Editing Chebyshev polynomials (section)

==Definitions==

=== Recurrence definition ===

The ''Chebyshev polynomials of the first kind'' can be defined by the recurrence relation
<math display="block">\begin{align}
 T_0(x) & = 1, \\
 T_1(x) & = x, \\
 T_{n+1}(x) & = 2 x\,T_n(x) - T_{n-1}(x).
\end{align}</math>

The ''Chebyshev polynomials of the second kind'' can be defined by the recurrence relation
<math display="block">\begin{align}
 U_0(x) & = 1, \\
 U_1(x) & = 2 x, \\
 U_{n+1}(x) & = 2 x\,U_n(x) - U_{n-1}(x),
\end{align}</math>
which differs from the above only by the rule for ''n=1''.

===Trigonometric definition===

The Chebyshev polynomials of the first and second kind can be defined as the unique polynomials satisfying
<math display="block">T_n(\cos\theta) = \cos(n\theta)</math>
and
<math display="block">U_n(\cos\theta) = \frac{\sin\big((n + 1)\theta\big)}{\sin\theta},</math>
for {{math|1=''n'' = 0, 1, 2, 3, …}}.

An equivalent way to state this is via exponentiation of a [[complex number]]: given a complex number {{math|1=''z'' = ''a'' + ''bi''}} with absolute value of one,
<math display="block">z^n = T_n(a) + ib U_{n-1}(a).</math>
Chebyshev polynomials can be defined in this form when studying [[trigonometric polynomials]].<ref>{{Cite journal |last=Schaeffer |first=A. C. |date=1941 |title=Inequalities of A. Markoff and S. Bernstein for polynomials and related functions |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-47/issue-8/Inequalities-of-A-Markoff-and-S-Bernstein-for-polynomials-and/bams/1183503783.full |journal=Bulletin of the American Mathematical Society |volume=47 |issue=8 |pages=565–579 |doi=10.1090/S0002-9904-1941-07510-5 |issn=0002-9904|doi-access=free }}</ref>

That {{math|cos{{nnbsp}}''nx''}} is an {{mvar|n}}th-[[degree of a polynomial|degree]] polynomial in {{math|cos{{nnbsp}}''x''}} can be seen by observing that {{math|cos{{nnbsp}}''nx''}} is the [[complex number|real part]] of one side of [[de Moivre's formula]]:
<math display="block">\cos n \theta + i \sin n \theta = (\cos \theta + i \sin \theta)^n.</math>
The real part of the other side is a polynomial in {{math|cos{{nnbsp}}''x''}} and {{math|sin{{nnbsp}}''x''}}, in which all powers of {{math|sin{{nnbsp}}''x''}} are [[parity (mathematics)|even]] and thus replaceable through the identity {{math|1=cos<sup>2</sup>{{nnbsp}}''x'' + sin<sup>2</sup>{{nnbsp}}''x'' = 1}}. By the same reasoning, {{math|sin{{nnbsp}}''nx''}} is the [[complex number|imaginary part]] of the polynomial, in which all powers of {{math|sin{{nnbsp}}''x''}} are [[parity (mathematics)|odd]] and thus, if one factor of {{math|sin{{nnbsp}}''x''}} is factored out, the remaining factors can be replaced to create a {{math|(''n''&nbsp;−&nbsp;1)}}st-degree polynomial in {{math|cos{{nnbsp}}''x''}}.

For ''x'' outside the interval [-1,1], the above definition implies
<math display="block">T_n(x) = \begin{cases}
 \cos(n \arccos x)                          & \text{ if }~ |x| \le 1, \\
 \cosh(n \operatorname{arcosh} x)           & \text{ if }~ x \ge 1, \\ 
 (-1)^n \cosh(n \operatorname{arcosh}(-x) ) & \text{ if }~ x \le -1.
\end{cases}</math>

===Commuting polynomials definition===

Chebyshev polynomials can also be characterized by the following theorem:<ref>{{cite journal|first=J. F. |last=Ritt |author-link=Joseph Ritt |doi=10.1090/S0002-9947-1922-1501189-9 |title=Prime and Composite Polynomials |journal=Trans. Amer. Math. Soc. |year=1922|volume=23 |pages=51–66 | url=https://www.ams.org/journals/tran/1922-023-01/S0002-9947-1922-1501189-9 |doi-access=free}}</ref>

If <math> F_n(x)</math> is a family of monic polynomials with coefficients in a field of characteristic <math>0</math> such that <math> \deg F_n(x) = n</math> and <math> F_m(F_n(x)) = F_n(F_m(x))</math> for all 
<math>m</math> and <math> n</math>, then, up to a simple change of variables, either <math> F_n(x) = x^n</math> for all <math> n</math> or 
<math>F_n(x) = 2\cdot T_n(x/2)</math> for all <math> n</math>.

===Pell equation definition===
The Chebyshev polynomials can also be defined as the solutions to the [[Pell equation]]:
<math display="block">T_n(x)^2 - \left(x^2 - 1\right) U_{n-1}(x)^2 = 1</math>
in a [[ring (mathematics)|ring]] {{math|''R''[''x'']}}.<ref>{{cite thesis |first=Jeroen |last=Demeyer |url=http://cage.ugent.be/~jdemeyer/phd.pdf |title=Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields |archive-url=https://web.archive.org/web/20070702185523/https://cage.ugent.be/~jdemeyer/phd.pdf |archive-date=2007-07-02 |degree=Ph.D. |year=2007 |page=70}}</ref> Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:
<math display="block">T_n(x) + U_{n-1}(x)\,\sqrt{x^2-1} = \left(x + \sqrt{x^2-1}\right)^n~. </math>

===Generating functions===

The [[generating function|ordinary generating function]] for {{mvar|T<sub>n</sub>}} is
<math display="block">\sum_{n=0}^\infty T_n(x)\,t^n = \frac{1 - tx}{1 - 2tx + t^2}.</math>

There are several other [[generating function]]s for the Chebyshev polynomials; the [[exponential generating function]] is
<math display="block">\begin{align}
\sum_{n=0}^\infty T_n(x) \frac{t^n}{n!}
  &= {\tfrac{1}{2}} \Bigl({\exp}\Bigl({\textstyle t\bigl(x - \sqrt{x^2 - 1}~\!\bigr)}\Bigr)
                      + {\exp}\Bigl({\textstyle t\bigl(x + \sqrt{x^2 - 1}~\!\bigr)}\Bigr)\Bigr) \\
  &= e^{tx} \cosh\left({\textstyle t\sqrt{x^2 - 1} }~\! \right).
\end{align}</math>

The generating function relevant for 2-dimensional [[potential theory]] and [[Cylindrical multipole moments|multipole expansion]] is
<math display="block">\sum\limits_{n=1}^\infty T_{n}(x)\,\frac{t^n}{n} = \ln\left(\frac{1}{\sqrt{1 - 2tx + t^2 }}\right).</math>

The ordinary generating function for {{mvar|U<sub>n</sub>}} is
<math display="block">\sum_{n=0}^\infty U_n(x)\,t^n = \frac{1}{1 - 2tx + t^2},</math>
and the exponential generating function is
<math display="block">
 \sum_{n=0}^\infty U_n(x) \frac{t^n}{n!} = e^{tx} \biggl(\!\cosh\left(t\sqrt{x^2 - 1}\right) + \frac{x}{\sqrt{x^2 - 1}} \sinh\left(t\sqrt{x^2 - 1}\right)\biggr).
</math>