Editing Chebyshev polynomials (section)

=== Even order modified Chebyshev polynomials ===
Some applications rely on Chebyshev polynomials but may be unable to accommodate the lack of a root at zero, which rules out the use of standard Chebyshev polynomials for these kinds of applications.  Even order [[Chebyshev filter]] designs using equally terminated passive networks are an example of this.<ref name=":022">{{Cite book |last=Saal |first=Rudolf |url=https://archive.org/details/handbuchzumfilte0000saal |title=Handbook of Filter Design |publisher=Allgemeine Elektricitais-Gesellschaft |date=January 1979 |isbn=3-87087-070-2 |edition=1st |location=Munich, Germany |pages=25, 26, 56–61, 116, 117 |language=English, German}}</ref>  However, even order Chebyshev polynomials may be modified to move the lowest roots down to zero while still maintaining the desirable Chebyshev equi-ripple effect.  Such modified polynomials contain two roots at zero, and may be referred to as even order modified Chebyshev polynomials.  Even order modified Chebyshev polynomials may be created from the [[Chebyshev nodes]] in the same manner as standard Chebyshev polynomials.
<math display="block">P_N = \prod_{i=1}^N(x-C_i)
</math>

where

* <math>P_N</math> is an ''N''-th order Chebyshev polynomial
* <math>C_i</math> is the ''i''-th Chebyshev node

In the case of even order modified Chebyshev polynomials, the [[Chebyshev nodes#Even order modified Chebyshev nodes|even order modified Chebyshev nodes]] are used to construct the even order modified  Chebyshev polynomials.
<math display="block">Pe_N = \prod_{i=1}^N(x-Ce_i)
</math>

where

* <math>P e_N</math> is an ''N''-th order even order modified Chebyshev polynomial
* <math>Ce_i</math> is the ''i''-th even order modified Chebyshev node

For example, the 4th order Chebyshev polynomial from the [[Chebyshev polynomials#Examples|example above]] is <math>X^4-X^2+.125
</math>, which by inspection contains no roots of zero.  Creating the polynomial from the even order modified Chebyshev nodes creates a 4th order even order modified Chebyshev polynomial of <math>X^4-.828427X^2
</math>, which by inspection contains two roots at zero, and may be used in applications requiring roots at zero.