open problems survey
#14
In generalization of (the already solved) TPID 6 and following this thread of Andrew:

Does the sequence of interpolating polynomials of the points \( (0,0),(1,y_1),\dots,(n,y_n) \) defined by \( y_n [4] n = n \) pointwise converge to a function \( f \) on (0,oo) (, satisfying \( f(n)=y_n \))?

If it converges:
a) is then the limit function \( f \) analytic, particularly at the point \( x=\eta \)?
b) For \( b\le \eta \) let b[4]x be the regular superexponential at the lower fixpoint. Is then f(x)[4]x = x for non-integer x with \( f(x)\le\eta \)?
c) For \( b> \eta \) let b[4]x be the super-exponential via Kneser/perturbed Fatou coordinates. Is then f(x)[4]x = x for non-integer x with \( f(x)>\eta \)?

To be more precise we can explicitely give the interpolating polynomials:
\( f_N(x) = \sum_{n=0}^N \left(x\\n\right) \sum_{m=0}^n \left(n\\m\right) (-1)^{n-m} y_m \),
the question of this post is whether
\( \lim_{n\to\infty} f_n(x) \) exists for each \( x>0 \).


Messages In This Thread
open problems survey - by bo198214 - 05/17/2008, 10:03 AM
Exponential Factorial, TPID 2 - by andydude - 05/26/2008, 03:24 PM
Existence of bounded b^z TPID 4 - by bo198214 - 10/08/2008, 04:22 PM
A conjecture on bounds. TPID 7 - by andydude - 10/23/2009, 05:27 AM
Logarithm reciprocal TPID 9 - by bo198214 - 07/20/2010, 05:50 AM
RE: open problems survey - by nuninho1980 - 10/31/2010, 09:50 PM
convergence of self-tetra-root polynomial interpolation. TPID 12 - by bo198214 - 05/31/2011, 04:54 PM
Tommy's conjecture TPID 16 - by tommy1729 - 06/07/2014, 10:44 PM
The third super-root TPID 18 - by andydude - 12/25/2015, 06:16 AM
RE: open problems survey - by JmsNxn - 08/23/2021, 11:54 PM
RE: open problems survey - by Gottfried - 07/04/2022, 11:10 AM
RE: open problems survey - by tommy1729 - 07/04/2022, 01:12 PM
RE: open problems survey - by Gottfried - 07/04/2022, 01:19 PM
RE: open problems survey - by leon - 10/16/2023, 01:26 PM
RE: open problems survey - by Catullus - 07/12/2022, 03:22 AM
RE: open problems survey - by JmsNxn - 07/12/2022, 05:39 AM
RE: open problems survey - by Catullus - 11/01/2022, 06:33 AM
RE: open problems survey - by Leo.W - 08/10/2022, 01:23 PM
RE: open problems survey - by tommy1729 - 08/12/2022, 01:28 AM
RE: open problems survey - by Leo.W - 08/12/2022, 05:26 AM
RE: open problems survey - by Catullus - 12/22/2022, 06:37 AM
RE: open problems survey - by leon - 10/10/2023, 04:04 PM

Possibly Related Threads…
Thread Author Replies Views Last Post
  open problems / Discussion Gottfried 8 19,390 06/26/2008, 07:20 PM
Last Post: bo198214



Users browsing this thread: 1 Guest(s)