Continued Fraction Approximants to the Brillouin-Wigner Perturbation Series

The Brillouin-Wigner series for the energy is converted into a continued fraction. Refinements on the Brillouin-Wigner formulas developed in recent publications are identified with alternate ($E^{\mathrm{}\left(n\right)\mathrm{}}$) approximants to the continued fraction. A second sequence of approximants [$E^{(n+\frac{1}{2})}$] occurs between successive terms of the $E^{\mathrm{}\left(n\right)\mathrm{}}$ sequence. These are useful in calculations as shown by an illustrative example, but do not possess the extremum property which is a valued characteristic of the first sequence. A general proof is given that the approximants $E^{\mathrm{}\left(n\right)\mathrm{}}$ are invariant under the $\mu$ transformation defined and verified for $n=1, 2, \mathrm{and} \infty$ in an earlier publication.