000223099906968

-0.0000851073857983

0.00150933332628

-0.00392984366170

0.00776958374083

-0.0115704367446

0

0.0000116121812620

-0.0000293586208479

-0.000109807488995

0.000120164536704

0.000290176034739

-0.000206232231635

-0.000276467343816

-0.000391754045290

0.00234414377342

-0.00520000569233

0.00884602555284

0

0.00000922094512360

0.0000684549777770

-0.0000798140615651

-0.000293289356200

0.000190169428300

0.000514515807562

-0.000249794344110

-0.000258744046274

-0.000884113280245

0.00332974863404

-0.00637985999133

0

-0.0000281326395956

0.0000407197995304

0.000254478523660

-0.000159415648134

-0.000643681175683

0.000244712936664

0.000808996669112

-0.000290514027042

-0.000120237194298

-0.00156152904832

0.00437759973853

0

-0.0000130744820957

-0.000174765277447

0.000107022219976

0.000723693305397

-0.000241293720810

-0.00121349305469

0.000247074481467

0.00115660459637

-0.000348228712042

0.000177531440051

-0.00238806973087

0

0.0000790678539611

-0.0000509153645345

-0.000691411780357

0.000185376174105

0.00169611543620

-0.000257736012247

-0.00202316585911

0.000159489699569

0.00153386513052

-0.000453577912925

0.000649502319867

0

0.0000125598083643

0.000520031268526

-0.0000878204186092

-0.00209831597086

0.000159818142150

0.00341809068358

-0.000102695878258

-0.00304489925501

-0.0000465076302668

0.00191880195570

-0.000639295364230

0

-0.000257444626675

-0.00000560163583526

0.00219132519453

0.0000774885874687

-0.00524993565328

-0.000238197148364

0.00609224515367

0.000347091607245

-0.00419971113343

-0.000382906372544

0.00229692073560

0

0.0000399463258851

-0.00179321941211

-0.000393926763669

0.00708866051564

0.00107649022665

-0.0112876740738

-0.00145536270256

0.00979926697250

0.00119787229708

-0.00536903759965

-0.000843150907368

0

0.000965154912485

0.000597931194494

-0.00803525350006

-0.00251127673418

0.0188906613302

0.00429420339159

-0.0214416358316

-0.00407840860992

0.0144407003456

0.00250329169161

-0.00641682419913

0

-0.000479398220790

0.00710879598324

0.00415391248559

-0.0276350570478

-0.0101591117622

0.0432073573046

0.0121303625147

-0.0367072852041

-0.00872608327064

0.0197254703316

0.00423995118032

0

-0.00413400035554

-0.00492686165030

0.0337822921646

0.0196419756453

-0.0781999520194

-0.0316735109546

0.0871838557014

0.0280285757842

-0.0574862641505

-0.0158949998739

0.0252069454158

0

0.00364643030317

-0.0321199817356

-0.0303987588850

0.123132859442

0.0717993863914

-0.189632330949

-0.0821936577995

0.158294897310

0.0561459477663

-0.0833102527369

-0.0258044757139

0

0.0200644656154

0.0355155275043

-0.161371990105

-0.138129365513

0.368719723749

0.216573813194

-0.405011847681

-0.185172406164

0.262477331560

0.100651849527

-0.112748061977

0

-0.0266575258227

0.163984344506

0.217099842002

-0.621211818806

-0.502402984261

0.944593968266

0.561221939319

-0.777011178359

-0.372049587016

0.402058565471

0.164786181361

0

-0.109433975865

-0.260796020227

0.868040322611

0.997694463252

-1.96144527296

-1.53571144363

2.12752918160

1.28414365605

-1.35896459404

-0.679330436962

0.574102779542

0

0.203327436701

-0.937854250819

-1.62757702735

3.51646920918

3.71182584274

-5.28880579131

-4.07561297163

4.29651204725

2.64641867680

-2.19161222503

-1.14304812752

0

0.665130619209

2.03725117010

-5.21187900185

-7.69651265113

11.6634597966

11.6847026371

-12.5142190515

-9.61214522598

7.89504962424

4.98627814056

-3.28846954268

0

-1.66290268242

5.95678065193

13.1278524887

-22.1339082828

-29.5986840636

32.9725282990

32.0714426086

-26.4976106584

-20.5007171703

13.3508288173

8.69043910681

0

-4.46640908999

-17.2004065386

34.6184714675

64.3249478880

-76.8111306867

-96.5868487442

81.6300989964

78.4396151915

-50.9456676853

-40.0779077720

20.9616625377

0

14.7351954384

-41.6407654187

-114.983036865

153.482786768

256.820864049

-226.702159524

-275.300078287

180.447707027

173.785467539

-89.9407771586

-72.5919140556

0

32.8328822406

157.885595430

-251.974932794

-585.440530393

554.789212202

871.054820739

-584.569327444

-699.977362563

361.335696650

353.288864141

-147.066594023

0

-141.988015420

317.215867693

1096.93776669

-1160.66255957

-2430.64419617

1701.15729804

2582.17675228

-1342.39810662

-1613.21386356

662.614306472

665.804867226

0

-261.445537846

-1578.11102327

1988.14780518

5808.95742632

-4346.51106854

-8575.75205606

4543.98663605

6830.48490615

-2784.11676454

-3412.43948801

1121.99572399

0

1488.67144258

-2602.46237198

-11400.3914328

9457.12874113

25088.3012505

-13761.5135901

-26447.6015871

10772.1967423

16378.6965829

-5269.35407760

-6692.11726412

0

2225.38586814

17167.5422612

-16775.7623465

-62790.1318537

36429.2371165

92071.4180506

-37801.2355725

-72775.7762113

22968.1109026

36043.7099027

-9169.48708622

0

-16965.9992298

22632.4324966

128919.990125

-81699.3259569

-282001.209389

118053.557193

295291.413488

-91687.1256079

-181491.326093

44455.6923008

73520.4777118

0

-19864.8060373

-202949.957825

148447.802757

738101.788937

-320192.510162

-1075878.94572

329766.603527

844756.046306

-198680.621951

-415253.578974

78563.7447446

0

209816.100420

-203538.981603

-1583257.02501

729670.792682

3444745.92555

-1046623.20368

-3585816.98087

806147.907721

2189367.41953

-387205.992395

-880311.779326

0

179952.210614

2601889.48175

-1331939.30794

-9415064.91099

2851431.65424

13651227.5125

-2911995.29981

-10655777.4807

1737625.06275

5203689.86150

-679539.726003

0

-2809660.13947

1804361.41661

21068440.9439

-6414541.20408

-45620175.4042

9117979.35471

47239086.9788

-6950397.41814

-28674195.3232

3298413.50452

11454373.4953

0

-1537617.51817

-36091847.4033

11232709.1805

130008060.821

-23794693.2562

-187610831.806

24005965.8180

145678442.909

-14121437.7770

-70728434.2325

5429606.42913

0

40646569.5053

-13834857.2340

-303052770.285

48467482.2723

653379950.065

-67774658.5637

-673377701.275

50662222.3521

406613987.318

-23479413.7152

-161490843.705

0

9497336.70101

540392137.825

-67078112.2567

-1938575327.49

138038281.362

2785541778.69

-134418266.711

-2152796870.90

75631332.4795

1039789385.27

-27460376.7453

0

-633756457.512

36321161.5318

4700532956.13

-112411455.910

-10094645940.5

133313249.347

10359297792.8

-77628094.3870

-6226116877.91

23149218.9697

2460005438.55

0

55726510.4799

-8712493396.23

-464543661.478

31137786218.3

1087458698.49

-44568140091.6

-1222476193.24

34298090791.4

809405223.168

-16488568951.5

-354311889.107

0

10624909421.6

2258110308.89

-78427159370.5

-8325790733.68

167824702223.

12318317185.2

-171557719099.

-9839260929.73

102672679190.

4931865970.96

-40378641054.5

0

-4479387389.03

150896297343.

33830524377.7

-537443820225.

-73730866273.1

766524921906.

76950043000.1

-587612884016.

-47153631547.1

281298641107.

19049913758.0

0

-191085306212.

-90492032637.6

1.40426147025E12

326278291737.

-2.99510410092E12

-471578616715.

3.05089282338E12

367059804326.

-1.81884436225E12

-178813905496.

712297299427.

0

147339758784.

-2.80099958068E12

-1.09524040267E12

9.94481572290E12

2.35767861839E12

-1.41375347119E13

-2.42709175388E12

1.07994799022E13

1.46464955320E12

-5.15000328031E12

-581666558076.

0

3.67833350467E12

2.68324418152E12

-2.69212194339E13

-9.59393858609E12

5.72461167767E13

1.37437470904E13

-5.81230177576E13

-1.05919222433E13

3.45290228970E13

5.10280681255E12

-1.34704718730E13

0

-4.19319940120E12

5.55998508341E13

3.09092946611E13

-1.96830270039E14

-6.61078904851E13

2.78974129793E14

6.75678356972E13

-2.12413893151E14

-4.04490518050E13

1.00938902710E14

1.59204641663E13

0

-7.56224969314E13

-7.43691124956E13

5.51372049332E14

2.64508276903E14

-1.16918343295E15

-3.76818353935E14

1.18353893902E15

2.88614340474E14

-7.00827624795E14

-1.38089771104E14

2.72448367408E14

0

1.15852751376E14

-1.17766671407E15

-8.48897778129E14

4.15784511805E15

1.80736068720E15

-5.87669966081E15

-1.83806072834E15

4.46120741857E15

1.09425993761E15

-2.11311591829E15

-4.28046930851E14

0

1.65696934682E15

2.05186154784E15

-1.20384785431E16

-7.26873720573E15

2.54614960359E16

1.03117358326E16

-2.57027203038E16

-7.86161269750E15

1.51743397354E16

3.74228431873E15

-5.88004687148E15

...

...

...

...

...

...

...

...

...

...

 

 

1.00000000000

0.433045596018

0.175813571676

0.0685214533424

0.0259260061301

0.00958557929620

0.00347798716295

0.00124213189878

0.000437626092140

0.000152362209100

0.0000524902552539

0.0000179137522579

 

The interesting thing is here, whether we can change order of summation. This is then interesting, if we always assume x=1 in d(x,h).

 

Since each column is associated with one power of the h-parameter, we may sum up each column to have only one term for a powerseries in h.

 

We see, that the sequence of entries along a column diverge after a local minimum and looking at coefficients of higher index it seems that they will grow at least hypergeometrically (like the factorials) .

I constructed a Noerlund-means method of summation which I can apply to the powerseries if I want to estimate the column-sums meaning we apply  x=1 at d(x,h) keeping h variable. That summation is parametrized to sum alternating series of order gamma(k)1.5 Remark: this is absolutely experimental and no proof for the correct value , for conversion to convergent series and/or regularity of the result is given so far. But because the results and also the intermediate values seem to be reasonable I give here the assumed column-sums in the brown row at the bottom of the table.

 

This would then give a powerseries for d(1,h) in terms of h:

 

                d(1,h) = 1 + 0.4330 h + 0.1758 h2 + 0.06852 h3 + ...

 

and for h=1 this should be e - 1 ~ 1.718 ... and this value is good approximated by the row-sum of the last (brown-marked) row.

 


Checking the convergence of the partial sums for integer and fractional heights (h=1,2,-1, 0.5)

 

A numerical check of

                U1 = POLY * V(1)
                Approx = Noerlund(0,0) * dV(1)* U1   // we need no acceleration for this because it's simply the exponential-series

showing the convergence of approx_k to the final value:

left the coefficients in U1, right the Noerlund-summed partial sums

 

U1

Partial sums: Approx:

 

0

0

1.00000000000

1.10803324100

0.500000000000

1.57457355300

0.166666666667

1.69632420459

0.0416666666667

1.71614032126

0.00833333333333

1.71816671191

0.00138888888889

1.71828087196

0.000198412698413

1.71828207238

0.0000248015873016

1.71828183500

0.00000275573192240

1.71828182789

0.000000275573192240

1.71828182845

0.0000000250521083854

1.71828182846

0.00000000208767569879

1.71828182846

1.60590438368E-10

1.71828182846

1.14707455977E-11

1.71828182846

7.64716373182E-13

1.71828182846

 

The same with h-parameter h= 2, so the second-iteration; d(x,2) = exp(exp(x)-1)-1 with x=1

                U21 = POLY * V(2)~
                Approx = Noerlund(0,0) * dV(1)* U21// again we need no acceleration-procedure for this

 

U21

Partial sums: Approx:

 

0

0

1.00000000000

1.60230732254

1.00000000000

2.77868485049

0.833333333333

3.64038609659

0.625000000000

4.13617161436

0.433333333333

4.38997896613

0.281944444444

4.50325607997

0.174007936508

4.54954546855

0.102678571429

4.56656903743

0.0582754629630

4.57238024057

0.0319596009700

4.57420919266

0.0169996091871

4.57474473371

0.00879662406138

4.57489192860

0.00443943225627

4.57492973020

0.00218975755744

4.57493886159

0.00105757104279

4.57494096334

0.000500896018318

4.57494141015

0.000232971039341

4.57494150301

0.000106534988039

4.57494152085

0.0000479488461554

4.57494152405

0.0000212602719146

4.57494152466

0.00000929459892442

4.57494152474

0.00000400953097900

4.57494152476

0.00000170787472022

4.57494152476

0.000000718768664806

4.57494152476

0.000000299047612615

4.57494152476

0.000000123065453419

4.57494152476

0.0000000501169612163

4.57494152476

 

The same with h-parameter h= -1, so the inverse-iteration; giving log(1+x) with x=1:

                U -11 = POLY * V(-1)~
                Approx = Noerlund(0,0.95) * dV(1)* U-11         // we use only small order of the Noerlundtransformation to accelerate convergence

 

U-11

Partial sums: Approx:

 

0

0

1.00000000000

0.444444444444

-0.500000000000

0.592592592593

0.333333333333

0.658436213992

-0.250000000000

0.680384087791

0.200000000000

0.688797439415

-0.166666666667

0.691601889956

0.142857142857

0.692623801540

-0.125000000000

0.692964438735

0.111111111111

0.693085511172

-0.100000000000

0.693125868651

0.0909090909091

0.693140005390

-0.0833333333333

0.693144717637

0.0769230769231

0.693146352716

-0.0714285714286

0.693146897743

0.0666666666667

0.693147085613

-0.0625000000000

0.693147148237

0.0588235294118

0.693147169718

-0.0555555555556

0.693147176879

0.0526315789474

0.693147179326

-0.0500000000000

0.693147180142

0.0476190476190

0.693147180420

-0.0454545454545

0.693147180513

0.0434782608696

0.693147180544

-0.0416666666667

0.693147180555

0.0400000000000

0.693147180558

-0.0384615384615

0.693147180559

0.0370370370370

0.693147180560

-0.0357142857143

0.693147180560

0.0344827586207

0.693147180560

-0.0333333333333

0.693147180560

 

The same with h-parameter h= 1/2, so giving the formal powerseries for the half-iteration, using x=1 

The approximation is very difficult now since we have a divergent series in the terms, whose characteristic is not yet really known, but seems to be more than hypergeometric, so a high order for Noerlund-summation is needed. However, for size = 64 I got an approximation using Eulersummation of order  ord=2.5 which is shown below.

                U0.51 = POLY * V(1/2)
                Approx = Noerlund(0, 2.5) * dV(1)* U0.51         // medium order of Noerlundsummation provides convergence – but it is only local

 

U0.51

Partial sums: Approx:

 

0

0

1.00000000000

0.162591050989

0.250000000000

0.373041564566

0.0208333333333

0.576651361395

0.E-810

0.751214327591

0.000260416666667

0.891123237110

-0.0000759548611111

0.998492144926

0.00000155009920635

1.07840852624

0.0000154041108631

1.13654884139

-0.00000907453910384

1.17810073526

-0.0000000828199706170

1.20737494332

0.00000360740727676

1.22775702511

-0.00000169514972633

1.24180758666

-0.00000133089916348

1.25141142987

0.00000177521444910

1.25792762288

0.000000370353976658

1.26232035257

-0.00000191475684776

1.26526469485

0.000000344673434042

1.26722815289

0.00000241913411616

1.26853149270

-0.00000147705874041

1.26939305279

-0.00000360462602023

1.26996042399

0.00000426030599723

1.27033276648

0.00000619401781838

1.27057634173

-0.0000126252925336

1.27073521306

-0.0000117360887110

1.27083855431

0.0000413952285774

1.27090560466

0.0000222030302120

1.27094900588

-0.000153108566769

1.27097703715

-0.0000278327871472

1.27099510405

0.000641018661899

1.27100672598

-0.000111307516319

1.27101418835

-0.00303026666274

1.27101897160

0.00167666962999

1.27102203256

0.0160951154545

1.27102398833

-0.0157084159784

1.27102523611

-0.0954804645039

1.27102603109

0.139489606827

1.27102653689

0.628520649401

1.27102685831

-1.27694165810

1.27102706231

-4.55956399021

1.27102719163

12.4027724564

1.27102727353

36.1854546816

1.27102732533

-129.305594720

1.27102735806

-311.608441223

1.27102737873

1453.71643376

1.27102739177

2883.75499733

1.27102739998

-17648.6056027

1.27102740515

-28323.2666121

1.27102740840

231312.842070

1.27102741045

289837.706925

1.27102741173

-3269335.96562

1.27102741254

-2992168.60724

1.27102741304

49750634.1587

1.27102741336

28980063.0330

1.27102741356

-813616473.772

1.27102741368

-201961594.949

1.27102741376

14271686431.9

1.27102741381

-1325490857.72

1.27102741384

-267978508283.

1.27102741386

119319788076.

1.27102741387

5.37563669599E12

1.27102741388

-4.37013046468E12

1.27102741388

-1.14977800862E14

1.27102741388

1.37951986894E14

1.27102741388

 

But using more terms it is obvious that that convergence is only local; So I used an experimental stronger parametrization of the Noerlund-summation (in my implementation with two parameters), nonetheless apparently arriving at the same value

                Approx = Noerlund(1.4, 1.3) * dV(1)* U0.51     // we need Noerlund-summation for this because series is strongly diverging

Using 256 terms we find the following approximation :

 

U0.51

Partial sums: Approx:

 

0

0

0

1

1.00000000000

0.4166666666666666666666667

2

0.250000000000

0.6802187646414115767312961

3

0.0208333333333

0.8543367076839841850527931

4

1.063167461E-204

0.9728122533370859110058801

5

0.000260416666667

1.055158890718194379537863

6

-0.0000759548611111

1.113322049145836683676599

7

0.00000155009920635

1.154927026644590189403069

8

0.0000154041108631

1.184995099232425260962484

9

-0.00000907453910384

1.206912382008609448210350

10

-0.0000000828199706170

1.223005413341123686437128

11

0.00000360740727676

1.234896974845728865717737

12

-0.00000169514972633

1.243733093659537186974143

13

-0.00000133089916348

1.250331573819650138051143

14

0.00000177521444910

1.255281207630202865239360

15

0.000000370353976658

1.259009183115809527489827

16

-0.00000191475684776

1.261827541981969525892891

17

0.000000344673434042

1.263965589964040800596140

18

0.00000241913411616

1.265592744460861863146730

19

-0.00000147705874041

1.266834787928400056087080

20

-0.00000360462602023

1.267785523004340843822270

21

0.00000426030599723

1.268515189788157734675139

22

0.00000619401781838

1.269076583543275748684661

23

-0.0000126252925336

1.269509526672548941751353

24

-0.0000117360887110

1.269844154811896911030546

25

0.0000413952285774

1.270103343096363738965483

26

0.0000222030302120

1.270304505488932369876442

27

-0.000153108566769

1.270460934623827685972399

28

-0.0000278327871472

1.270582803292161357418455

29

0.000641018661899

1.270677915673107836555729

30

-0.000111307516319

1.270752272718673998185568

31

-0.00303026666274

1.270810498998789451747707

32

0.00167666962999

1.270856165904167737114013

33

0.0160951154545

1.270892037054775901782266

34

-0.0157084159784

1.270920255131584852660284

35

-0.0954804645039

1.270942484470710617007680

36

0.139489606827

1.270960020154843030156470

37

0.628520649401

1.270973871664026359126703

38

-1.27694165810

1.270984827158658745978229

39

-4.55956399021

1.270993502982183114841320

40

12.4027724564

1.271000381858227299430625

41

36.1854546816

1.271005842420900556793057

42

-129.305594720

1.271010182086980195864213

 

 

 

 

 

 

242

1.99391580834E202

1.271027413889951521419677

243

-1.89349634501E202

1.271027413889951521420391

244

-7.29686595931E204

1.271027413889951521421002

245

9.45085000863E204

1.271027413889951521421524

246

2.71412390180E207

1.271027413889951521421972

247

-4.46257691886E207

1.271027413889951521422355

248

-1.02596030513E210

1.271027413889951521422683

249

2.04869812900E210

1.271027413889951521422964

250

3.94079346151E212

1.271027413889951521423204

251

-9.27402089917E212

1.271027413889951521423411

252

-1.53792247833E215

1.271027413889951521423587

253

4.17365586728E215

1.271027413889951521423739

254

6.09718075121E217

1.271027413889951521423868

255

-1.87702245373E218

1.271027413889951521423980

 

We seem to arrive at a value d( 1, 0.5) ~ 1.27102741388995152142.., which reinserted into that powerseries (and again be Noerlund-summed) gives a reasonable approximation to d(d(1,0.5),0.5) = d(1,1) .

However, it is possible that this is still only a local approximation. It seems, that the growthrate of the terms is even more than hypergeometric; a plot of the quotients of successive coefficients indicates strongly, that in the average the coefficients grow dependent on the index k like O((k!)1+δ) (with oscillations) where if δ>0 significantly the used Noerlund-summation is not strong enough. The following plot shows the growthrate by the formula

                 

 

Note: I used dxp(x,h) in the program for the drawing of the graphic instead of d(x,h)

Trend: (estimated using Excel)
                ln(ck/ck-1) = 1.0722ln(k)-2.9185
                 ck ~(-0.054)k (k!)1.07 

 


 

Gottfried Helms

(Last update at 12.2.2011 // 3.Nov.2010)


References:

[Comtet]        Comtet, L;
Advanced Combinatorics (1974 edition)
D. Reidel Publishing Company, Dordrecht - Holland

[A&S]             Abramowitz, M. and Stegun, I.A.;
Handbook of mathematical functions 9th printing
page 824; online at http://www.math.sfu.ca/~cbm/aands/

[Erdös]           Erdös, Paul , Jabotinsky, Eri;
On analytic iteration
J. Anal. Math. 8, 361-376 (1961)  (also online at digicenter göttingen)

[Baker]           Baker, I.N.;
Zusammensetzung ganzer Funktionen,
Math Zeitschr. Bd. 69 pp 121-163 (1958)  (also online at digicenter göttingen)


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