Editing Chebyshev polynomials (section)

==Properties==

===Symmetry===

<math display="block">\begin{align}
 T_n(-x) &= (-1)^n\, T_n(x),\\[1ex]
 U_n(-x) &= (-1)^n\, U_n(x).
\end{align}</math>

That is, Chebyshev polynomials of even order have [[even and odd functions|even symmetry]] and therefore contain only even powers of {{mvar|x}}. Chebyshev polynomials of odd order have [[even and odd functions|odd symmetry]] and therefore contain only odd powers of {{mvar|x}}.

===Roots and extrema===
A Chebyshev polynomial of either kind with degree {{mvar|n}} has {{mvar|n}} different [[simple root]]s, called '''Chebyshev roots''', in the interval {{closed-closed|−1, 1}}. The roots of the Chebyshev polynomial of the first kind are sometimes called [[Chebyshev nodes]] because they are used as ''nodes'' in polynomial interpolation. Using the trigonometric definition and the fact that:
<math display="block">\cos\left((2k+1)\frac{\pi}{2}\right)=0</math>
one can show that the roots of {{mvar|T<sub>n</sub>}} are:
<math display="block"> x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1.</math>
Similarly, the roots of {{mvar|U<sub>n</sub>}} are:
<math display="block"> x_k = \cos\left(\frac{k}{n+1}\pi\right),\quad k=1,\ldots,n.</math>
The [[Maxima and minima|extrema]] of {{mvar|T<sub>n</sub>}} on the interval {{math|−1 ≤ ''x'' ≤ 1}} are located at:
<math display="block"> x_k = \cos\left(\frac{k}{n}\pi\right),\quad k=0,\ldots,n.</math>

One unique property of the Chebyshev polynomials of the first kind is that on the interval {{math|−1 ≤ ''x'' ≤ 1}} all of the [[Maxima and minima|extrema]] have values that are either −1 or 1. Thus these polynomials have only two finite [[Critical value (critical point)|critical value]]s, the defining property of [[Shabat polynomial]]s. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:
<math display="block">\begin{align}
T_n(1) &= 1 \\
T_n(-1) &= (-1)^n \\
U_n(1) &= n+1 \\
U_n(-1) &= (-1)^n (n+1).
\end{align}</math>

The [[Maxima and minima|extrema]] of <math>T_n(x)</math> on the interval <math>-1 \leq x \leq 1</math> where <math>n>0</math> are located at <math>n+1</math> values of <math>x</math>. They are <math> \pm 1</math>, or <math>  \cos\left(\frac{2\pi k}{d}\right)</math> where <math>d > 2</math>, <math>d \;|\; 2n</math>, <math>0 < k < d/2</math> and <math>(k, d) = 1</math>, i.e., <math>k</math> and <math>d</math> are relatively prime numbers.

Specifically ([[Minimal polynomial of 2cos(2pi/n)]]<ref name=Gurtas>{{cite journal |first1=Y. Z. |last1=Gürtaş |title=Chebyshev Polynomials and the minimal polynomial of <math>\cos (2 \pi/n)</math> |year=2017 |journal=American Mathematical Monthly |volume=124 |number=1 |pages=74–78 |doi=10.4169/amer.math.monthly.124.1.74|s2cid=125797961 }}</ref><ref name=Wolfram0>{{cite journal |first1=D. A. |last1=Wolfram |title=Factoring Chebyshev polynomials of the first and second kinds with minimal polynomials of <math>\cos (2 \pi /d )</math> |year=2022 |journal=American Mathematical Monthly |volume=129 |number=2 |pages=172–176 |doi=10.1080/00029890.2022.2005391|s2cid=245808448 }}</ref>) when <math>n</math> is even: 
* <math>T_n(x) = 1</math> if <math>x = \pm 1</math>, or <math>d > 2</math> and <math>2n/d</math> is even. There are <math>n/2 + 1</math> such values of <math>x</math>.
* <math>T_n(x) = -1</math> if  <math>d > 2</math> and <math>2n/d</math> is odd. There are <math>n/2</math> such values of <math>x</math>.

When <math>n</math> is odd: 
* <math>T_n(x) = 1</math> if <math>x = 1</math>, or <math>d > 2</math> and <math>2n/d</math> is even. There are <math>(n+1)/2</math> such values of <math>x</math>.
* <math>T_n(x) = -1</math> if  <math>x = -1</math>, or <math>d > 2</math> and <math>2n/d</math> is odd. There are <math>(n+1)/2</math> such values of <math>x</math>.

===Differentiation and integration===
The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it can be shown that:
<math display="block">\begin{align}
\frac{\mathrm{d}T_n}{\mathrm{d}x} &= n U_{n - 1} \\
\frac{\mathrm{d}U_n}{\mathrm{d}x} &= \frac{(n + 1)T_{n + 1} - x U_n}{x^2 - 1} \\
\frac{\mathrm{d}^2 T_n}{\mathrm{d}x^2} &= n\, \frac{n T_n - x U_{n - 1}}{x^2 - 1} = n\, \frac{(n + 1)T_n - U_n}{x^2 - 1}.
\end{align}</math>

The last two formulas can be numerically troublesome due to the division by zero ({{Sfrac|0|0}} [[indeterminate form]], specifically) at {{math|1=''x'' = 1}} and {{math|1=''x'' = −1}}. By [[L'Hôpital's rule]]:
<math display="block">\begin{align}
\left. \frac{\mathrm{d}^2 T_n}{\mathrm{d}x^2} \right|_{x = 1} \!\! &= \frac{n^4 - n^2}{3}, \\
\left. \frac{\mathrm{d}^2 T_n}{\mathrm{d}x^2} \right|_{x = -1} \!\! &= (-1)^n \frac{n^4 - n^2}{3}.
\end{align}</math>

More generally,
<math display="block">\left.\frac{\mathrm{d}^p T_n}{\mathrm{d}x^p} \right|_{x = \pm 1} \!\! = (\pm 1)^{n+p}\prod_{k=0}^{p-1}\frac{n^2-k^2}{2k+1}~,</math>
which is of great use in the numerical solution of [[eigenvalue]] problems.

Also, we have:
<math display="block">\frac{\mathrm{d}^p}{\mathrm{d}x^p}\,T_n(x) = 2^p\,n\mathop{{\sum}'}_{0\leq k\leq n-p\atop k \,\equiv\, n-p \pmod 2}
\binom{\frac{n+p-k}{2}-1}{\frac{n-p-k}{2}}\frac{\left(\frac{n+p+k}{2}-1\right)!}{\left(\frac{n-p+k}{2}\right)!}\,T_k(x),~\qquad p \ge 1,</math>
where the prime at the summation symbols means that the term contributed by {{math|1=''k'' = 0}} is to be halved, if it appears.

Concerning integration, the first derivative of the {{mvar|T<sub>n</sub>}} implies that:
<math display="block">\int U_n\, \mathrm{d}x = \frac{T_{n + 1}}{n + 1}</math>
and the recurrence relation for the first kind polynomials involving derivatives establishes that for {{math|''n'' ≥ 2}}:
<math display="block">\int T_n\, \mathrm{d}x = \frac{1}{2}\,\left(\frac{T_{n + 1}}{n + 1} - \frac{T_{n - 1}}{n - 1}\right) = \frac{n\,T_{n + 1}}{n^2 - 1} - \frac{x\,T_n}{n - 1}.</math>

The last formula can be further manipulated to express the integral of {{mvar|T<sub>n</sub>}} as a function of Chebyshev polynomials of the first kind only:
<math display="block">\begin{align}
\int T_n\, \mathrm{d}x &= \frac{n}{n^2 - 1} T_{n + 1} - \frac{1}{n - 1} T_1 T_n \\
&= \frac{n}{n^2 - 1}\,T_{n + 1} - \frac{1}{2(n - 1)}\,(T_{n + 1} + T_{n - 1}) \\
&= \frac{1}{2(n + 1)}\,T_{n + 1} - \frac{1}{2(n - 1)}\,T_{n - 1}.
\end{align}</math>

Furthermore, we have:
<math display="block">\int_{-1}^1 T_n(x)\, \mathrm{d}x =
\begin{cases}
\frac{(-1)^n + 1}{1 - n^2} & \text{ if }~ n \ne 1 \\
0                          & \text{ if }~ n = 1.
\end{cases}</math>

===Products of Chebyshev polynomials===
The Chebyshev polynomials of the first kind satisfy the relation:
<math display="block">T_m(x)\,T_n(x) = \tfrac{1}{2}\!\left(T_{m+n}(x) + T_{|m-n|}(x)\right)\!,\qquad \forall m,n \ge 0,</math>
which is easily proved from the [[List of trigonometric identities#Product-to-sum and sum-to-product identities|product-to-sum formula]] for the cosine:
<math display="block">2 \cos \alpha \, \cos \beta = \cos (\alpha + \beta) + \cos (\alpha - \beta).</math>
For {{math|1=''n'' = 1}} this results in the already known recurrence formula, just arranged differently, and with {{math|1=''n'' = 2}} it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest {{mvar|m}}) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:
<math display="block">\begin{align}
  T_{2n}(x) &= 2\,T_n^2(x)           - T_0(x) &&= 2 T_n^2(x) - 1, \\
T_{2n+1}(x) &= 2\,T_{n+1}(x)\,T_n(x) - T_1(x) &&= 2\,T_{n+1}(x)\,T_n(x) - x, \\
T_{2n-1}(x) &= 2\,T_{n-1}(x)\,T_n(x) - T_1(x) &&= 2\,T_{n-1}(x)\,T_n(x) - x .
\end{align}</math>

The polynomials of the second kind satisfy the similar relation:
<math display="block"> T_m(x)\,U_n(x) = \begin{cases}
\frac{1}{2}\left(U_{m+n}(x) + U_{n-m}(x)\right),  & ~\text{ if }~ n \ge m-1,\\
\\
\frac{1}{2}\left(U_{m+n}(x) - U_{m-n-2}(x)\right), & ~\text{ if }~ n \le m-2.
\end{cases} </math>
(with the definition {{math|''U''<sub>−1</sub> ≡ 0}} by convention ). They also satisfy:
<math display="block"> U_m(x)\,U_n(x) = \sum_{k=0}^n\,U_{m-n+2k}(x) = \sum_\underset{\text{ step 2 }}{p=m-n}^{m+n} U_p(x)~.</math>
for {{math|''m'' ≥ ''n''}}.
For {{math|1=''n'' = 2}} this recurrence reduces to:
<math display="block"> U_{m+2}(x) = U_2(x)\,U_m(x) - U_m(x) - U_{m-2}(x) = U_m(x)\,\big(U_2(x) - 1\big) - U_{m-2}(x)~,</math>
which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether {{mvar|m}} starts with 2 or 3.

===Composition and divisibility properties===
The trigonometric definitions of {{math|''T''<sub>''n''</sub>}} and {{math|''U''<sub>''n''</sub>}} imply the composition or nesting properties:<ref>{{citation|last1=Rayes|first1=M. O.|last2=Trevisan|first2=V.|last3=Wang|first3=P. S.|title=Factorization properties of chebyshev polynomials|journal=Computers & Mathematics with Applications|volume=50|issue=8–9|year=2005|pages=1231–1240|doi=10.1016/j.camwa.2005.07.003|doi-access=free}}</ref>
<math display="block">\begin{align}
T_{mn}(x) &= T_m(T_n(x)),\\
U_{mn-1}(x) &= U_{m-1}(T_n(x))U_{n-1}(x).
\end{align}
</math>
For {{math|''T''<sub>''mn''</sub>}} the order of composition may be reversed, making the family of polynomial functions {{math|''T''<sub>''n''</sub>}} a [[commutative]] [[semigroup]] under composition.

Since {{math|''T''<sub>''m''</sub>(''x'')}} is divisible by {{mvar|x}} if {{mvar|m}} is odd, it follows that {{math|''T''<sub>''mn''</sub>(''x'')}} is divisible by {{math|''T''<sub>''n''</sub>(''x'')}} if {{mvar|m}} is odd. Furthermore, {{math|''U''<sub>''mn''−1</sub>(''x'')}} is divisible by {{math|''U''<sub>''n''−1</sub>(''x'')}}, and in the case that {{mvar|m}} is even, divisible by {{math|''T''<sub>''n''</sub>(''x'')''U''<sub>''n''−1</sub>(''x'')}}.

===Orthogonality===
Both {{mvar|T<sub>n</sub>}} and {{mvar|U<sub>n</sub>}} form a sequence of [[orthogonal polynomials]]. The polynomials of the first kind {{mvar|T<sub>n</sub>}} are orthogonal with respect to the weight:
<math display="block">\frac{1}{\sqrt{1 - x^2}},</math>
on the interval {{closed-closed|−1, 1}}, i.e. we have:
<math display="block">\int_{-1}^1 T_n(x)\,T_m(x)\,\frac{\mathrm{d}x}{\sqrt{1-x^2}} = 
\begin{cases}
0             & ~\text{ if }~ n \ne m, \\[5mu]
\pi           & ~\text{ if }~ n=m=0, \\[5mu]
\frac{\pi}{2} & ~\text{ if }~ n=m \ne 0.
\end{cases}</math>

This can be proven by letting {{math|1=''x'' = cos ''θ''}} and using the defining identity {{math|1=''T''<sub>''n''</sub>(cos ''θ'') = cos(''nθ'')}}.

Similarly, the polynomials of the second kind {{mvar|U<sub>n</sub>}} are orthogonal with respect to the weight:
<math display="block">\sqrt{1-x^2}</math>
on the interval {{closed-closed|−1, 1}}, i.e. we have:
<math display="block">\int_{-1}^1 U_n(x)\,U_m(x)\,\sqrt{1-x^2} \,\mathrm{d}x =
\begin{cases}
0             & ~\text{ if }~ n \ne m, \\[5mu]
\frac{\pi}{2} & ~\text{ if }~ n = m.
\end{cases}</math>

(The measure {{math|{{sqrt|1 − ''x''<sup>2</sup>}} d''x''}} is, to within a normalizing constant, the [[Wigner semicircle distribution]].)

These orthogonality properties follow from the fact that the Chebyshev polynomials solve the [[Chebyshev equation|Chebyshev differential equations]]:
<math display="block">\begin{align}
(1 - x^2)T_n'' - xT_n' + n^2 T_n &= 0, \\[1ex]
(1 - x^2)U_n'' - 3xU_n' + n(n + 2) U_n &= 0,
\end{align}</math>which are [[Sturm–Liouville problem|Sturm–Liouville differential equations]]. It is a general feature of such [[differential equation]]s that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions to [[Sturm–Liouville problem|those equations]].)

The {{mvar|T<sub>n</sub>}} also satisfy a discrete orthogonality condition:
<math display="block">\sum_{k=0}^{N-1}{T_i(x_k)\,T_j(x_k)} =
\begin{cases}
0           & ~\text{ if }~ i \ne j, \\[5mu]
N           & ~\text{ if }~ i = j = 0, \\[5mu]
\frac{N}{2} & ~\text{ if }~ i = j \ne 0,
\end{cases} </math>
where {{mvar|N}} is any integer greater than {{math|max(''i'', ''j'')}},{{sfn|Mason|Handscomb|2002}} and the {{math|''x''<sub>''k''</sub>}} are the {{mvar|N}} [[Chebyshev nodes]] (see above) of {{math|''T''<sub>''N''&thinsp;</sub>(''x'')}}:
<math display="block">x_k = \cos\left(\pi\,\frac{2k+1}{2N}\right) \quad ~\text{ for }~ k = 0, 1, \dots, N-1.</math>

For the polynomials of the second kind and any integer {{math|''N'' > ''i'' + ''j''}} with the same Chebyshev nodes {{math|''x''<sub>''k''</sub>}}, there are similar sums:
<math display="block">\sum_{k=0}^{N-1}{U_i(x_k)\,U_j(x_k)\left(1-x_k^2\right)} =
\begin{cases}
0           & \text{ if }~ i \ne j, \\[5mu]
\frac{N}{2} & \text{ if }~ i = j,
\end{cases}</math>
and without the weight function:
<math display="block">\sum_{k=0}^{N-1}{ U_i(x_k) \, U_j(x_k) } =
\begin{cases}
0                         & ~\text{ if }~ i \not\equiv j \pmod{2}, \\[5mu]
N \cdot (1 + \min\{i,j\}) & ~\text{ if }~ i \equiv j\pmod{2}.
\end{cases} </math>

For any integer {{math|''N'' > ''i'' + ''j''}}, based on the {{mvar|N}} zeros of {{math|''U''<sub>''N''&thinsp;</sub>(''x'')}}:
<math display="block">y_k = \cos\left(\pi\,\frac{k+1}{N+1}\right) \quad ~\text{ for }~ k=0, 1, \dots, N-1,</math>
one can get the sum:
<math display="block">\sum_{k=0}^{N-1}{U_i(y_k)\,U_j(y_k)(1-y_k^2)} =
\begin{cases}
0 & ~\text{ if } i \ne j, \\[5mu]
\frac{N+1}{2} & ~\text{ if } i = j,
\end{cases}</math>
and again without the weight function:
<math display="block">\sum_{k=0}^{N-1}{U_i(y_k)\,U_j(y_k)} =
\begin{cases}
0  & ~\text{ if }~ i \not\equiv j \pmod{2}, \\[5mu]
\bigl(\min\{i,j\} + 1\bigr)\bigl(N-\max\{i,j\}\bigr)  & ~\text{ if }~ i \equiv j\pmod{2}.
\end{cases}</math>

===Minimal {{math|∞}}-norm===
For any given {{math|''n'' ≥ 1}}, among the polynomials of degree {{mvar|n}} with leading coefficient 1 ([[monic polynomial|monic]] polynomials):
<math display="block">f(x) = \frac{1}{\,2^{n-1}\,}\,T_n(x)</math>
is the one of which the maximal absolute value on the interval {{closed-closed|−1, 1}} is minimal.

This maximal absolute value is:
<math display="block">\frac1{2^{n-1}}</math>
and {{math|{{abs|''f''(''x'')}}}} reaches this maximum exactly {{math|''n'' + 1}} times at:
<math display="block">x = \cos \frac{k\pi}{n}\quad\text{for }0 \le k \le n.</math>

{{Math proof | proof =
Let's assume that {{math|''w<sub>n</sub>''(''x'')}} is a polynomial of degree {{mvar|n}} with leading coefficient 1 with maximal absolute value on the interval {{closed-closed|−1, 1}} less than {{math|1&thinsp;/&thinsp;2<sup>''n''&nbsp;−&thinsp;1</sup>}}.

Define
<math display="block">f_n(x) = \frac{1}{\,2^{n-1}\,}\,T_n(x) - w_n(x)</math>

Because at extreme points of {{mvar|T<sub>n</sub>}} we have
<math display="block">\begin{align}
|w_n(x)| &< \left|\frac1{2^{n-1}}T_n(x)\right| \\
f_n(x) &> 0 \qquad \text{ for }~ x = \cos \frac{2k\pi}{n} ~&&\text{ where } 0 \le 2k \le n \\
f_n(x) &< 0 \qquad \text{ for }~ x = \cos \frac{(2k + 1)\pi}{n} ~&&\text{ where } 0 \le 2k + 1 \le n
\end{align}</math>

From the [[intermediate value theorem]], {{math|''f<sub>n</sub>''(''x'')}} has at least {{mvar|n}} roots.  However, this is impossible, as {{math|''f<sub>n</sub>''(''x'')}} is a polynomial of degree {{math|''n'' − 1}}, so the [[fundamental theorem of algebra]] implies it has at most {{math|''n'' − 1}} roots.
}}

====Remark====
By the [[equioscillation theorem]], among all the polynomials of degree {{math|≤&thinsp;''n''}}, the polynomial {{mvar|f}} minimizes {{math|{{norm|&thinsp;''f''&thinsp;}}<sub>∞</sub>}} on {{closed-closed|−1, 1}} [[if and only if]] there are {{math|''n'' + 2}} points {{math|−1 ≤ ''x''<sub>0</sub> < ''x''<sub>1</sub> < ⋯ < ''x''<sub>''n'' + 1</sub> ≤ 1}} such that {{math|1={{abs|&thinsp;''f''(''x<sub>i</sub>'')}} = {{norm|&thinsp;''f''&thinsp;}}<sub>∞</sub>}}.

Of course, the null polynomial on the interval {{closed-closed|−1, 1}} can be approximated by itself and minimizes the {{math|∞}}-norm.

Above, however, {{math|{{abs|&thinsp;''f''&thinsp;}}}} reaches its maximum only {{math|''n'' + 1}} times because we are searching for the best polynomial of degree {{math|''n'' ≥ 1}} (therefore the theorem evoked previously cannot be used).

===Chebyshev polynomials as special cases of more general polynomial families===
The Chebyshev polynomials are a special case of the ultraspherical or [[Gegenbauer polynomials]] <math>C_n^{(\lambda)}(x)</math>, which themselves are a special case of the [[Jacobi polynomials]] <math>P_n^{(\alpha,\beta)}(x)</math>:
<math display="block">\begin{align}
T_n(x) &= \frac{n}{2} \lim_{q \to 0}  \frac{1}{q}\,C_n^{(q)}(x) \qquad ~\text{ if }~ n \ge 1, \\
 &= \frac{1}{\binom{n-\frac{1}{2}}{n}} P_n^{\left(-\frac{1}{2}, -\frac{1}{2}\right)}(x) = \frac{2^{2n}}{\binom{2n}{n}} P_n^{\left(-\frac{1}{2}, -\frac{1}{2}\right)}(x)~,
\\[2ex]
U_n(x) & = C_n^{(1)}(x)\\
 &= \frac{n+1}{\binom{n+\frac{1}{2}}{n}} P_n^{\left(\frac{1}{2}, \frac{1}{2}\right)}(x) = \frac{2^{2n+1}}{\binom{2n+2}{n+1}} P_n^{\left(\frac{1}{2}, \frac{1}{2}\right)}(x)~.
\end{align}</math>

Chebyshev polynomials are also a special case of [[Dickson polynomial]]s:
<math display="block">D_n(2x\alpha,\alpha^2)= 2\alpha^{n}T_n(x) \, </math> 
<math display="block">E_n(2x\alpha,\alpha^2)= \alpha^{n}U_n(x). \, </math>
In particular, when <math>\alpha=\tfrac{1}{2}</math>, they are related by <math>D_n(x,\tfrac{1}{4}) = 2^{1-n}T_n(x)</math> and <math>E_n(x,\tfrac{1}{4}) = 2^{-n}U_n(x)</math>.

===Other properties===
The curves given by {{math|''y'' {{=}} ''T''<sub>''n''</sub>(''x'')}}, or equivalently, by the parametric equations {{math|''y'' {{=}} ''T''<sub>''n''</sub>(cos ''θ'') {{=}} cos ''nθ''}}, {{math|''x'' {{=}} cos ''θ''}}, are a special case of [[Lissajous curve]]s with frequency ratio equal to {{mvar|n}}.

Similar to the formula:
<math display="block">T_n(\cos\theta) = \cos(n\theta),</math>
we have the analogous formula:
<math display="block">T_{2n+1}(\sin\theta) = (-1)^n \sin\left(\left(2n+1\right)\theta\right).</math>

For {{math|''x'' ≠ 0}}:
<math display="block">T_n\!\left(\frac{x + x^{-1}}{2}\right) = \frac{x^n+x^{-n}}{2}</math>
and:
<math display="block">x^n = T_n\! \left(\frac{x+x^{-1}}{2}\right)
+ \frac{x-x^{-1}}{2}\ U_{n-1}\!\left(\frac{x+x^{-1}}{2}\right),</math>
which follows from the fact that this holds by definition for {{math|1=''x'' = ''e<sup>iθ</sup>''}}.

There are relations between [[Legendre polynomial]]s and Chebyshev polynomials

<math>\sum_{k=0}^{n}P_{k}\left(x\right)T_{n-k}\left(x\right) = \left(n+1\right)P_{n}\left(x\right)</math>

<math>\sum_{k=0}^{n}P_{k}\left(x\right)P_{n-k}\left(x\right) = U_{n}\left(x\right)</math>

These identities can be proven using generating functions and discrete convolution

====Chebyshev polynomials as determinants====

From their definition by recurrence it follows that the Chebyshev polynomials can be obtained as [[determinant]]s of special [[tridiagonal matrix|tridiagonal matrices]] of size <math>k \times k</math>:
<math display="block">T_k(x) = \det
\begin{bmatrix}
         x &      1 &      0 & \cdots & 0      \\
         1 &     2x &      1 & \ddots & \vdots \\
         0 &      1 &     2x & \ddots & 0      \\
    \vdots & \ddots & \ddots & \ddots & 1      \\
         0 & \cdots &      0 &      1 & 2x
\end{bmatrix},</math>
and similarly for <math>U_k</math>.