11/21/2007, 07:51 AM
Gottfried Wrote:I get for the sum of both by my matrix-method
Code:.
Sb(x) + Rb(x) = V(x)~ *Mb[,1] + V(x)~ * Lb[,1]
= V(x)~ * (Mb + Lb)[,1]
= V(x)~ * I [,1]
= V(x)~ * [0,1,0,0,...]~
= x
Sb(x) + Rb(x) = x
This is what I have the most trouble understanding. First what is your [,1] notation mean? I understand "~" is transpose, and that Bb is the Bell matrix \( Bb = B_x[s^x] \). Second, what I can't see, or is not obvious to me at least, is why:
\( (I + Bb^{-1})^{-1} + (I + Bb)^{-1} = I \)
Is there any reason why this should be so? Can this be proven?
Wait, I just implemented it in Mathematica, and you're right! (as right as can be without a complete proof). Cool! This may just be the single most bizarre theorem in the theory of tetration and/or divergent series.
Andrew Robbins
