Editing Chebyshev polynomials (section)

== Families of polynomials related to Chebyshev polynomials ==
Polynomials denoted <math>C_n(x)</math> and <math>S_n(x)</math> closely related to Chebyshev polynomials are sometimes used. They are defined by:{{sfn|Hochstrasser|1972|p=778}}
<math display="block">C_n(x) = 2T_n\left(\frac{x}{2}\right),\qquad S_n(x) = U_n\left(\frac{x}{2}\right)</math>
and satisfy:
<math display="block">C_n(x) = S_n(x) - S_{n-2}(x).</math>
A. F. Horadam called the polynomials <math>C_n(x)</math> '''Vieta–Lucas polynomials''' and denoted them <math>v_n(x)</math>. He called the polynomials <math>S_n(x)</math> '''Vieta–Fibonacci polynomials''' and denoted them {{nowrap|<math>V_n(x)</math>.}}<ref>{{citation|last=Horadam|first=A. F.|title=Vieta polynomials|journal=Fibonacci Quarterly|volume=40|issue=3|year=2002|url=https://www.fq.math.ca/Scanned/40-3/horadam2.pdf|pages=223–232}}</ref> Lists of both sets of polynomials are given in [[François Viète|Viète's]] ''Opera Mathematica'', Chapter IX, Theorems VI and VII.<ref>{{cite book|last=Viète|first=François|title=Francisci Vietae Opera mathematica : in unum volumen congesta ac recognita / opera atque studio Francisci a Schooten|year=1646|publisher=Bibliothèque nationale de France|url=https://gallica.bnf.fr/ark:/12148/bpt6k107597d.pdf}}</ref> The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of <math>i</math> and a shift of index in the case of the latter, equal to [[Fibonacci polynomials|Lucas and Fibonacci polynomials]] {{math|''L''<sub>''n''</sub>}} and {{math|''F''<sub>''n''</sub>}} of imaginary argument.

'''Shifted Chebyshev polynomials''' of the first and second kinds are related to the Chebyshev polynomials by:{{sfn|Hochstrasser|1972|p=778}}
<math display="block">T_n^*(x) = T_n(2x-1),\qquad U_n^*(x) = U_n(2x-1).</math>

When the argument of the Chebyshev polynomial satisfies {{math|2''x'' − 1 ∈ {{closed-closed|−1, 1}}}} the argument of the shifted Chebyshev polynomial satisfies {{math|''x'' ∈ {{closed-closed|0, 1}}}}. Similarly, one can define shifted polynomials for generic intervals {{closed-closed|''a'', ''b''}}.

Around 1990 the terms "third-kind" and "fourth-kind" came into use in connection with Chebyshev polynomials, although the polynomials denoted by these terms had an earlier development under the name '''airfoil polynomials'''. According to J. C. Mason and G. H. Elliott, the terminology "third-kind" and "fourth-kind" is due to [[Walter Gautschi]], "in consultation with colleagues in the field of orthogonal polynomials."<ref name=MasonElliott1993>{{citation|last1=Mason|first1=J. C.|last2=Elliott | first2=G. H.|title= Near-minimax complex approximation by four kinds of Chebyshev polynomial expansion | journal = J. Comput. Appl. Math. | volume = 46| pages = 291–300| year = 1993| issue = 1–2 | doi = 10.1016/0377-0427(93)90303-S | doi-access = free}}</ref> The '''Chebyshev polynomials of the third kind''' are defined as:
<math display="block">V_n(x)=\frac{\cos\left(\left(n+\frac{1}{2}\right)\theta\right)}{\cos\left(\frac{\theta}{2}\right)}=\sqrt\frac{2}{1+x}T_{2n+1}\left(\sqrt\frac{x+1}{2}\right)</math>
and the '''Chebyshev polynomials of the fourth kind''' are defined as:
<math display="block">W_n(x)=\frac{\sin\left(\left(n+\frac{1}{2}\right)\theta\right)}{\sin\left(\frac{\theta}{2}\right)}=U_{2n}\left(\sqrt\frac{x+1}{2}\right),</math>
where <math>\theta=\arccos x</math>.<ref name=MasonElliott1993/><ref name=DesmaraisBland1995>{{citation|last1=Desmarais|first1=Robert N.|last2=Bland|first2=Samuel R.|title=Tables of properties of airfoil polynomials|publisher=National Aeronautics and Space Administration|work=NASA Reference Publication 1343|url=https://ntrs.nasa.gov/citations/19960001864|year=1995}}</ref>
They coincide with the [[Dirichlet kernel]].

In the airfoil literature <math>V_n(x)</math> and <math>W_n(x)</math> are denoted <math>t_n(x)</math> and <math>u_n(x)</math>. The polynomial families <math>T_n(x)</math>, <math>U_n(x)</math>, <math>V_n(x)</math>, and <math>W_n(x)</math> are orthogonal with respect to the weights:
<math display="block">\left(1-x^2\right)^{-1/2},\quad\left(1-x^2\right)^{1/2},\quad(1-x)^{-1/2}(1+x)^{1/2},\quad(1+x)^{-1/2}(1-x)^{1/2}</math>
and are proportional to Jacobi polynomials <math>P_n^{(\alpha,\beta)}(x)</math> with:<ref name="DesmaraisBland1995" />
<math display="block">(\alpha,\beta)=\left(-\frac{1}{2},-\frac{1}{2}\right),\quad(\alpha,\beta)=\left(\frac{1}{2},\frac{1}{2}\right),\quad(\alpha,\beta)=\left(-\frac{1}{2},\frac{1}{2}\right),\quad(\alpha,\beta)=\left(\frac{1}{2},-\frac{1}{2}\right).</math>

All four families satisfy the recurrence <math>p_n(x)=2xp_{n-1}(x)-p_{n-2}(x)</math> with <math>p_0(x) = 1</math>, where <math>p_n = T_n</math>, <math>U_n</math>, <math>V_n</math>, or <math>W_n</math>, but they differ according to whether <math>p_1(x)</math> equals <math>x</math>, <math>2x</math>, <math>2x-1</math>, or {{nowrap|<math>2x+1</math>.}}<ref name=MasonElliott1993/>

=== 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.