True or False Logarithm

[Gottfried]

(...)
Then, formally, the coefficientsvector X could be computed by the product

X = (LI~ * DI * LI) * Z

But LI~ * DI * LI could imply divergent dot-products, (I didn't actually test this here) so we leave it with two separate factors:
(...)

Hmm, I just tried this with base 2 and 3, and actually the entries of X seem to be computable. I get

for the row r=0 \( X[0] = - \sum_{k=0}^{\infty} \frac1{b^k - 1}
\)

for a row r>0 \( X[r] = - (-1)^r \frac{b^r }{b^r -1}* \prod_{k=1}^r \frac 1{b^k-1} \)

Of course the prod-expression can be rewritten as q-factorial multiplied by powers of (b-1)

\( X[r] = - (-1)^r * \frac{b^r}{b^r-1}*\frac 1{(b -1)^r * r !_b } \)
and then

\( falselog(b^m) = -\sum_{k=1}^{\infty}\frac1{b^k-1}
- \sum_{r=1}^{\infty} (-1)^r \frac{b^r}{(b^r-1) * r!_b * (b-1)^r}*(b^m)^r \)

correct for positive integer m and wrong for other m.
(But now it seems that I drifted far away from Henryk's formula, sorry)

Gottfried

(I forgot it, but we had this already: here )