I want to know how to synthesise a passive LC circuit to implement an arbitrary known transfer function. (I understand that not all functions are suitable, so let's assume \$T(s)\$ is a positive real function; all poles and zeros are in the left half plane; in conjugate pairs where appropriate, etc. etc. If it simplifies things to assume \$T(s)={1\over{D(s)}}\$ for polynomial \$D(s)\$; \$R_S=R_L\$; or similar, that's fine. I assume Cauer ladder-style networks throughout for concreteness, but I'm not attached to that topology if an alternative is easier to answer.)
I am following Electronic Filter Design Handbook 4th Ed. (Williams and Taylor), and by performing the procedure described there for expanding Equation (1-4):$$Z_{11} = {{D(s)-s^n}\over{D(s)+s^n}}$$as a continued fraction, I can successfully replicate Table 11-2 (LC element values for Butterworth filters of arbitrary order). This is exactly what I want, but I am interested in other transfer functions in general and not only Butterworth.
An example of a transfer function I am unable to synthesise is the 5th order Chebyshev low-pass filter (0.5 dB ripple) in Example 2-20. I can reproduce the pole positions listed in the book (\$-0.105699\pm0.954967i, -0.276724\pm0.590202i, -0.34205\$), and if I apply Equation (1-4) then I obtain:$$Z_{11}={{1.1069s^4+1.72666s^3+1.10185s^2+0.597734s+0.13417}\over{2s^5+1.1069s^4+1.72666s^3+1.10185s^2+0.597734s+0.13417}}$$Following the same Example 1-1 method that seems to works perfectly for all Butterworth transfer functions, I invert the expression to find the continued fraction, and get as far as:$$Z_{11}={1\over{1.80685s-1.81853+{{2.87577s^3+2.02559s^2+1.44231s+0.378163}\over{1.1069s^4+1.72666s^3+1.10185s^2+0.597734s+0.13417}}}}$$At this point I'm stuck. I know the \$1.80685s\$ term is correct, since it agrees with the first element in the solution to Example 2-20, but I do not understand how to handle the \$-1.81853\$ constant term in the denominator. Such terms had never appeared when transforming Butterworth functions, where each intermediate level of the continued fraction involved only a multiple of \$s\$, indicating a pure reactance? I can absorb the constant into the continuation of the fraction, but that makes little difference, since then the subsequent quotient is not a multiple of \$s\$.
I would appreciate any advice on how to proceed.
(Please note that I am looking for more than just the values for the Chebyshev example; I realise it would be very easy to find those in any standard table, and also that for those particular functions, closed-form solutions are known. I would like to understand how to compute components for transfer functions that are neither Butterworth nor Chebyshev, but are too obscure to appear in tables or to have names.)