Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the Gauss Notation for generalized continued fractions.
Description of the Bauer-Muir transformation
(Based on pp. 76-77 of Lisa Lorentzen and Haakon Waadeland's book (A), chapter II, theorems 11, and Theorem 7; also section 5 of J. Mc Laughlin, and Nancy J. Wyshinski's online paper (C), and section 5 of Bruce C. Berndt, Sen-Shan Huang, Jaebum Sohn, and Seung Hwan Son's online paper (B) ).
Given a convergent c.f. $\underset{n=1}{\overset{\infty }{\mathbb{K}}}(a_{n}/b_{n})=\lim_{n\rightarrow \infty }A_{n}/B_{n}$ and a sequence ${w_{n}}$ we can construct a new c.f $\underset{n=1}{\overset{\infty }{\mathbb{K}}}({c_{n}/d_{n}})=\lim_{n\rightarrow \infty }C_{n}/D_{n}$ (if convergent) which is its Bauer-Muir transform with respect to ${w_{n}}$.
By theorem 11, chapter II, of Lisa Lorentzen and Haakon Waadeland's book (A), pp.76-77, the relations between $A_{n},B_{n}$ and $C_{n},D_{n}$ are given by:
$C_{n}=A_{n}+A_{n-1}w_{n}$, $D_{n}=B_{n}+B_{n-1}w_{n}$, with the initial conditions $C_{-1}=1,D_{-1}=0.$
If for $n\geq 1$, $\lambda_{n}=a_{n}-w_{n-1}(b_{n}+w_{n})\neq 0$, then
$$c_{n}=a_{n-1}\lambda_{n}/\lambda_{n-1},$$
$$d_{n}=b_{n}+w_{n}-w_{n-2}\lambda_{n}/\lambda _{n-1},$$
for $n\geq 2,$ and
$$\underset{n=1}{\overset{\infty }{\mathbb{K}}}(a_{n}/b_{n})=w_{0}+\dfrac{\lambda_{1}}{b_{1}+w_{1}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}(c_{n}/d_{n})}.$$
The elements of $\underset{n=1}{\overset{\infty }{\mathbb{K}}}(a_{n}/b_{n})$ are computed by an application of Theorem 7, chapter II, of Lisa Lorentzen and Haakon Waadeland's book (A), which transforms a sequence into a continued fraction. I was able to derive $c_{n}$ but not $d_{n}$.
Example: Application to the $\log (1+t)$ expansion. By choosing $w_{0}=w_{1}=0,w_{n}=(n-1)t$, for $n\geq 2$, one can derive
$$\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( n^{2}t/\left( \left( n+1\right) -kt\right) \right) =\dfrac{t}{2+\underset{n=3}{\overset{\infty }{% \mathbb{K}}}\left( \left( n-2\right) ^{2}t/\left( n-\left( n-3\right) t\right) \right) }.$$
Hence (see Wikipedia) the expansion
$$\log (1+t)=\displaystyle\sum_{n=1}^{\infty }\dfrac{(-1)^{n-1}t^{n}}{n}=\dfrac{t}{1+% \underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( n^{2}t/\left( \left( n+1\right) -nt\right) \right) },$$
can be improved with respect to the convergence speed to this one
$$\log (1+t)=\dfrac{t}{1+\dfrac{t}{2+\underset{n=3}{\overset{\infty }{\mathbb{% K}}}\left( \left( n-2\right) ^{2}t/\left( n-\left( n-3\right) t\right) \right) }}.$$
Derivation of $c_n$
Here is how I got $c_n$. In page 77 of reference A it is proved that
$$C_{n-1}D_n-D_{n-1}C_n=(A_{n-1}B_{n-2}-A_{n-2}B_{n-1})\lambda_n.$$
Hence
$$C_{n-2}D_{n-1}-D_{n-2}C_{n-1}=(A_{n-2}B_{n-3}-A_{n-3}B_{n-2})\lambda_{n-1}.$$
For $n\ge 2$ Theorem 7 of reference A (derived from the fundamental c.f. recurrence) gives
$$c_n=\dfrac{C_{n-1}D_n-D_{n-1}C_n}{-(C_{n-2}D_{n-1}-D_{n-2}C_{n-1})}=\dfrac{(A_{n-1}B_{n-2}-A_{n-2}B_{n-1})\lambda_n}{-(A_{n-2}B_{n-3}-A_{n-3}B_{n-2})\lambda_{n-1}}.$$
By the determinant formula we have
$$A_{n-1}B_{n-2}-A_{n-2}B_{n-1}=-a_{n-1}(A_{n-2}B_{n-3}-A_{n-3}B_{n-2}).$$
Thus
$$c_n=a_{n-1}\dfrac{\lambda_n}{\lambda_{n-1}}.$$
QUESTION 1: How does one prove
$$d_{n}=b_{n}+w_{n}-w_{n-2}\lambda_{n}/\lambda_{n-1}$$
from
$$d_{n}=\dfrac{C_{n}D_{n-2}-D_{n}C_{n-2}}{C_{n}D_{n-1}-D_{n}C_{n-1}}\qquad\text{for}\quad n\ge 2$$
and
$\lambda_{n}=a_{n}-w_{n-1}(b_{n}+w_{n})$, $C_{n}=A_{n}+A_{n-1}w_{n}$, $D_{n}=B_{n}+B_{n-1}w_{n}$ ?
References
A - Lisa Lorentzen and Haakon Waadeland, Continued Fractions and Applications, North-Holland, Amsterdam, 1992. (pdf file of pp. 76-77)
B - Bruce C. Berndt, Sen-Shan Huang, Jaebum Sohn, and Seung Hwan Son, A Transformation Formula in Rogers--Ramanujan Continued Fraction. (section 5)
C - J. Mc Laughlin, and Nancy J. Wyshinski, Real numbers with polynomial continued fraction expansions, arXiv, 2004.