A New Representation of Hurwitz's Determinants in the Expansion of Certain Ladder Filters

Given a real strictly Hurwitz polynomial
H_{n}(s) = a_{0}\prod_{\upsilon = 1}^{n} (s - s_{\upsilon}), n = 3, 4, \cdots ,
the standard method of calculating the continued fraction expansion of
\{[odd H_{n}(s)]/ [even H_{n}(s)]\}^{\pm 1}
about its pole at infinity uses Routh's scheme or Hurwitz's determinants
\Delta_{r}, r = 1, 2, \cdots , n
, in the coefficients of
H_{n}(s)
(on the equivalence of the two, see [2]). In filter theory, cases are often encountered where knowledge of the zeros of
H_{n}(s)
precedes that of its coefficients, and one would then prefer to have formulas for the coefficients in the above continued fraction expansion directly in terms of the former rather than the latter. This is achieved by expressing
\Delta_{r}
as bialternants in the zeros of
H_{n}(s)
and reads
\Delta_{r} = (-)^{r(r+1)/2} a_{0}^{r}A(0, 1, \cdots , n - r - 1, n - r + 1, \cdots , n + r - 1)/A(0, 1, \cdots , n - 1)
, where the alternant in the denominator is the Vandermonde in
s_{1}, s_{2}, \cdots , s_{n}
, whereas the alternant in the numerator is obtained from it on replacing the exponents
0, 1, \cdots , n - 1
by
0, 1, \cdots, n - r - 1, n - r + 1, \cdots , n + r - 1
. Examples include
H_{n}(s) = \prod_{\upsilon = 1}^{n} [s - j \exp (2 \upsilon - 1)j \pi /2n]
and
H_{n}(s) = (s + 1)^{n}
.