Does anyone have taylor series approximations for tetration and slog base e^(1/e)?

[sheldonison]

I ask because I want to observe how logarithmic semi-operators behave for bases less than or equal to \( e^{1/e} \). I haven't a clue where I might going about getting these. I think it was Sheldon who posted coefficients for me before, but they were base 2, and the graphs didn't look pretty; so I just wonder if more erratic bases will give different results. I know that \( log_{e^{1/e}}(x) \) has a fix point at x = e, so I wonder if that might change anything and might shift the hump that appears with base 2.

If anybody wonders what I'm talking about it's \( 0 \le q \le 1 \) and \( r:f(x) = f^{\alpha r}(x) = f^{[r]}(x) \):
\( x\, \{-q\}\, y = {\small (1-q):}\log_{b}({\small (q-1):}\log_{b}x+ {\small (q-1):}\log_{b}y) \) which behaves as addition
\( x\, \{1-q\}\, y = {\small (q-1):}\log_{b}({\small (1-q):}\log_{b}x + {\small (1-q):}\log_{b}y) \) which behaves as multiplication
\( x\, \{2-q\}\, y = {\small (q-1):}\log_{b}(y [{\small (1-q):}\log_{b}x]) \) which behaves as exponentiation

Thanks for reading this. Any help would be greatly appreciated.

Hey James,

Hey Andy, haven't heard from you in awhile! I've been laying low for awhile, trying to learn more background math so I can talk more intelligently on these forums, and maybe eventually write a paper....

I'll have to take a pass on James's larger problem, but I think I can get you a series for base \( \eta = e^{1/e} \). Jaydfox (Jay Daniels) called the upper superfunction for this base cheta(z). For all real(z), cheta(z) is > e, and increasing, and cheta(z) is entire. Henryk has made numerous other posts on this function, with references to earlier work, which is mathematically equivalent to iterating exp(z)-1. I believe Gottfried may have posted results for the lower superfunction base eta. I'm using Jay's suggestion to normalize cheta(0)=2*e. First off a caveat: I feel like I have some understanding of all of the other bases for tetratation. Even though cheta(z) was the very first super exponential base I explored, there many things I don't understand cheta(z). Unlike all other bases, it has no periodicity or pseudo periodicity. I'm using Newton interpolation algorithm here, that seems to work quite well. Before I knew how to calculate sexp(z) for any other bases, I was very curious about cheta(z), and cheta(z) in the complex plane, and this is the algorithm I used to investigate it.

I center the Newton polynomial at cheta(-95) with 25 sample points on either side. So that the Newton_polynomial(0)=cheta(-95). This gives a consistent well behaved polynomial, whose 50th coefficient is 5E-101, with consistent accurate values for values of cheta in the neighborhood of cheta(-95), with an accuracy radius of at least 10 units, in either direction.

For the results I'm posting here centered at cheta(0)=2e, I iterate the exponent of that function 95 times, to make a unit circle in the complex plane centered around cheta(0), from which a taylor series can be generated. It appears to work; I've haven't posted it before. Initialized to 67 digits accuracy in pari-gp, the algorithm seems to give results with nearly 50 decimal digits of accuracy. Here is the Taylor Series. a0=2e, printed to 32 digits.

Code:

0 5.4365636569180904707205749427053
1 1.1771399745582020467487064927981
2 0.47791083712959936964236746127117
3 0.18626062152494972692276478391796
4 0.070474191198539960880465202693624
5 0.026056306225434063913977558720610
6 0.0094541495787515083484748872855356
7 0.0033764647774015865179387607261247
8 0.0011895908149927411979137386055855
9 0.00041416349743994006206357899506395
10 0.00014268359371573690572984247219736
11 0.000048694763765091835931424063371768
12 0.000016477512451260383444394568944931
13 0.0000055326597652388183384746557130853
14 0.0000018445541337171731425492507600409
15 0.00000061095142258861804599507950586002
16 0.00000020113633929013309964268387743384
17 0.000000065845717087468591004558852969906
18 0.000000021442747870947309095492187967455
19 0.0000000069485439464512255857882560746267
20 0.0000000022412832385662916992460615895339
21 0.00000000071978893862885391677345614556987
22 2.3020973206030329181894361145544 E-10
23 7.3341040297826856350206259498267 E-11
24 2.3278852998967291568233733165642 E-11
25 7.3628505815581778431734554753314 E-12
26 2.3209857992934250244177110110812 E-12
27 7.2930204918450243443324949177246 E-13
28 2.2846097982451633980559339079833 E-13
29 7.1358033041146574466639840152247 E-14
30 2.2225543653988499920641676567938 E-14
31 6.9038181736676583445161044386747 E-15
32 2.1389425595138842758935272382177 E-15
33 6.6103445086593382475520541375449 E-16
34 2.0379986212392242975901900360689 E-16
35 6.2686731343734042989649316156714 E-17
36 1.9238630883507697992098052847583 E-17
37 5.8915865956656546031878586526468 E-18
38 1.8004488782209858871936565169317 E-18
39 5.4909672937838092357031986642134 E-19
40 1.6713366863796135261320082560915 E-19
41 5.0775249729439841733736059243593 E-20
42 1.5397061803790039357741194269061 E-20
43 4.6606303423839124073481231546437 E-21
44 1.4082983164868122075336349130200 E-21
45 4.2482389646419479670428084576358 E-22
46 1.2794035465663755925461443200159 E-22
47 3.8468885494278543290882325049087 E-23
48 1.1548705835256123872218006706955 E-23
49 3.4617530598681323751620914079008 E-24
50 1.0361306324224779753428626830104 E-24
51 3.0967381932148655440014679355037 E-25
52 9.2423246088852631003347605140667 E-26
53 2.7546045204596969212253367181553 E-26
54 8.1988428443303828566587256203397 E-27
55 2.4371065951483711807719660406315 E-27
56 7.2349910531853903620035897727885 E-28
57 2.1451384654610549641820852967050 E-28
58 6.3524081182337135717494628619063 E-29
59 1.8788780939059811381889624923360 E-29
60 5.5506897778611357646729108553389 E-30
61 1.6379251020900380753695185487417 E-30
62 4.8278087421120487722463748941123 E-31
63 1.4214279502607189609788849783519 E-31
64 4.1804970061336688633898121429008 E-32
65 1.2281981008321929711060382301444 E-32
66 3.6045845774315208753223996337347 E-33
67 1.0568098942930644804288412600335 E-33
68 3.0952921594935758380591422397535 E-34

Now, for the inverse function I am also centering the series around z=2*e. So, you need to substitute y=z-2*e, and this is the Taylor series for the inverse function, \( \text{cheta}^{-1}(z-2e) \).
- Sheldon

Code:

0 -2.8394844317361184366323169886 E-46
1 0.84951664340115092455978946179380
2 -0.29299658208028424957861348941347
3 0.10509962628439971957233240144673
4 -0.038157724678352862884934417549581
5 0.013925154552107876079062165565924
6 -0.0050952225123435103030773323796416
7 0.0018671879930152709206427694717335
8 -0.00068490296133904125688872118488805
9 0.00025139034517175714464979472515850
10 -0.000092313189599831207400315327952637
11 0.000033909513426122719199099928773847
12 -0.000012459097214278183406543059927027
13 0.0000045786159697733865794175565753500
14 -0.0000016828557401554335864965747890457
15 0.00000061860247032573763467393855008516
16 -0.00000022741483226518418491094851146220
17 0.000000083610506876659067250809903046197
18 -0.000000030741996153004058819109207794717
19 0.000000011303887549718411176045726840015
20 -0.0000000041566598052944547999518543402356
21 0.0000000015285478957891868574437546094353
22 -0.00000000056212008538080452772480392355221
23 2.0672482156588402950705541420527 E-10
24 -7.6027015127751986795782560107348 E-11
25 2.7961060314772058262329305454437 E-11
26 -1.0283679197507576516530461271023 E-11
27 3.7822616925408613984109447810316 E-12
28 -1.3911114500866907702507009276973 E-12
29 5.1165688309305946581953915277227 E-13
30 -1.8819219526223455358781434824655 E-13
31 6.9219700998269053341103265344978 E-14
32 -2.5460249489325270551084541622129 E-14
33 9.3648317462180677670443574299830 E-15
34 -3.4446198869227001149385448196462 E-15
35 1.2670282251343045884635356804874 E-15
36 -4.6605227237301411241081317439108 E-16
37 1.7142969554422764221773785470395 E-16
38 -6.3058018366578812057806434170850 E-17
39 2.3195151116741095091654975044205 E-17
40 -8.5321114627106034923460832031476 E-18
41 3.1384707379328624159531298493402 E-18
42 -1.1544675657668811615413235021795 E-18
43 4.2466584398146493466987617123853 E-19
44 -1.5621212453791332783782145294831 E-19
45 5.7462410024361900551890874499463 E-20
46 -2.1137543536539174110135441193169 E-20
47 7.7754704178615174823047812681216 E-21
48 -2.8602251382786823028314838146849 E-21
49 1.0521437215593346136619106291665 E-21
50 -3.8703580282475950589132824159248 E-22
51 1.4237323972688916255261870725321 E-22
52 -5.2372906974743051783073917784620 E-23
53 1.9265754323196651962839735628669 E-23
54 -7.0870634936127301616271498413745 E-24
55 2.6070391010756576820822103151750 E-24
56 -9.5902434635083516682157243528384 E-25
57 3.5278697393657245345167652687742 E-25
58 -1.2977655109484285361881296652688 E-25
59 4.7739806502062836993834657752311 E-26
60 -1.7561668807408021340395405357838 E-26
61 6.4602835243675049421151226060374 E-27
62 -2.3765010449423382541883982332437 E-27
63 8.7422867197032152849307293297797 E-28
64 -3.2159749628188831946088316129999 E-28
65 1.1830437957766415405064649862914 E-28
66 -4.3520062549876002351263488987314 E-29
67 1.6009534383143544576118352603138 E-29
68 -5.8893636455378353194841863940980 E-30
69 2.1664990350170076098431454251477 E-30
70 -7.9698301848454300290272438589004 E-31
71 2.9318385870247681422532261029951 E-31
72 -1.0785280589301633949989045005489 E-31
73 3.9675574722314662059345458319316 E-32
74 -1.4595378961050141920140942761514 E-32
75 5.3691789767797473462281237059548 E-33
76 -1.9751529348915986487843860664041 E-33
77 7.2659750890193416485885457517150 E-34
78 -2.6729288396177802100060770186772 E-34