Chebyshev polynomials

Plot of the first five T_n Chebyshev polynomials of the first kind

The Chebyshev polynomials of the first kind are obtained from the recurrence relation

${\displaystyle {\begin{aligned}T_{0}(x)&=1\\T_{1}(x)&=x\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x).\end{aligned}}}$

The ordinary generating function for T_n is

${\displaystyle \sum _{n=0}^{\infty }T_{n}(x)\,t^{n}={\frac {1-tx}{1-2tx+t^{2}}}.}$

There are several other generating functions for the Chebyshev polynomials; the exponential generating function is

${\displaystyle \sum _{n=0}^{\infty }T_{n}(x){\frac {t^{n}}{n!}}={\frac {1}{2}}\!\left(e^{t\left(x-{\sqrt {x^{2}-1}}\right)}+e^{t\left(x+{\sqrt {x^{2}-1}}\right)}\right)=e^{tx}\cosh \left(t{\sqrt {x^{2}-1}}\right).}$

The generating function relevant for 2-dimensional potential theory and multipole expansion is

${\displaystyle \sum \limits _{n=1}^{\infty }T_{n}(x)\,{\frac {t^{n}}{n}}=\ln \left({\frac {1}{\sqrt {1-2tx+t^{2}}}}\right).}$

Plot of the first five U_n Chebyshev polynomials of the second kind

The Chebyshev polynomials of the second kind are defined by the recurrence relation

${\displaystyle {\begin{aligned}U_{0}(x)&=1\\U_{1}(x)&=2x\\U_{n+1}(x)&=2x\,U_{n}(x)-U_{n-1}(x).\end{aligned}}}$

Notice that the two sets of recurrence relations are identical, except for ${\displaystyle T_{1}(x)=x}$ vs. ${\displaystyle U_{1}(x)=2x}$. The ordinary generating function for U_n is

${\displaystyle \sum _{n=0}^{\infty }U_{n}(x)\,t^{n}={\frac {1}{1-2tx+t^{2}}},}$

and the exponential generating function is

${\displaystyle \sum _{n=0}^{\infty }U_{n}(x){\frac {t^{n}}{n!}}=e^{tx}\!\left(\!\cosh \left(t{\sqrt {x^{2}-1}}\right)+{\frac {x}{\sqrt {x^{2}-1}}}\sinh \left(t{\sqrt {x^{2}-1}}\right)\!\right).}$

As described in the introduction, the Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying

${\displaystyle T_{n}(x)={\begin{cases}\cos(n\arccos x)&{\text{ if }}~|x|\leq 1\\\cosh(n\operatorname {arcosh} x)&{\text{ if }}~x\geq 1\\(-1)^{n}\cosh(n\operatorname {arcosh} (-x))&{\text{ if }}~x\leq -1\end{cases}}}$

or, in other words, as the unique polynomials satisfying

${\displaystyle T_{n}(\cos \theta )=\cos(n\theta )}$

for n = 0, 1, 2, 3, … which as a technical point is a variant (equivalent transpose) of Schröder's equation. That is, T_n(x) is functionally conjugate to n x, codified in the nesting property below.

The polynomials of the second kind satisfy:

${\displaystyle U_{n-1}(\cos \theta )\sin \theta =\sin(n\theta ),}$

or

${\displaystyle U_{n}(\cos \theta )={\frac {\sin {\big (}(n+1)\,\theta {\big )}}{\sin \theta }},}$

which is structurally quite similar to the Dirichlet kernel D_n(x):

${\displaystyle D_{n}(x)={\frac {\sin \left((2n+1){\dfrac {x}{2}}\,\right)}{\sin {\dfrac {x}{2}}}}=U_{2n}\!\!\left(\cos {\frac {x}{2}}\right).}$

(The Dirichlet kernel, in fact, coincides with what is now known as the Chebyshev polynomial of the fourth kind.)

That cos nx is an nth-degree polynomial in cos x can be seen by observing that cos nx is the real part of one side of de Moivre's formula. The real part of the other side is a polynomial in cos x and sin x, in which all powers of sin x are even and thus replaceable through the identity cos² x + sin² x = 1. By the same reasoning, sin nx is the imaginary part of the polynomial, in which all powers of sin x are odd and thus, if one factor of sin x is factored out, the remaining factors can be replaced to create a (n−1)st-degree polynomial in cos x.

The identity is quite useful in conjunction with the recursive generating formula, inasmuch as it enables one to calculate the cosine of any integer multiple of an angle solely in terms of the cosine of the base angle.

The first two Chebyshev polynomials of the first kind are computed directly from the definition to be

${\displaystyle T_{0}(\cos \theta )=\cos(0\theta )=1}$

and

${\displaystyle T_{1}(\cos \theta )=\cos \theta ,}$

while the rest may be evaluated using a specialization of the product-to-sum identity

${\displaystyle 2\cos n\theta \cos \theta =\cos \lbrack (n+1)\theta \rbrack +\cos \lbrack (n-1)\theta \rbrack }$

as, for example,

${\displaystyle {\begin{aligned}T_{2}(\cos \theta )&=\cos 2\theta =2\cos \theta \cos \theta -1=2\cos ^{2}\theta -1.\\T_{3}(\cos \theta )&=\cos 3\theta =2\cos \theta \cos 2\theta -\cos \theta =4\cos ^{3}\theta -3\cos \theta .\end{aligned}}}$

Conversely, an arbitrary integer power of trigonometric functions may be expressed as a linear combination of trigonometric functions using Chebyshev polynomials

${\displaystyle \cos ^{n}\theta =2^{1-n}\!\mathop {\mathop {{\sum }'} _{j=0}^{n}} _{n-j\,{\text{even}}}\!\!{\binom {n}{\tfrac {n-j}{2}}}\,T_{j}(\cos \theta ),}$

where the prime at the summation symbol indicates that the contribution of j = 0 needs to be halved if it appears, and ${\displaystyle T_{j}(\cos \theta )=\cos(j\theta )}$.

An immediate corollary is the expression of complex exponentiation in terms of Chebyshev polynomials: given z = a + bi,

${\displaystyle {\begin{aligned}z^{n}&=|z|^{n}\!\left(\cos \left(n\arccos {\frac {a}{|z|}}\right)+i{\frac {b}{|b|}}\sin \left(n\arccos {\frac {a}{|z|}}\right)\right)\\&=|z|^{n}\,T_{n}\!\!\!\;\left({\frac {a}{|z|}}\right)+ib|z|^{n-1}\ U_{n-1}\!\!\!\;\left({\frac {a}{|z|}}\right).\end{aligned}}}$

Chebyshev polynomials can also be characterized by the following theorem:[3]

If ${\displaystyle F_{n}(x)}$ is a family of monic polynomials with coefficients in a field of characteristic ${\displaystyle 0}$ such that ${\displaystyle \deg F_{n}(x)=n}$ and ${\displaystyle F_{m}(F_{n}(x))=F_{n}(F_{m}(x))}$ for all ${\displaystyle m}$ and ${\displaystyle n}$, then, up to a simple change of variables, either ${\displaystyle F_{n}(x)=x^{n}}$ for all ${\displaystyle n}$ or ${\displaystyle F_{n}(x)=2*T_{n}(x/2)}$ for all ${\displaystyle n}$.

The Chebyshev polynomials can also be defined as the solutions to the Pell equation

${\displaystyle T_{n}(x)^{2}-\left(x^{2}-1\right)U_{n-1}(x)^{2}=1}$

in a ring R[x].[4] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

${\displaystyle T_{n}(x)+U_{n-1}(x)\,{\sqrt {x^{2}-1}}=\left(x+{\sqrt {x^{2}-1}}\right)^{n}~.}$

${\displaystyle {\begin{aligned}T_{n}(-x)&=(-1)^{n}\,T_{n}(x)={\begin{cases}T_{n}(x)\quad &~{\text{ for }}~n~{\text{ even}}\\\\-T_{n}(x)\quad &~{\text{ for }}~n~{\text{ odd}}\end{cases}}\\\\\\U_{n}(-x)&=(-1)^{n}\,U_{n}(x)={\begin{cases}U_{n}(x)\quad &~{\text{ for }}~n~{\text{ even}}\\\\-U_{n}(x)\quad &~{\text{ for }}~n~{\text{ odd}}\end{cases}}\\\end{aligned}}}$

That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of x. Chebyshev polynomials of odd order have odd symmetry and therefore contain only odd powers of x.

A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [−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

${\displaystyle \cos \left((2k+1){\frac {\pi }{2}}\right)=0}$

one can show that the roots of T_n are

${\displaystyle x_{k}=\cos \left({\frac {\pi (k+1/2)}{n}}\right),\quad k=0,\ldots ,n-1.}$

Similarly, the roots of U_n are

${\displaystyle x_{k}=\cos \left({\frac {k}{n+1}}\pi \right),\quad k=1,\ldots ,n.}$

The extrema of T_n on the interval −1 ≤ x ≤ 1 are located at

${\displaystyle x_{k}=\cos \left({\frac {k}{n}}\pi \right),\quad k=0,\ldots ,n.}$

One unique property of the Chebyshev polynomials of the first kind is that on the interval −1 ≤ x ≤ 1 all of the extrema have values that are either −1 or 1. Thus these polynomials have only two finite critical values, the defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:

${\displaystyle T_{n}(1)=1}$

${\displaystyle T_{n}(-1)=(-1)^{n}}$

${\displaystyle U_{n}(1)=n+1}$

${\displaystyle U_{n}(-1)=(-1)^{n}(n+1).}$

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

Specifically,[10][11] when ${\displaystyle n}$ is even,

• ${\displaystyle T_{n}(x)=1}$ if ${\displaystyle x=\pm 1}$, or ${\displaystyle d>2}$ and ${\displaystyle 2n/d}$ is even. There are ${\displaystyle n/2+1}$ such values of ${\displaystyle x}$.
• ${\displaystyle T_{n}(x)=-1}$ if ${\displaystyle d>2}$ and ${\displaystyle 2n/d}$ is odd. There are ${\displaystyle n/2}$ such values of ${\displaystyle x}$.

When ${\displaystyle n}$ is odd,

• ${\displaystyle T_{n}(x)=1}$ if ${\displaystyle x=1}$, or ${\displaystyle d>2}$ and ${\displaystyle 2n/d}$ is even. There are ${\displaystyle (n+1)/2}$ such values of ${\displaystyle x}$.
• ${\displaystyle T_{n}(x)=-1}$ if ${\displaystyle x=-1}$, or ${\displaystyle d>2}$ and ${\displaystyle 2n/d}$ is odd. There are ${\displaystyle (n+1)/2}$ such values of ${\displaystyle x}$.

This result has been generalized to solutions of ${\displaystyle U_{n}(x)\pm 1=0}$,[11] and to ${\displaystyle V_{n}(x)\pm 1=0}$ and ${\displaystyle W_{n}(x)\pm 1=0}$ for Chebyshev polynomials of the third and fourth kinds, respectively.[12]

The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it can be shown that:

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} T_{n}}{\mathrm {d} x}}&=nU_{n-1}\\{\frac {\mathrm {d} U_{n}}{\mathrm {d} x}}&={\frac {(n+1)T_{n+1}-xU_{n}}{x^{2}-1}}\\{\frac {\mathrm {d} ^{2}T_{n}}{\mathrm {d} x^{2}}}&=n\,{\frac {nT_{n}-xU_{n-1}}{x^{2}-1}}=n\,{\frac {(n+1)T_{n}-U_{n}}{x^{2}-1}}.\end{aligned}}}$

The last two formulas can be numerically troublesome due to the division by zero (0/0 indeterminate form, specifically) at x = 1 and x = −1. It can be shown that:

${\displaystyle {\begin{aligned}\left.{\frac {\mathrm {d} ^{2}T_{n}}{\mathrm {d} x^{2}}}\right|_{x=1}\!\!&={\frac {n^{4}-n^{2}}{3}},\\\left.{\frac {\mathrm {d} ^{2}T_{n}}{\mathrm {d} x^{2}}}\right|_{x=-1}\!\!&=(-1)^{n}{\frac {n^{4}-n^{2}}{3}}.\end{aligned}}}$

More general formula states:

${\displaystyle \left.{\frac {d^{p}T_{n}}{dx^{p}}}\right|_{x=\pm 1}\!\!=(\pm 1)^{n+p}\prod _{k=0}^{p-1}{\frac {n^{2}-k^{2}}{2k+1}}~,}$

which is of great use in the numerical solution of eigenvalue problems.

Also, we have

${\displaystyle {\frac {\mathrm {d} ^{p}}{\mathrm {d} x^{p}}}\,T_{n}(x)=2^{p}\,n\mathop {{\sum }'} _{0\leq k\leq n-p \atop k\,\equiv \,n-p{\pmod {2}}}{\binom {{\frac {n+p-k}{2}}-1}{\frac {n-p-k}{2}}}{\frac {\left({\frac {n+p+k}{2}}-1\right)!}{\left({\frac {n-p+k}{2}}\right)!}}\,T_{k}(x),~\qquad p\geq 1,}$

where the prime at the summation symbols means that the term contributed by k = 0 is to be halved, if it appears.

Concerning integration, the first derivative of the T_n implies that

${\displaystyle \int U_{n}\,\mathrm {d} x={\frac {T_{n+1}}{n+1}}}$

and the recurrence relation for the first kind polynomials involving derivatives establishes that for n ≥ 2

${\displaystyle \int T_{n}\,\mathrm {d} x={\frac {1}{2}}\,\left({\frac {T_{n+1}}{n+1}}-{\frac {T_{n-1}}{n-1}}\right)={\frac {n\,T_{n+1}}{n^{2}-1}}-{\frac {x\,T_{n}}{n-1}}.}$

The last formula can be further manipulated to express the integral of T_n as a function of Chebyshev polynomials of the first kind only:

${\displaystyle {\begin{aligned}\int T_{n}\,\mathrm {d} x&={\frac {n}{n^{2}-1}}T_{n+1}-{\frac {1}{n-1}}T_{1}T_{n}\\&={\frac {n}{n^{2}-1}}\,T_{n+1}-{\frac {1}{2(n-1)}}\,(T_{n+1}+T_{n-1})\\&={\frac {1}{2(n+1)}}\,T_{n+1}-{\frac {1}{2(n-1)}}\,T_{n-1}.\end{aligned}}}$

Furthermore, we have

${\displaystyle \int _{-1}^{1}T_{n}(x)\,\mathrm {d} x={\begin{cases}{\frac {(-1)^{n}+1}{1-n^{2}}}&{\text{ if }}~n\neq 1\\0&{\text{ if }}~n=1.\end{cases}}}$

The Chebyshev polynomials of the first kind satisfy the relation

${\displaystyle T_{m}(x)\,T_{n}(x)={\tfrac {1}{2}}\!\left(T_{m+n}(x)+T_{|m-n|}(x)\right)\!,\qquad \forall m,n\geq 0,}$

which is easily proved from the product-to-sum formula for the cosine,

${\displaystyle 2\cos \alpha \,\cos \beta =\cos(\alpha +\beta )+\cos(\alpha -\beta ).}$

For n = 1 this results in the already known recurrence formula, just arranged differently, and with n = 2 it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest 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:

${\displaystyle {\begin{aligned}T_{2n}(x)&=2\,T_{n}^{2}(x)-T_{0}(x)&&=2T_{n}^{2}(x)-1,\\T_{2n+1}(x)&=2\,T_{n+1}(x)\,T_{n}(x)-T_{1}(x)&&=2\,T_{n+1}(x)\,T_{n}(x)-x,\\T_{2n-1}(x)&=2\,T_{n-1}(x)\,T_{n}(x)-T_{1}(x)&&=2\,T_{n-1}(x)\,T_{n}(x)-x.\end{aligned}}}$

The polynomials of the second kind satisfy the similar relation

${\displaystyle T_{m}(x)\,U_{n}(x)={\begin{cases}{\frac {1}{2}}\left(U_{m+n}(x)+U_{n-m}(x)\right),&~{\text{ if }}~n\geq m-1,\\\\{\frac {1}{2}}\left(U_{m+n}(x)-U_{m-n-2}(x)\right),&~{\text{ if }}~n\leq m-2.\end{cases}}}$

(with the definition U₋₁ ≡ 0 by convention ). They also satisfy

${\displaystyle U_{m}(x)\,U_{n}(x)=\sum _{k=0}^{n}\,U_{m-n+2k}(x)=\sum _{\underset {\text{ step 2 }}{p=m-n}}^{m+n}U_{p}(x)~.}$

for m ≥ n. For n = 2 this recurrence reduces to

${\displaystyle U_{m+2}(x)=U_{2}(x)\,U_{m}(x)-U_{m}(x)-U_{m-2}(x)=U_{m}(x)\,{\big (}U_{2}(x)-1{\big )}-U_{m-2}(x)~,}$

which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether m starts with 2 or 3.

The trigonometric definitions of T_n and U_n imply the composition or nesting properties[13]

${\displaystyle {\begin{aligned}T_{mn}(x)&=T_{m}(T_{n}(x)),\\U_{mn-1}(x)&=U_{m-1}(T_{n}(x))U_{n-1}(x).\end{aligned}}}$

For T_mn the order of composition may be reversed, making the family of polynomial functions T_n a commutative semigroup under composition.

Since T_m(x) is divisible by x if m is odd, it follows that T_mn(x) is divisible by T_n(x) if m is odd. Furthermore, U_mn−1(x) is divisible by U_n−1(x), and in the case that m is even, divisible by T_n(x)U_n−1(x).

Both T_n and U_n form a sequence of orthogonal polynomials. The polynomials of the first kind T_n are orthogonal with respect to the weight

${\displaystyle {\frac {1}{\sqrt {1-x^{2}}}},}$

on the interval [−1, 1], i.e. we have:

${\displaystyle \int _{-1}^{1}T_{n}(x)\,T_{m}(x)\,{\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}}={\begin{cases}0&~{\text{ if }}~n\neq m,\\\\\pi &~{\text{ if }}~n=m=0,\\\\{\frac {\pi }{2}}&~{\text{ if }}~n=m\neq 0.\end{cases}}}$

This can be proven by letting x = cos θ and using the defining identity T_n(cos θ) = cos(nθ).

Similarly, the polynomials of the second kind U_n are orthogonal with respect to the weight

${\displaystyle {\sqrt {1-x^{2}}}}$

on the interval [−1, 1], i.e. we have:

${\displaystyle \int _{-1}^{1}U_{n}(x)\,U_{m}(x)\,{\sqrt {1-x^{2}}}\,\mathrm {d} x={\begin{cases}0&~{\text{ if }}~n\neq m,\\{\frac {\pi }{2}}&~{\text{ if }}~n=m.\end{cases}}}$

(The measure √1 − x² dx is, to within a normalizing constant, the Wigner semicircle distribution.)

These orthogonality properties follow from the fact that the Chebyshev polynomials solve the Chebyshev differential equations

${\displaystyle (1-x^{2})y''-xy'+n^{2}y=0,}$

${\displaystyle (1-x^{2})y''-3xy'+n(n+2)y=0,}$

which are Sturm–Liouville differential equations. It is a general feature of such differential equations that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions to those equations.)

The T_n also satisfy a discrete orthogonality condition:

${\displaystyle \sum _{k=0}^{N-1}{T_{i}(x_{k})\,T_{j}(x_{k})}={\begin{cases}0&~{\text{ if }}~i\neq j,\\N&~{\text{ if }}~i=j=0,\\{\frac {N}{2}}&~{\text{ if }}~i=j\neq 0,\end{cases}}}$

where N is any integer greater than max(i, j),[7] and the x_k are the N Chebyshev nodes (see above) of T_N(x):

${\displaystyle x_{k}=\cos \left(\pi \,{\frac {2k+1}{2N}}\right)\quad ~{\text{ for }}~k=0,1,\dots ,N-1.}$

For the polynomials of the second kind and any integer N > i + j with the same Chebyshev nodes x_k, there are similar sums:

${\displaystyle \sum _{k=0}^{N-1}{U_{i}(x_{k})\,U_{j}(x_{k})\left(1-x_{k}^{2}\right)}={\begin{cases}0&{\text{ if }}~i\neq j,\\{\frac {N}{2}}&{\text{ if }}~i=j,\end{cases}}}$

and without the weight function:

${\displaystyle \sum _{k=0}^{N-1}{U_{i}(x_{k})\,U_{j}(x_{k})}={\begin{cases}0&~{\text{ if }}~i\not \equiv j{\pmod {2}},\\N\cdot (1+\min\{i,j\})&~{\text{ if }}~i\equiv j{\pmod {2}}.\end{cases}}}$

For any integer N > i + j, based on the N zeros of U_N(x):

${\displaystyle y_{k}=\cos \left(\pi \,{\frac {k+1}{N+1}}\right)\quad ~{\text{ for }}~k=0,1,\dots ,N-1,}$

one can get the sum:

${\displaystyle \sum _{k=0}^{N-1}{U_{i}(y_{k})\,U_{j}(y_{k})(1-y_{k}^{2})}={\begin{cases}0&~{\text{ if }}i\neq j,\\{\frac {N+1}{2}}&~{\text{ if }}i=j,\end{cases}}}$

and again without the weight function:

${\displaystyle \sum _{k=0}^{N-1}{U_{i}(y_{k})\,U_{j}(y_{k})}={\begin{cases}0&~{\text{ if }}~i\not \equiv j{\pmod {2}},\\{\bigl (}\min\{i,j\}+1{\bigr )}{\bigl (}N-\max\{i,j\}{\bigr )}&~{\text{ if }}~i\equiv j{\pmod {2}}.\end{cases}}}$

For any given n ≥ 1, among the polynomials of degree n with leading coefficient 1 (monic polynomials),

${\displaystyle f(x)={\frac {1}{\,2^{n-1}\,}}\,T_{n}(x)}$