Closest approximation for #MarkovNumbers yet 😁. Used a combination of my own #snakegraph theory and my own #chebyshevpolynomial theory to get there.

I can use the #ChebyshevPolynomial formula to calculate those Markov Numbers precisely. If I can show the difference between any two such approximations is greater than the sum of their ranges of error, #Unicity follows.

Oh and n + m = N (obvi)

Alright, this is pretty beautiful. I have gotten the #ChebyshevPolynomials *out* of the equation. They were of magnificent utility while they lasted. Au revoir, #ChebyshevPolynomial... hopefully this will enlighten #MarkovNumber #Unicity somehow. It's a pretty fantastic equation. #MarkovNumbers

Ugh, I am stuck stuck stuck. I have a feeling that if I can re-build identity II to apply to negative subscripts (using Identity I again) I'll get some sort of weird infinite series that may be of utility. Tick, brain! #MarkovNumbers #Unicity #ChebyshevPolynomial #MarkovNumber #ChebyshevPolynomials

(2) Given that there is only one such sequence {z_j}, "c" only appears once on the tree.

Like holy fuck. #MarkovNumbers #MarkovNumber #Chebyshev #ChebyshevPolynomial #ChebyshevPolynomials #Unicity

Okay, actually, this is a total proof outline. Given "c" and treating "b" as a variable, identity III demands strict {z_j} values for each and every "j," in order for the reduced coefficient on each (1.5b)^i to line up with the #ChebyshevPolynomial. #MarkovNumber #MarkovNumbers #Unicity #Chebyshev

#MarkovNumbers #Unicity #Chebyshev #ChebyshevPolynomial #MarkovNumber this is getting close to the generating function for Chebyshev...

#MarkovNumbers #Unicity #Chebyshev #ChebyshevPolynomial #MarkovNumber this is getting close to the generating function for Chebyshev...