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 \( x>0 \).
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 \( x>0 \).