# 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 1.10803 0.5 1.57457 0.166667 1.69632 0.0416667 1.71614 0.00833333 1.71817 0.00138889 1.71828 0.000198413 1.71828 2.48016e-05 1.71828 2.75573e-06 1.71828 2.75573e-07 1.71828 2.50521e-08 1.71828 2.08768e-09 1.71828 1.6059e-10 1.71828 1.14707e-11 1.71828 7.64716e-13 1.71828

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 1.60231 1 2.77868 0.833333 3.64039 0.625 4.13617 0.433333 4.38998 0.281944 4.50326 0.174008 4.54955 0.102679 4.56657 0.0582755 4.57238 0.0319596 4.57421 0.0169996 4.57474 0.00879662 4.57489 0.00443943 4.57493 0.00218976 4.57494 0.00105757 4.57494 0.000500896 4.57494 0.000232971 4.57494 0.000106535 4.57494 4.79488e-05 4.57494 2.12603e-05 4.57494 9.2946e-06 4.57494 4.00953e-06 4.57494 1.70787e-06 4.57494 7.18769e-07 4.57494 2.99048e-07 4.57494 1.23065e-07 4.57494 5.0117e-08 4.57494

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 0.444444 -0.5 0.592593 0.333333 0.658436 -0.25 0.680384 0.2 0.688797 -0.166667 0.691602 0.142857 0.692624 -0.125 0.692964 0.111111 0.693086 -0.1 0.693126 0.0909091 0.69314 -0.0833333 0.693145 0.0769231 0.693146 -0.0714286 0.693147 0.0666667 0.693147 -0.0625 0.693147 0.0588235 0.693147 -0.0555556 0.693147 0.0526316 0.693147 -0.05 0.693147 0.047619 0.693147 -0.0454545 0.693147 0.0434783 0.693147 -0.0416667 0.693147 0.04 0.693147 -0.0384615 0.693147 0.037037 0.693147 -0.0357143 0.693147 0.0344828 0.693147 -0.0333333 0.693147

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 0.162591 0.25 0.373042 0.0208333 0.576651 0 0.751214 0.000260417 0.891123 -7.59549e-05 0.998492 1.5501e-06 1.07841 1.54041e-05 1.13655 -9.07454e-06 1.1781 -8.282e-08 1.20737 3.60741e-06 1.22776 -1.69515e-06 1.24181 -1.3309e-06 1.25141 1.77521e-06 1.25793 3.70354e-07 1.26232 -1.91476e-06 1.26526 3.44673e-07 1.26723 2.41913e-06 1.26853 -1.47706e-06 1.26939 -3.60463e-06 1.26996 4.26031e-06 1.27033 6.19402e-06 1.27058 -1.26253e-05 1.27074 -1.17361e-05 1.27084 4.13952e-05 1.27091 2.2203e-05 1.27095 -0.000153109 1.27098 -2.78328e-05 1.271 0.000641019 1.27101 -0.000111308 1.27101 -0.00303027 1.27102 0.00167667 1.27102 0.0160951 1.27102 -0.0157084 1.27103 -0.0954805 1.27103 0.13949 1.27103 0.628521 1.27103 -1.27694 1.27103 -4.55956 1.27103 12.4028 1.27103 36.1855 1.27103 -129.306 1.27103 -311.608 1.27103 1453.72 1.27103 2883.75 1.27103 -17648.6 1.27103 -28323.3 1.27103 231313 1.27103 289838 1.27103 -3.26934e+06 1.27103 -2.99217e+06 1.27103 4.97506e+07 1.27103 2.89801e+07 1.27103 -8.13616e+08 1.27103 -2.01962e+08 1.27103 1.42717e+10 1.27103 -1.32549e+09 1.27103 -2.67979e+11 1.27103 1.1932e+11 1.27103 5.37564e+12 1.27103 -4.37013e+12 1.27103 -1.14978e+14 1.27103 1.37952e+14 1.27103

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;
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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