03/02/2021, 05:09 AM
(This post was last modified: 03/02/2021, 10:32 PM by sheldonison.)

(03/01/2021, 11:22 PM)MphLee Wrote: I'm probably missing some key piece of the puzzle (terminology). Are you talking about a kind of inverse Schroeder-like function right? A confortable abuse of name similar to how we can call inverse Abel-like function of ?

In a strict sense, I don't see how is a Schroeder function of or of .

Sorry for the confusion; yes you are correct. The Schroeder like function is Schroeder like only in that it has a formal power series beginning with x+a2x^2 ... and a multiplier at zero, with the multiplier=e. Since it is a formal series, we can get the formal inverse and generate a function and then it turns out the function we're iterating is actually

This is the function James is actually iterating when he generates

Schroeder function and inverse formal definition using f(x)

this works for n=2,3,4 ....

The FPS (formal power series) approach is another intriguing approach to understanding , and the iterated functions. The FPS approach would need more effort to make it rigorous; and the effort to make the FPS rigorous might become increasingly daunting for the iterated phi series for n>2. Even though is entire, f has singularities where the derivative of is equal to zero. Here is the Taylor series for f; the function we are actually iterating to generate , which has a fixed point of

Code:

`{f=`

x^ 1* 2.71828182845905

+x^ 2* 1.71828182845905

+x^ 3* 0.775624792750073

+x^ 4* 0.191889268327428

+x^ 5* 0.0520249429156080

+x^ 6* 0.00599242247026314

+x^ 7* 0.00182349994116415

+x^ 8* 9.81807721872041 E-5

+x^ 9* -5.19018256906951 E-5

+x^10* 7.84647429007181 E-5

+x^11* -5.26096655278693 E-5

+x^12* 3.02110037576056 E-5

+x^13* -1.51896837385654 E-5

+x^14* 6.77204817742090 E-6

+x^15* -2.59325526178607 E-6 ...

Here are the first few Taylor series coefficients of the function which is entire. We can generate the individual terms with a closed form in terms of "e", but I don't have a generic equation for the closed form. The higher order pentation, and hexation also have similar formal series representations, which I have also generated.

Code:

`x`

+x^ 2* 0.367879441171442

+x^ 3* 0.117454709986170

+x^ 4* 0.0324612092929206

+x^ 5* 0.00811730704942829

+x^ 6* 0.00188547471967479

+x^ 7* 0.000413224905195451

+x^ 8* 8.63482541739982 E-5

+x^ 9* 1.73333585608164 E-5

+x^10* 3.36137276288664 E-6

+x^11* 6.32477784106711 E-7

+x^12* 1.15869533017107 E-7

+x^13* 2.07255547935482 E-8

+x^14* 3.62795192280962 E-9

+x^15* 6.22699185709248 E-10 + ...

Finally, for completeness here are the first few Taylor series terms of the formal series for

Code:

`x`

+x^ 2* -0.367879441171442

+x^ 3* 0.153215856487055

+x^ 4* -0.0653506857689096

+x^ 5* 0.0275282379807258

+x^ 6* -0.0111894054323465

+x^ 7* 0.00428067464337933

+x^ 8* -0.00147921549686095

+x^ 9* 0.000417655162777504

+x^10* -5.90712473237761 E-5

+x^11* -3.60198809495273 E-5

+x^12* 4.43758017488974 E-5

+x^13* -3.14471384470661 E-5

+x^14* 1.81562489170478 E-5

+x^15* -9.15769831156020 E-6

- Sheldon