http://math.eretrandre.org/tetrationforum/ Wed, 20 Aug 2008 04:51:29 +0200 MyBB http://math.eretrandre.org/tetrationforum/showthread.php?tid=197 Wed, 13 Aug 2008 19:53:11 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=197
recently I posted this in sci.math.research. I think a copy of this here would be good, too (I've posted a similar msg already in the thread "matrix-method", but may be not everyone expects such general things there) The progress is mentioned in part 3.

------------------------------------------------------------
Tetration: Progress in fractional iteration?

In 1958 I.N.Baker proved in [1], that the powerseries for
fractional iterates of the function exp(x)-1 have
convergence-radius zero. P.Erdös / E.Jabotinsky followed
in [2] with the stronger statement "The function exp(x) - 1
was shown by I. N. Baker [L] to have no real non-integer iterates."

Attempts to define fractional iterates of exp(x)-1 or more
general t^x-1 in the context of the "tetration"-discussion
are since rated with a portion of suspicion...

However - even if a series has convergence-radius zero it
may be summed using a technique of divergent summation; one
other example for zero-convergence-radius is the series
f(x) = 0! - 1!x + 2!x^2 - 3!x^3 + ... - ...
to which a summation-method was applied by L.Euler.

The extremely simple Euler-transformation, for instance,
allows to sum the alternating geometric series up to any
order by transforming the original series of coefficients
a_k into coefficients b_k, which form then a conventionally
summable series.

I seem to have found a similar simple procedure for the
functions U(x) = exp(x)-1 and Ut(x) = t^x - 1, and especially
their fractional iterates, just using the Stirlingnumbers 2nd kind
analoguously to Euler's binomial-coefficients.
This transformation seems to transform the diverging sequences
of coefficients a_k (having also nonperiodic change of sign) even
of fractional iterates into the converging sequence of b_k,
if the base t is greater than exp(1.5) - which are the especially
difficult cases since the iterates diverge for bases >2.

A short technical report is at
http://go.helms-net.de/math/tetdocs/Coef...Height.htm

It reflects only some initial findings, but I think, it
gives already a wider perspective - let's see.

Comments/critics/corrections are much appreciated -

Gottfried Helms

[1] Baker, Irvine Noel; Zusammensetzungen ganzer Funktionen
1958; Mathematische Zeitschrift, Vol 69, Pg 121-163,
[2] Erdös, Paul, Jabotinsky, Eri; On analytical iteration
1961; J. Anal. Math. 8, 361-376

====================================================]]>
recently I posted this in sci.math.research. I think a copy of this here would be good, too (I've posted a similar msg already in the thread "matrix-method", but may be not everyone expects such general things there) The progress is mentioned in part 3.

------------------------------------------------------------
Tetration: Progress in fractional iteration?

In 1958 I.N.Baker proved in [1], that the powerseries for
fractional iterates of the function exp(x)-1 have
convergence-radius zero. P.Erdös / E.Jabotinsky followed
in [2] with the stronger statement "The function exp(x) - 1
was shown by I. N. Baker [L] to have no real non-integer iterates."

Attempts to define fractional iterates of exp(x)-1 or more
general t^x-1 in the context of the "tetration"-discussion
are since rated with a portion of suspicion...

However - even if a series has convergence-radius zero it
may be summed using a technique of divergent summation; one
other example for zero-convergence-radius is the series
f(x) = 0! - 1!x + 2!x^2 - 3!x^3 + ... - ...
to which a summation-method was applied by L.Euler.

The extremely simple Euler-transformation, for instance,
allows to sum the alternating geometric series up to any
order by transforming the original series of coefficients
a_k into coefficients b_k, which form then a conventionally
summable series.

I seem to have found a similar simple procedure for the
functions U(x) = exp(x)-1 and Ut(x) = t^x - 1, and especially
their fractional iterates, just using the Stirlingnumbers 2nd kind
analoguously to Euler's binomial-coefficients.
This transformation seems to transform the diverging sequences
of coefficients a_k (having also nonperiodic change of sign) even
of fractional iterates into the converging sequence of b_k,
if the base t is greater than exp(1.5) - which are the especially
difficult cases since the iterates diverge for bases >2.

A short technical report is at
http://go.helms-net.de/math/tetdocs/Coef...Height.htm

It reflects only some initial findings, but I think, it
gives already a wider perspective - let's see.

Comments/critics/corrections are much appreciated -

Gottfried Helms

[1] Baker, Irvine Noel; Zusammensetzungen ganzer Funktionen
1958; Mathematische Zeitschrift, Vol 69, Pg 121-163,
[2] Erdös, Paul, Jabotinsky, Eri; On analytical iteration
1961; J. Anal. Math. 8, 361-376

====================================================]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=196 Sat, 02 Aug 2008 15:43:05 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=196
http://www.congreso.us.es/holomorphic/

Gottfried]]>
http://www.congreso.us.es/holomorphic/

Gottfried]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=195 Thu, 24 Jul 2008 15:25:54 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=195
I am currently reviewing the problem of divergence of powerseries (for the fractional iteration of Ut°h(x), but real bases).

First I produced some plots with few different bases to get a better overview over the general characterisitcs of the divergence of the powerseries.

Today I found a start to overcome that nasty type of divergence: using a Stirling-transform (as I call it at the moment) of the powerseries I get nicely convergent series even for the fractional case (just had only time to check this for one (divergent) base b=exp(2)). This is a big progress for my implementation of diagonalization!

Look at
http://go.helms-net.de/math/tetdocs/Coef...Height.htm

Gottfried]]>
I am currently reviewing the problem of divergence of powerseries (for the fractional iteration of Ut°h(x), but real bases).

First I produced some plots with few different bases to get a better overview over the general characterisitcs of the divergence of the powerseries.

Today I found a start to overcome that nasty type of divergence: using a Stirling-transform (as I call it at the moment) of the powerseries I get nicely convergent series even for the fractional case (just had only time to check this for one (divergent) base b=exp(2)). This is a big progress for my implementation of diagonalization!

Look at
http://go.helms-net.de/math/tetdocs/Coef...Height.htm

Gottfried]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=194 Mon, 21 Jul 2008 18:26:22 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=194
Using base b=2, then the fixpoint-shift is performed by

t = 0.824678546142 + 1.56743212385*I
u = 0.571623609127 + 1.08646115737*I = log(t)

such that b = t^(1/t) = exp(u/t) .


Using the fixpointshift

x' = x/t-1 x"=(x+1)*t

and the according triangular Bell-matrix Ut I could now compute some fractional iterates, which was not possible before.

There might be some mistake in my diagonalization-procedure due to fractional powers if the argument is complex. Because I could not locate the error yet I employed my new exponential polynomial interpolation. *In principle* this seems a start, the error seems to not to occur here.
(For the method see : http://go.helms-net.de/math/tetdocs/Expo...lation.pdf )

To check: Ut begins as follows

Code:
1                           .                        .                        .                        .                      .
  0      0.57162361+1.0864612*I                        .                        .                        .                      .
  0    -0.42682215+0.62104685*I  -0.85364430+1.2420937*I                        .                        .                      .
  0   -0.30624163-0.036240215*I  -1.8374498-0.21744129*I  -1.8374498-0.21744129*I                        .                      .
  0  -0.033920340-0.088358850*I  -0.47488476-1.2370239*I   -1.2211322-3.1809186*I  -0.81408817-2.1206124*I                      .
  0   0.015321758-0.017472227*I  0.45965275-0.52416682*I    2.2982637-2.6208341*I    3.6772220-4.1933346*I  1.8386110-2.0966673*I
  ...     ....

The computed matrix POLY begins as follows

Code:
0                           .                          .                          .                            .                           .
  0                   1.0000000                          .                          .                            .                           .
  0    -0.34295950+0.39829087*I    0.34295950-0.39829087*I                          .                            .                           .
  0    0.011293278-0.29398501*I   0.082028794+0.54639055*I  -0.093322072-0.25240554*I                            .                           .
  0    0.086289977+0.13933724*I   -0.34931427-0.28800518*I    0.39760954+0.14818664*I  -0.13458524+0.00048129196*I                           .
  0  -0.080900876-0.038017321*I  0.33173844+0.0088923457*I   -0.42115742+0.18129664*I      0.18386237-0.21507654*I  -0.013542508+0.062904872*I
  ...      ....


so we have to compute the column-vector A_h of coefficents for Ut°h(x)

Code:
´
A_h         = POLY * V(u^h)
V(x)~ * A_h = Ut°h(x)

Check:
For h=1 we get the beginning of A_1

Code:
´
A_1:
                           0
      0.57162361+1.0864612*I
    -0.42682215+0.62104685*I
   -0.30624163-0.036240215*I
  -0.033920340-0.088358850*I
   0.015321758-0.017472227*I
   ...

which is of course the same as the 2nd column in Ut itself.

The vector A_0.5 for Ut°0.5(x') begins with first 64 coefficients as follows and seems to have a slowly divergent sequence of entries.

Here is Ut°0.5(x') given as powerseries in x':

Code:
´
                                       0 +           (0.94849474+0.57272914*I)*x'
  +       (0.075363236+0.32629361*I)*x'^2 +      (-0.017446691+0.022563745*I)*x'^3
  +   (-0.0034925347-0.0020992659*I)*x'^4 +  (-0.000015974734+0.0036674771*I)*x'^5
  +    (0.0045440026-0.0055371510*I)*x'^6 +      (-0.010881477+0.012802630*I)*x'^7
  +     (0.0094010926-0.027893929*I)*x'^8 +      (0.0068992435+0.039389807*I)*x'^9
  +    (-0.029755850-0.034623204*I)*x'^10 +      (0.043933293+0.013888987*I)*x'^11
  +    (-0.041102377+0.011322125*I)*x'^12 +      (0.024005637-0.028919052*I)*x'^13
  +   (-0.0011784702+0.033375740*I)*x'^14 +     (-0.019605161-0.025397896*I)*x'^15
  +    (0.033453991+0.0081087366*I)*x'^16 +     (-0.036878144+0.014840833*I)*x'^17
  +     (0.027006852-0.038148976*I)*x'^18 +    (-0.0036781248+0.053912245*I)*x'^19
  +    (-0.027708275-0.053842125*I)*x'^20 +      (0.056195979+0.033961262*I)*x'^21
  +   (-0.069317259+0.0017624491*I)*x'^22 +      (0.058909002-0.041734262*I)*x'^23
  +    (-0.025253063+0.070928405*I)*x'^24 +     (-0.022366540-0.076595337*I)*x'^25
  +     (0.068379966+0.053024259*I)*x'^26 +    (-0.095592452-0.0041201605*I)*x'^27
  +     (0.090634618-0.056569702*I)*x'^28 +      (-0.049205371+0.10866605*I)*x'^29
  +     (-0.020385356-0.13073050*I)*x'^30 +       (0.097340351+0.10806611*I)*x'^31
  +     (-0.15348021-0.040025855*I)*x'^32 +       (0.16266104-0.056519720*I)*x'^33
  +      (-0.11153240+0.15003727*I)*x'^34 +      (0.0074207583-0.20364656*I)*x'^35
  +       (0.12034367+0.18834832*I)*x'^36 +      (-0.22729198-0.095748995*I)*x'^37
  +      (0.26774097-0.054555400*I)*x'^38 +       (-0.21197038+0.21665590*I)*x'^39
  +      (0.060973606-0.33029389*I)*x'^40 +        (0.14726472+0.34088251*I)*x'^41
  +      (-0.34428288-0.22181063*I)*x'^42 +      (0.45090197-0.0090434807*I)*x'^43
  +      (-0.40560503+0.28635743*I)*x'^44 +        (0.19241344-0.51222367*I)*x'^45
  +       (0.14274919+0.58702754*I)*x'^46 +       (-0.49615266-0.44787066*I)*x'^47
  +       (0.73375762+0.10126897*I)*x'^48 +       (-0.73663932+0.36383272*I)*x'^49
  +       (0.45073248-0.79217029*I)*x'^50 +        (0.077418484+1.0077057*I)*x'^51
  +      (-0.69596341-0.87761013*I)*x'^52 +         (1.1830357+0.37401427*I)*x'^53
  +       (-1.3174698+0.39105319*I)*x'^54 +         (0.96551733-1.1773648*I)*x'^55
  +       (-0.15336955+1.6810426*I)*x'^56 +        (-0.90616777-1.6389445*I)*x'^57
  +        (1.8539135+0.94127137*I)*x'^58 +        (-2.2912467+0.28977904*I)*x'^59
  +         (1.9273758-1.6940103*I)*x'^60 +        (-0.71865274+2.7591658*I)*x'^61
  +        (-1.0565265-2.9866144*I)*x'^62 +          (2.8316963+2.0905013*I)*x'^63
  + ...

The series converges very slowly for x=1 (x'=1/t -1), if at all, and is very poorly Euler-summable.

But using more appropriate example-values for x, such that x'=1,x'=1.5 or the like we get some meaningful values, well approximatable (needs only small orders for Euler-summation).

Some results:

Code:
x'            =  1
  y=Ut°0.5(x')  =  1.0035209 + 0.92089313*I  abs(y) = 1.3620200
  z=Ut°0.5(y)   = -0.17532145 + 1.5674321*I  abs(z) = 1.5772067
  check: t^x'-1 = -0.17532145 + 1.5674321*I

  x  =          =  1.6493571  + 3.1348642*I  abs(x) = 3.5422807
  y" =Tb°0.5(x) =  0.20882326 + 3.8998239*I  abs(y")= 3.9054108
  z" =Tb°0.5(y")= -1.7767488  + 2.5852553*I  abs(z")= 3.1369382
  
  check:2^x     = -1.7767488  + 2.5852553*I

-----------------------------------------------------------------

  x'            =  1.2500000
  y=Ut°0.5(x')  =  1.2626569  + 1.2681218*I  abs(y)= 1.7895349
  z=Ut°0.5(y)   = -0.56863584 + 1.9971715*I  abs(z)= 2.0765454
  check: t^x'-1 = -0.56863584 + 1.9971715*I

  x  =          =  1.8555267  + 3.5267223*I  abs(x )= 3.9850658
  y" =Tb°0.5(x) = -0.12173015 + 4.5923540*I  abs(y")= 4.5939670
  z" =Tb°0.5(y")= -2.7746940  + 2.3231585*I  abs(z")= 3.6188385

  check:2^x     =  -2.7746940 + 2.3231585*I

-----------------------------------------------------------------

  x'            =   1.5000000
  y=Ut°0.5(x')  =   1.5206787  + 1.6658045*I  abs(y) = 2.2555195
  z=Ut°0.5(y)   =  -1.1387426  + 2.3530211*I  abs(z) = 2.6140855
  check: t^x'-1 =  -1.1387426  + 2.3530211*I

  x  =          =   2.0616964  + 3.9185803*I  abs(x )= 4.4278509
  y" =Tb°0.5(x) =  -0.53228577 + 5.3247459*I  abs(y")= 5.3512847
  z" =Tb°0.5(y")=  -3.8026189  + 1.7230164*I  abs(z")= 4.1747690

  check:2^x     =  -3.8026189  + 1.7230164*I

-----------------------------------------------------------------


Anyway - things are still not yet fully satisfying. The range for parameters is very much limited, and there are still incompatibilities.
The ugliest problem is the divergence with complex numbers involved - the common summation-methods are simply not taylored for complex series. Sigh...

Gottfried]]>
Using base b=2, then the fixpoint-shift is performed by

t = 0.824678546142 + 1.56743212385*I
u = 0.571623609127 + 1.08646115737*I = log(t)

such that b = t^(1/t) = exp(u/t) .


Using the fixpointshift

x' = x/t-1 x"=(x+1)*t

and the according triangular Bell-matrix Ut I could now compute some fractional iterates, which was not possible before.

There might be some mistake in my diagonalization-procedure due to fractional powers if the argument is complex. Because I could not locate the error yet I employed my new exponential polynomial interpolation. *In principle* this seems a start, the error seems to not to occur here.
(For the method see : http://go.helms-net.de/math/tetdocs/Expo...lation.pdf )

To check: Ut begins as follows

Code:
1                           .                        .                        .                        .                      .
  0      0.57162361+1.0864612*I                        .                        .                        .                      .
  0    -0.42682215+0.62104685*I  -0.85364430+1.2420937*I                        .                        .                      .
  0   -0.30624163-0.036240215*I  -1.8374498-0.21744129*I  -1.8374498-0.21744129*I                        .                      .
  0  -0.033920340-0.088358850*I  -0.47488476-1.2370239*I   -1.2211322-3.1809186*I  -0.81408817-2.1206124*I                      .
  0   0.015321758-0.017472227*I  0.45965275-0.52416682*I    2.2982637-2.6208341*I    3.6772220-4.1933346*I  1.8386110-2.0966673*I
  ...     ....

The computed matrix POLY begins as follows

Code:
0                           .                          .                          .                            .                           .
  0                   1.0000000                          .                          .                            .                           .
  0    -0.34295950+0.39829087*I    0.34295950-0.39829087*I                          .                            .                           .
  0    0.011293278-0.29398501*I   0.082028794+0.54639055*I  -0.093322072-0.25240554*I                            .                           .
  0    0.086289977+0.13933724*I   -0.34931427-0.28800518*I    0.39760954+0.14818664*I  -0.13458524+0.00048129196*I                           .
  0  -0.080900876-0.038017321*I  0.33173844+0.0088923457*I   -0.42115742+0.18129664*I      0.18386237-0.21507654*I  -0.013542508+0.062904872*I
  ...      ....


so we have to compute the column-vector A_h of coefficents for Ut°h(x)

Code:
´
A_h         = POLY * V(u^h)
V(x)~ * A_h = Ut°h(x)

Check:
For h=1 we get the beginning of A_1

Code:
´
A_1:
                           0
      0.57162361+1.0864612*I
    -0.42682215+0.62104685*I
   -0.30624163-0.036240215*I
  -0.033920340-0.088358850*I
   0.015321758-0.017472227*I
   ...

which is of course the same as the 2nd column in Ut itself.

The vector A_0.5 for Ut°0.5(x') begins with first 64 coefficients as follows and seems to have a slowly divergent sequence of entries.

Here is Ut°0.5(x') given as powerseries in x':

Code:
´
                                       0 +           (0.94849474+0.57272914*I)*x'
  +       (0.075363236+0.32629361*I)*x'^2 +      (-0.017446691+0.022563745*I)*x'^3
  +   (-0.0034925347-0.0020992659*I)*x'^4 +  (-0.000015974734+0.0036674771*I)*x'^5
  +    (0.0045440026-0.0055371510*I)*x'^6 +      (-0.010881477+0.012802630*I)*x'^7
  +     (0.0094010926-0.027893929*I)*x'^8 +      (0.0068992435+0.039389807*I)*x'^9
  +    (-0.029755850-0.034623204*I)*x'^10 +      (0.043933293+0.013888987*I)*x'^11
  +    (-0.041102377+0.011322125*I)*x'^12 +      (0.024005637-0.028919052*I)*x'^13
  +   (-0.0011784702+0.033375740*I)*x'^14 +     (-0.019605161-0.025397896*I)*x'^15
  +    (0.033453991+0.0081087366*I)*x'^16 +     (-0.036878144+0.014840833*I)*x'^17
  +     (0.027006852-0.038148976*I)*x'^18 +    (-0.0036781248+0.053912245*I)*x'^19
  +    (-0.027708275-0.053842125*I)*x'^20 +      (0.056195979+0.033961262*I)*x'^21
  +   (-0.069317259+0.0017624491*I)*x'^22 +      (0.058909002-0.041734262*I)*x'^23
  +    (-0.025253063+0.070928405*I)*x'^24 +     (-0.022366540-0.076595337*I)*x'^25
  +     (0.068379966+0.053024259*I)*x'^26 +    (-0.095592452-0.0041201605*I)*x'^27
  +     (0.090634618-0.056569702*I)*x'^28 +      (-0.049205371+0.10866605*I)*x'^29
  +     (-0.020385356-0.13073050*I)*x'^30 +       (0.097340351+0.10806611*I)*x'^31
  +     (-0.15348021-0.040025855*I)*x'^32 +       (0.16266104-0.056519720*I)*x'^33
  +      (-0.11153240+0.15003727*I)*x'^34 +      (0.0074207583-0.20364656*I)*x'^35
  +       (0.12034367+0.18834832*I)*x'^36 +      (-0.22729198-0.095748995*I)*x'^37
  +      (0.26774097-0.054555400*I)*x'^38 +       (-0.21197038+0.21665590*I)*x'^39
  +      (0.060973606-0.33029389*I)*x'^40 +        (0.14726472+0.34088251*I)*x'^41
  +      (-0.34428288-0.22181063*I)*x'^42 +      (0.45090197-0.0090434807*I)*x'^43
  +      (-0.40560503+0.28635743*I)*x'^44 +        (0.19241344-0.51222367*I)*x'^45
  +       (0.14274919+0.58702754*I)*x'^46 +       (-0.49615266-0.44787066*I)*x'^47
  +       (0.73375762+0.10126897*I)*x'^48 +       (-0.73663932+0.36383272*I)*x'^49
  +       (0.45073248-0.79217029*I)*x'^50 +        (0.077418484+1.0077057*I)*x'^51
  +      (-0.69596341-0.87761013*I)*x'^52 +         (1.1830357+0.37401427*I)*x'^53
  +       (-1.3174698+0.39105319*I)*x'^54 +         (0.96551733-1.1773648*I)*x'^55
  +       (-0.15336955+1.6810426*I)*x'^56 +        (-0.90616777-1.6389445*I)*x'^57
  +        (1.8539135+0.94127137*I)*x'^58 +        (-2.2912467+0.28977904*I)*x'^59
  +         (1.9273758-1.6940103*I)*x'^60 +        (-0.71865274+2.7591658*I)*x'^61
  +        (-1.0565265-2.9866144*I)*x'^62 +          (2.8316963+2.0905013*I)*x'^63
  + ...

The series converges very slowly for x=1 (x'=1/t -1), if at all, and is very poorly Euler-summable.

But using more appropriate example-values for x, such that x'=1,x'=1.5 or the like we get some meaningful values, well approximatable (needs only small orders for Euler-summation).

Some results:

Code:
x'            =  1
  y=Ut°0.5(x')  =  1.0035209 + 0.92089313*I  abs(y) = 1.3620200
  z=Ut°0.5(y)   = -0.17532145 + 1.5674321*I  abs(z) = 1.5772067
  check: t^x'-1 = -0.17532145 + 1.5674321*I

  x  =          =  1.6493571  + 3.1348642*I  abs(x) = 3.5422807
  y" =Tb°0.5(x) =  0.20882326 + 3.8998239*I  abs(y")= 3.9054108
  z" =Tb°0.5(y")= -1.7767488  + 2.5852553*I  abs(z")= 3.1369382
  
  check:2^x     = -1.7767488  + 2.5852553*I

-----------------------------------------------------------------

  x'            =  1.2500000
  y=Ut°0.5(x')  =  1.2626569  + 1.2681218*I  abs(y)= 1.7895349
  z=Ut°0.5(y)   = -0.56863584 + 1.9971715*I  abs(z)= 2.0765454
  check: t^x'-1 = -0.56863584 + 1.9971715*I

  x  =          =  1.8555267  + 3.5267223*I  abs(x )= 3.9850658
  y" =Tb°0.5(x) = -0.12173015 + 4.5923540*I  abs(y")= 4.5939670
  z" =Tb°0.5(y")= -2.7746940  + 2.3231585*I  abs(z")= 3.6188385

  check:2^x     =  -2.7746940 + 2.3231585*I

-----------------------------------------------------------------

  x'            =   1.5000000
  y=Ut°0.5(x')  =   1.5206787  + 1.6658045*I  abs(y) = 2.2555195
  z=Ut°0.5(y)   =  -1.1387426  + 2.3530211*I  abs(z) = 2.6140855
  check: t^x'-1 =  -1.1387426  + 2.3530211*I

  x  =          =   2.0616964  + 3.9185803*I  abs(x )= 4.4278509
  y" =Tb°0.5(x) =  -0.53228577 + 5.3247459*I  abs(y")= 5.3512847
  z" =Tb°0.5(y")=  -3.8026189  + 1.7230164*I  abs(z")= 4.1747690

  check:2^x     =  -3.8026189  + 1.7230164*I

-----------------------------------------------------------------


Anyway - things are still not yet fully satisfying. The range for parameters is very much limited, and there are still incompatibilities.
The ugliest problem is the divergence with complex numbers involved - the common summation-methods are simply not taylored for complex series. Sigh...

Gottfried]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=193 Fri, 18 Jul 2008 14:28:53 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=193 http://math.eretrandre.org/tetrationforum/showthread.php?tid=191 Wed, 16 Jul 2008 23:40:06 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=191
*bo goes around distributing a bottle champaign into glasses ... *
we have reached the 2000 posts!!!

Tata

*ggg*]]>
*bo goes around distributing a bottle champaign into glasses ... *
we have reached the 2000 posts!!!

Tata

*ggg*]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=190 Tue, 15 Jul 2008 13:52:00 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=190
It also seems to back the diagonalization-method from another point of view.

I've not seen this before (nor in a more common serial representation) - may be someone recognizes it though I used the matrix-notation.

It is at
http://go.helms-net.de/math/tetdocs/Expo...lation.pdf

and -if of interest here- I'd upload it to the forum-resources.

Gottfried]]>
It also seems to back the diagonalization-method from another point of view.

I've not seen this before (nor in a more common serial representation) - may be someone recognizes it though I used the matrix-notation.

It is at
http://go.helms-net.de/math/tetdocs/Expo...lation.pdf

and -if of interest here- I'd upload it to the forum-resources.

Gottfried]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=189 Mon, 14 Jul 2008 22:49:28 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=189
Without checking the whole forum through, I just wanted to throw in something from my experiences concerning tetration.
I hear there are some different possible solutions to this function, and I'm too curious if the following example is one of them:
While dabbling in this field of maths, I found a formula for 2^^x:
For 0 < x < 1, with n = 0.345627*(2-x), 2^^x ~~ [x*(2^n-1)+1]^(1/n)
(it's not 100% accurate, but ... about 99%, maybe some slight adjustments can fix that)
I could try and waste hours finding other parameters for other bases, but before I do that, I considered obtaining professional info from professional mathematicians.

Thanks in advance,
martin]]>
Without checking the whole forum through, I just wanted to throw in something from my experiences concerning tetration.
I hear there are some different possible solutions to this function, and I'm too curious if the following example is one of them:
While dabbling in this field of maths, I found a formula for 2^^x:
For 0 < x < 1, with n = 0.345627*(2-x), 2^^x ~~ [x*(2^n-1)+1]^(1/n)
(it's not 100% accurate, but ... about 99%, maybe some slight adjustments can fix that)
I could try and waste hours finding other parameters for other bases, but before I do that, I considered obtaining professional info from professional mathematicians.

Thanks in advance,
martin]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=188 Mon, 14 Jul 2008 09:17:01 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=188 Here I cite myself with two articles in sci.math (a little bit edited), melting them here into one.


(...)

b) Interpolation: in a previous post I discussed a likely difference of methods, when different interpolation-approaches depending on the h-parameter are assumed.
You gave a polynomial interpolation approach, which I think is somehow natural.
But the coefficients at -for instance- x^1 with increasing h

(1,1,1,1,1,...)

or at x^2

(0,1/2,2/2,3/2,4/2,... )

can also be interpolated including -for instance- a sine-function of h; as well as the coefficients at higher powers of x.

I don't mean to play a game of obfuscation here: the reason for my being "not-completely-satisfied" is, that -using U-tetration as application of T-tetration with fixpoint-shift- the common tetration (T-tetration in my wording) seems to be dependent on the selection of the fixpoint, if fractional iterations are computed.

So - while the matrix-based (diagonalization) method using U-tetration for each fixed base only may be consistent, when the polynomial interpolation-approach is applied, then still the cross-base-relations might be "imperfect"...

Hmm - I must be vague this way, because I still don't have hard data at hand to see these differences (they are said to be small, may be smaller than my approximation-accuracy) and so seem currently to be too small to be able to experiment with this problem effectively.

At least there is one consolidation: in my previous post I mentioned the different interpolation-method using the binomial-expansion and values of the powertower-function themselves (as can be seen for instance in [1],[2] or [3]) Here I found different results in my first comparision (using insufficient approximation) - however, a new computation indicates now, that the results of this method and of the diagonalization may come out to be the same (as expected) [4]


(second letter)

(...)
Perhaps I should explain this a bit more.

The fixpoint-substitution, which relates T and U-tetration is, for a base b=t^(1/t)



for the integer case of h; for fractional this is then assumed.

We try, using the most simple case, base b=sqrt(2) = 2^(1/2) = 4^(1/4)

So let and such that

--------------------------

We expect then, for general height h,


so the U-tetration for base 2 and for base 4 must give "compatible" results for their fractional interpolations - this is what I meant with "cross-base-relations"

The series, which occur with these U-tetrates are all divergent, and I can assign values only to a certain accuracy - while a summation-method were needed, which allows arbitrary accuracy: to first quantify the difference according to the different fixpoint-shifts and then second to formulate a hypothese for a correction-term, which is worth to work on.



[1] Comtet, Louis; Advanced Combinatorics,

[2] Woon, S.C.; Analytic Continuation of Operators —
Operators acting complex s-times
Chap 9 (online available in arXiv-org)

[3] Robbins, Andrew; (forum-message binomial-method=Woon-method)
http://math.eretrandre.org/tetrationforu...19#pid2319

[4] Helms, Gottfried; (binomial-method approximative equal to diagonalization)
http://math.eretrandre.org/tetrationforu...21#pid2321]]> Here I cite myself with two articles in sci.math (a little bit edited), melting them here into one.


(...)

b) Interpolation: in a previous post I discussed a likely difference of methods, when different interpolation-approaches depending on the h-parameter are assumed.
You gave a polynomial interpolation approach, which I think is somehow natural.
But the coefficients at -for instance- x^1 with increasing h

(1,1,1,1,1,...)

or at x^2

(0,1/2,2/2,3/2,4/2,... )

can also be interpolated including -for instance- a sine-function of h; as well as the coefficients at higher powers of x.

I don't mean to play a game of obfuscation here: the reason for my being "not-completely-satisfied" is, that -using U-tetration as application of T-tetration with fixpoint-shift- the common tetration (T-tetration in my wording) seems to be dependent on the selection of the fixpoint, if fractional iterations are computed.

So - while the matrix-based (diagonalization) method using U-tetration for each fixed base only may be consistent, when the polynomial interpolation-approach is applied, then still the cross-base-relations might be "imperfect"...

Hmm - I must be vague this way, because I still don't have hard data at hand to see these differences (they are said to be small, may be smaller than my approximation-accuracy) and so seem currently to be too small to be able to experiment with this problem effectively.

At least there is one consolidation: in my previous post I mentioned the different interpolation-method using the binomial-expansion and values of the powertower-function themselves (as can be seen for instance in [1],[2] or [3]) Here I found different results in my first comparision (using insufficient approximation) - however, a new computation indicates now, that the results of this method and of the diagonalization may come out to be the same (as expected) [4]


(second letter)

(...)
Perhaps I should explain this a bit more.

The fixpoint-substitution, which relates T and U-tetration is, for a base b=t^(1/t)



for the integer case of h; for fractional this is then assumed.

We try, using the most simple case, base b=sqrt(2) = 2^(1/2) = 4^(1/4)

So let and such that

--------------------------

We expect then, for general height h,


so the U-tetration for base 2 and for base 4 must give "compatible" results for their fractional interpolations - this is what I meant with "cross-base-relations"

The series, which occur with these U-tetrates are all divergent, and I can assign values only to a certain accuracy - while a summation-method were needed, which allows arbitrary accuracy: to first quantify the difference according to the different fixpoint-shifts and then second to formulate a hypothese for a correction-term, which is worth to work on.



[1] Comtet, Louis; Advanced Combinatorics,

[2] Woon, S.C.; Analytic Continuation of Operators —
Operators acting complex s-times
Chap 9 (online available in arXiv-org)

[3] Robbins, Andrew; (forum-message binomial-method=Woon-method)
http://math.eretrandre.org/tetrationforu...19#pid2319

[4] Helms, Gottfried; (binomial-method approximative equal to diagonalization)
http://math.eretrandre.org/tetrationforu...21#pid2321]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=186 Mon, 30 Jun 2008 16:01:39 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=186
We have a very simple formula for computing the -th iterate of an arbitrary function:



This series does not always converge but at least if has an attracting fixed point reachable from then converges, because then is bounded, say ]]>
We have a very simple formula for computing the -th iterate of an arbitrary function:



This series does not always converge but at least if has an attracting fixed point reachable from then converges, because then is bounded, say ]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=184 Sat, 28 Jun 2008 19:30:27 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=184
This is not a thread for asking questions nor for discussing the FAQ.
If you have a question open a new thread in General Discussion and Questions after you read the current tetration FAQ.
If you have suggestions/concerns/questions regarding the tetration FAQ post them in the Tetration FAQ discussion.]]>
This is not a thread for asking questions nor for discussing the FAQ.
If you have a question open a new thread in General Discussion and Questions after you read the current tetration FAQ.
If you have suggestions/concerns/questions regarding the tetration FAQ post them in the Tetration FAQ discussion.]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=183 Wed, 25 Jun 2008 14:35:40 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=183
I just found this article, which seem to be relevant to our subject when just skimming through. Although the powertower is not mentioned this seems also to mean functions whose infinite iterations converge to a fixpoint (if I didn't misread this article)

Die algebraischen Funktionen, deren Iteration die Identität liefert.
Hornich, Hans
in: Monatshefte für Mathematik Vol 52
Pg 311 - 322

http://gdz.sub.uni-goettingen.de/dms/res...N002469863

This is in german language. It may be as well relevant for the tetra-series-problem, since the infinite iterations to the two fixpoints are involved in computation of the series.

Gottfried]]>
I just found this article, which seem to be relevant to our subject when just skimming through. Although the powertower is not mentioned this seems also to mean functions whose infinite iterations converge to a fixpoint (if I didn't misread this article)

Die algebraischen Funktionen, deren Iteration die Identität liefert.
Hornich, Hans
in: Monatshefte für Mathematik Vol 52
Pg 311 - 322

http://gdz.sub.uni-goettingen.de/dms/res...N002469863

This is in german language. It may be as well relevant for the tetra-series-problem, since the infinite iterations to the two fixpoints are involved in computation of the series.

Gottfried]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=182 Sat, 21 Jun 2008 15:24:40 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=182
It t is polar angle, to plot series 1-1+1-1.. I plotted

r(t)=1-t+t^2-t^3+t^4 ..........+t^(2n)and
r(t)=1-t+t^2-t^3+t^4-t^5 ......- t^(2n+1)

the resulting series were symmetric and their combined graphs showed obvious inclination to converge towards 1/2 on vertical axis which is a "value" usually asigned to these osciullating series.

Given the limited power of mu system, I only usead up to 100 terms in each series, so pictures are still quite far from convergence point.

In similar way, I tried to plot:

1+1+1+... =-1/2
1-2+4-8+ =1/3
1-2+3-4+5.... = 1/4
1+2+3+4+5.. = -1/12

They all show clear tendency to converge graphically to these values polar plots.



The last series (-1/12) seems to need much more terms than I could produce to be sure where they converge.

In this plot, all mentioned series are plotted together, but no decription on graph- this is very easy to duplicate.

After that I tried plotting harmonic series. Alternating ones converged to ln2, as they should, divergent harmonic show after 100 terms makes me think that they either have value y= -ln(2) but these need much more terms to be evaluated, if possible. It might be that thses series have both r and phase value in sum.


I have rotated argument by replacing t with t-pi/2+1 to see if it helps to see convergence on y axis.In last 2 graphs I have added series in 1/t as well- that seems to improve localization of sum on graph as graphs now converge to that point from 4 directions.

Invserse prime summation seems to lead to sqrt(3) -1 as mod ® so summation can be (-(sqrt(3)-1)) plus phase, but again with my limited precision not easy to be sure, as well as i havent found out how to really calculate exact phase- though numerically it must be possible to evaluate very accurately.


This polar presentation seems a good way to show graphically both convergent and divergent infinite sums. The phase seems to be an important part and feature of especially divergent summation.

Ivars]]>
It t is polar angle, to plot series 1-1+1-1.. I plotted

r(t)=1-t+t^2-t^3+t^4 ..........+t^(2n)and
r(t)=1-t+t^2-t^3+t^4-t^5 ......- t^(2n+1)

the resulting series were symmetric and their combined graphs showed obvious inclination to converge towards 1/2 on vertical axis which is a "value" usually asigned to these osciullating series.

Given the limited power of mu system, I only usead up to 100 terms in each series, so pictures are still quite far from convergence point.

In similar way, I tried to plot:

1+1+1+... =-1/2
1-2+4-8+ =1/3
1-2+3-4+5.... = 1/4
1+2+3+4+5.. = -1/12

They all show clear tendency to converge graphically to these values polar plots.



The last series (-1/12) seems to need much more terms than I could produce to be sure where they converge.

In this plot, all mentioned series are plotted together, but no decription on graph- this is very easy to duplicate.

After that I tried plotting harmonic series. Alternating ones converged to ln2, as they should, divergent harmonic show after 100 terms makes me think that they either have value y= -ln(2) but these need much more terms to be evaluated, if possible. It might be that thses series have both r and phase value in sum.


I have rotated argument by replacing t with t-pi/2+1 to see if it helps to see convergence on y axis.In last 2 graphs I have added series in 1/t as well- that seems to improve localization of sum on graph as graphs now converge to that point from 4 directions.

Invserse prime summation seems to lead to sqrt(3) -1 as mod ® so summation can be (-(sqrt(3)-1)) plus phase, but again with my limited precision not easy to be sure, as well as i havent found out how to really calculate exact phase- though numerically it must be possible to evaluate very accurately.


This polar presentation seems a good way to show graphically both convergent and divergent infinite sums. The phase seems to be an important part and feature of especially divergent summation.

Ivars]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=180 Tue, 17 Jun 2008 14:02:32 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=180
If cardinality of Natural numbers is Aleph_0, and their powerset P(N) has cardinality of continuum, |P(N)|= c = 2^Aleph_0 which is also the cardinality of R, then:

x^x has cardinality of R^R = c^c= cardinality of power set of Reals=|P(R )|=2^c=2^2^Aleph_0 then for each step of tetration we have to add 2, so via recursion:

Cardinality

|x[4]1|=|P(N)|= |R|=2^Aleph_0

|x[4]n|= |P(x[4]n-1|= 2^| x[4]n-1| and generally:


|x[4]n|= |P(x[4]n-1|= (2[4]n)^| x[4]1|

What would be cardinality for infinite tetration , known to converge to real values or give complex values by analytic continuation? And what is the cardinality of a number obtained as a result of tetration, even for finite n?


Some references here:
Cardinality of continuum


Another question is , what is than the cardinality of:

x[4]y, x[4]I? From previous considerations, can it be (for y>1):

|x[4]y|= |P(x[4]y-1|=2^?

What is cardinality of I^I?

And what would such cardinalities mean?

Similarly the same question arises when trying to understand what is fractional or real Cartesian product of sets? May be the answer is in tetration.




Ivars]]>
If cardinality of Natural numbers is Aleph_0, and their powerset P(N) has cardinality of continuum, |P(N)|= c = 2^Aleph_0 which is also the cardinality of R, then:

x^x has cardinality of R^R = c^c= cardinality of power set of Reals=|P(R )|=2^c=2^2^Aleph_0 then for each step of tetration we have to add 2, so via recursion:

Cardinality

|x[4]1|=|P(N)|= |R|=2^Aleph_0

|x[4]n|= |P(x[4]n-1|= 2^| x[4]n-1| and generally:


|x[4]n|= |P(x[4]n-1|= (2[4]n)^| x[4]1|

What would be cardinality for infinite tetration , known to converge to real values or give complex values by analytic continuation? And what is the cardinality of a number obtained as a result of tetration, even for finite n?


Some references here:
Cardinality of continuum


Another question is , what is than the cardinality of:

x[4]y, x[4]I? From previous considerations, can it be (for y>1):

|x[4]y|= |P(x[4]y-1|=2^?

What is cardinality of I^I?

And what would such cardinalities mean?

Similarly the same question arises when trying to understand what is fractional or real Cartesian product of sets? May be the answer is in tetration.




Ivars]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=179 Wed, 11 Jun 2008 06:46:43 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=179
First lets start with some recent realizations about Abel functions, Julia functions, and topological conjugacy. So, I'll use the notation, and thus the topological conjugacy between "exp" and "dxp" can be expressed as:

which can also be found in this thread. The Abel functions and Julia functions can be related as:

which, as they apply to exp/dxp, can be proven (and is proven later) to imply:



Second, I think the process of finding iterated-dxp is well understood by now, so I'll start with that. One recent observation in this thread has been that Szekeres' Julia functions and Jabotinsky's L-functions (or iterative logarithm) are actually the same functions, which has opened my eyes to a whole new approach to iteration. With this in mind, not only can we express the Abel function as:

or the logarithm of the Schroeder function, but we can also express it as:

which also serves to emphasize the fact that Abel functions are only determined up to a constant, and that a solution (to the Abel functional equation) can always be generalized to , which is also true of integrals.

Iterated-dxp can be expressed as the hyperbolic iteration of as:

and since the Julia function of dxp is also the iterative logarithm of dxp:

which evaluates to the power series:

We can now find the Abel function of dxp by integrating its Julia function, but we need to find the reciprocal first. Finding the reciprocal of a power series can be a tedious task, so I've done the work for you:

and since , we solve for by equating the coefficients of x, and the solution to these equations is:


Lastly, we can relate these findings back to the super-logarithm.

Let be the Abel function of dxp.
Let .
Then is an Abel function of exp.
Proof.




. []

This means that which relates back to the super-logarithm as follows:


:)

Andrew Robbins]]>
First lets start with some recent realizations about Abel functions, Julia functions, and topological conjugacy. So, I'll use the notation, and thus the topological conjugacy between "exp" and "dxp" can be expressed as:

which can also be found in this thread. The Abel functions and Julia functions can be related as:

which, as they apply to exp/dxp, can be proven (and is proven later) to imply:



Second, I think the process of finding iterated-dxp is well understood by now, so I'll start with that. One recent observation in this thread has been that Szekeres' Julia functions and Jabotinsky's L-functions (or iterative logarithm) are actually the same functions, which has opened my eyes to a whole new approach to iteration. With this in mind, not only can we express the Abel function as:

or the logarithm of the Schroeder function, but we can also express it as:

which also serves to emphasize the fact that Abel functions are only determined up to a constant, and that a solution (to the Abel functional equation) can always be generalized to , which is also true of integrals.

Iterated-dxp can be expressed as the hyperbolic iteration of as:

and since the Julia function of dxp is also the iterative logarithm of dxp:

which evaluates to the power series:

We can now find the Abel function of dxp by integrating its Julia function, but we need to find the reciprocal first. Finding the reciprocal of a power series can be a tedious task, so I've done the work for you:

and since , we solve for by equating the coefficients of x, and the solution to these equations is:


Lastly, we can relate these findings back to the super-logarithm.

Let be the Abel function of dxp.
Let .
Then is an Abel function of exp.
Proof.




. []

This means that which relates back to the super-logarithm as follows:


:)

Andrew Robbins]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=178 Sat, 07 Jun 2008 22:35:01 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=178
quickfur Wrote:
It makes one wonder if the inverse of tetration would also create new numbers... I suspect it must've come up in this forum before, right?

Just interested as it's something i've wondered too, so i'd like to see the original post.]]>
quickfur Wrote:
It makes one wonder if the inverse of tetration would also create new numbers... I suspect it must've come up in this forum before, right?

Just interested as it's something i've wondered too, so i'd like to see the original post.]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=177 Sun, 01 Jun 2008 18:09:21 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=177
when updating the "basics of diagonalization" article, meant for Ivars, I detected a problem, which may have consequences in the diagonalization-approach and the attempt to get fractional powers by matrix-logarithm and/or diagonalization. Or, with the fixpoint-shift - I can't estimate it.
It looks like a minor curiosity, but makes me think about the completeness of the matrix-instruments, which we apply.

See the last (short) chapter in Intro_in_Diagonalization.pdf

Gottfried]]>
when updating the "basics of diagonalization" article, meant for Ivars, I detected a problem, which may have consequences in the diagonalization-approach and the attempt to get fractional powers by matrix-logarithm and/or diagonalization. Or, with the fixpoint-shift - I can't estimate it.
It looks like a minor curiosity, but makes me think about the completeness of the matrix-instruments, which we apply.

See the last (short) chapter in Intro_in_Diagonalization.pdf

Gottfried]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=175 Sun, 01 Jun 2008 14:54:52 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=175
Similar situation arises when dealing with tetration and higher operations today. One can try to generalize the usefulness of exp, log functions to increase/reduce order of operations by 1. We denote summation with [1], multiplication with [2], exponentiation with [3] and tetration with [4] etc.

Let us call normal logarithm 2_log, and normal exponentiation 2_exp. Then:

2_exp(a+b) = 2_exp(a[1]b)=2_exp(a)*2_exp(b) = 2_exp(a)[2]2_exp(b)

2_log (a*b) = 2_log(a)+2_log(b) = (2_log(a))[1](2_log(b))

We can form now more logarithms and exponetiations, requiring:

3_log(a^b) =3_log(a[3]b)= 3_log(a)*3_log(b) = 3_log(a)[2]3_log(b)

4_log(a[4]b)= 4_log(a)[3]4_log(b) = (4_log(a))^(4_log(b))

n_log(a[n]b)=n_log(a)[n-1] n_log(b)

This 3 log will have the same value for a^b, and b^a which would only be true for roots of equation a^b=b^a. Otherwise, 3_log must have either sign difference or index to indicate order of a and b.

Inversely, 3_exp, 4_exp and n_exp:

3_exp(a*b) =3_exp(a[2]b) = (3_exp(a))^(3_exp(b)) = 3_exp(a)[3]3_exp(b)

4_exp(a^b) = 4_exp(a[3]b)=4_exp(a)[4]4_exp(b)

n_exp(a[n-1]b)=n_exp(a)[n]n_exp(b)

Since operations above and including tetration are all too fast for numerical analysis, n_log defined in such way as above would not help much, but we could look for (3, 2) - log which reduces the order of infinity by 2 and generally for log that reduces order of infinity by as much as we need (e.g. - m). So far we stick to integer (n,m) values of orders of infinity since iteration to intermediate should be possible once the properties of functions are figured out.

(3, 2)_log (a^b) = (3, 2)_log (a[3]b)= (3,2)_log (a)+(3,2)_log (b)= (3,2)_log(a)[1](3,2)_log(b)

(4, 2)_log (a[4]b) = (4, 2)_log (a[4]b)= (4,2)_log (a)*(4,2)_log (b)= (4,2)_log(a)[2](4,2)_log(b)

In general:

(n,m)_log(a[n]b)=(n,m)_log(a) [n-m] (n,m)_log(b)

(n,m)_exp(a[n]b)=(n,m)_exp(a)[n+m] (n,m)_exp(b)

We may ask if there exists natural basis for such logarithms and exponentials, series expansion of some kind, is it related to n-factorials, and how to deal with noncommmutative/non-associative and multivalue properties of these functions arising from that.The development of the nature of such basis itself, (even if only to jump over one order of infinity, but more so over 2 or more) should be very informative.

After that we may form a 2-d table of few available values arising from (n,m) as corresponding natural basis and some values of numbers in those basis along 3rd d.

But most useful would probably be the possibility to have rules for dealing with those functions analogous to rules dealing with log ,exp based on calculus.

Which might mean most likely generalization of calculus and infinitesimal analysis-similarly how its basic form appeared soon after logarithms were invented.]]>
Similar situation arises when dealing with tetration and higher operations today. One can try to generalize the usefulness of exp, log functions to increase/reduce order of operations by 1. We denote summation with [1], multiplication with [2], exponentiation with [3] and tetration with [4] etc.

Let us call normal logarithm 2_log, and normal exponentiation 2_exp. Then:

2_exp(a+b) = 2_exp(a[1]b)=2_exp(a)*2_exp(b) = 2_exp(a)[2]2_exp(b)

2_log (a*b) = 2_log(a)+2_log(b) = (2_log(a))[1](2_log(b))

We can form now more logarithms and exponetiations, requiring:

3_log(a^b) =3_log(a[3]b)= 3_log(a)*3_log(b) = 3_log(a)[2]3_log(b)

4_log(a[4]b)= 4_log(a)[3]4_log(b) = (4_log(a))^(4_log(b))

n_log(a[n]b)=n_log(a)[n-1] n_log(b)

This 3 log will have the same value for a^b, and b^a which would only be true for roots of equation a^b=b^a. Otherwise, 3_log must have either sign difference or index to indicate order of a and b.

Inversely, 3_exp, 4_exp and n_exp:

3_exp(a*b) =3_exp(a[2]b) = (3_exp(a))^(3_exp(b)) = 3_exp(a)[3]3_exp(b)

4_exp(a^b) = 4_exp(a[3]b)=4_exp(a)[4]4_exp(b)

n_exp(a[n-1]b)=n_exp(a)[n]n_exp(b)

Since operations above and including tetration are all too fast for numerical analysis, n_log defined in such way as above would not help much, but we could look for (3, 2) - log which reduces the order of infinity by 2 and generally for log that reduces order of infinity by as much as we need (e.g. - m). So far we stick to integer (n,m) values of orders of infinity since iteration to intermediate should be possible once the properties of functions are figured out.

(3, 2)_log (a^b) = (3, 2)_log (a[3]b)= (3,2)_log (a)+(3,2)_log (b)= (3,2)_log(a)[1](3,2)_log(b)

(4, 2)_log (a[4]b) = (4, 2)_log (a[4]b)= (4,2)_log (a)*(4,2)_log (b)= (4,2)_log(a)[2](4,2)_log(b)

In general:

(n,m)_log(a[n]b)=(n,m)_log(a) [n-m] (n,m)_log(b)

(n,m)_exp(a[n]b)=(n,m)_exp(a)[n+m] (n,m)_exp(b)

We may ask if there exists natural basis for such logarithms and exponentials, series expansion of some kind, is it related to n-factorials, and how to deal with noncommmutative/non-associative and multivalue properties of these functions arising from that.The development of the nature of such basis itself, (even if only to jump over one order of infinity, but more so over 2 or more) should be very informative.

After that we may form a 2-d table of few available values arising from (n,m) as corresponding natural basis and some values of numbers in those basis along 3rd d.

But most useful would probably be the possibility to have rules for dealing with those functions analogous to rules dealing with log ,exp based on calculus.

Which might mean most likely generalization of calculus and infinitesimal analysis-similarly how its basic form appeared soon after logarithms were invented.]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=174 Sun, 01 Jun 2008 12:38:43 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=174
Tr (n) = n+(n-1) + (n-2) + ....2+1

n!= n*(n-1)*(n-2)*...*2*1

I am looking for properties of [3] analogue of factorial, its extension to complex numbers and generating function. There finite number of such factorials for each n, as exponentiation in not commutative, with 2 border cases:

Smallest:

Exponential factorial : n^(n-1)^(n-2)^ ...^2^1

Biggest:

3-factorial ( my invention for name) = 2^3^4^......(n-1)^n

Notice 1 is EXCLUDED from factorial here as it will destroy the whole idea giving result 1 for all n.

I am also interested in [0] analogue for factorial, [i] analogue , [-n] analogue - no idea yet what it means, even. With n-tations over [3] - [4] factorial, it is simpler though the number of factorials increase rapidly, but at least it is clear what it means:

4-factorial = 2[4]3[4]4[4].............(n-1)[4]n

I wonder if 2 stays, or shall we start from 3-just by analogy:


[1] factorial includes 0 as it is summation, but not negative numbers
[2] factorial includes 1 as it is multiplication, but not 0
[3] factorial includes 2 as it is exponentiation, but not 1

Conjecture :

[4] factorial includes 3?? as it is tetration, but not 2???
[5] factorial includes 4?? as it is pentation, but not 3???

[0] factorial includes -1? etc..


Ivars]]>
Tr (n) = n+(n-1) + (n-2) + ....2+1

n!= n*(n-1)*(n-2)*...*2*1

I am looking for properties of [3] analogue of factorial, its extension to complex numbers and generating function. There finite number of such factorials for each n, as exponentiation in not commutative, with 2 border cases:

Smallest:

Exponential factorial : n^(n-1)^(n-2)^ ...^2^1

Biggest:

3-factorial ( my invention for name) = 2^3^4^......(n-1)^n

Notice 1 is EXCLUDED from factorial here as it will destroy the whole idea giving result 1 for all n.

I am also interested in [0] analogue for factorial, [i] analogue , [-n] analogue - no idea yet what it means, even. With n-tations over [3] - [4] factorial, it is simpler though the number of factorials increase rapidly, but at least it is clear what it means:

4-factorial = 2[4]3[4]4[4].............(n-1)[4]n

I wonder if 2 stays, or shall we start from 3-just by analogy:


[1] factorial includes 0 as it is summation, but not negative numbers
[2] factorial includes 1 as it is multiplication, but not 0
[3] factorial includes 2 as it is exponentiation, but not 1

Conjecture :

[4] factorial includes 3?? as it is tetration, but not 2???
[5] factorial includes 4?? as it is pentation, but not 3???

[0] factorial includes -1? etc..


Ivars]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=173 Sat, 31 May 2008 17:11:04 +0200 http://math.eretrandre.org/tetrationforum/showthread.php?tid=173