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Introduction to Linear Algebra with Mathematica

Glossary

Preface

This section presents some properties of the most remarkable and useful in numerical computations Chebyshev polynomials of first kind Tn(x), second kind Un(x), third kind Vn(x), and fourth kind Wn(x). All four Chebyshev polynomials are eigenfunctions of the corresponding singular Sturm--Liouville problems. However, the last two Chebyshev polynomials of the third kind and the fourth kind are not so popular in applications.

Since the denominator in the generating functions for every Chebyshev polynomial is the same, the Chebyshev polynomials of four kinds satisfy the same difference equation of the second order:

\begin{equation} \label{EqCheb3.1} C(n) = 2x\,C(n-1) - C(n-2) , \qquad n=2,3,4,\ldots ; \end{equation}
subject to the initial values:
\begin{equation} \label{EqCheb3.2} T_0 = U_0 = V_0 = W_0 = 1, \qquad T_1 (x) = x, \quad U_1 (x) = 2x, \quad V_1 (x) = 2x-1, \quad W_1 (x) = 2x+1 . \end{equation}

 


        T(0,x) = 1
        T(1,x) = x
        T(n+1,x) = 2x T(n,x) - T(n-1,x)


       U(0,x) = 1
        U(1,x) = 2x
        U(n+1,x) = 2x U(n,x) - U(n-1,x)


        V(0,x) = 1
        V(1,x) = 2x-1
        V(n+1,x) = 2x V(n,x) - V(n-1,x)


        W(0,x) = 1
        W(1,x) = 2x+1
        W(n+1,x) = 2x W(n,x) - W(n-1,x)
There are several versions for calculating Chebyshev polynomials with Mathematica. For example, using Nest:
Expand[Nest[{#[[2]], 2*x*#[[2]] - #[[1]]} &, {1, x}, 3]]
f[{a_, b_}] = {b, 2*x*b - a}
Expand[Nest[f, {1, x}, 3]]

Chebyshev polynomials of the first kind

We can use the standard Mathematica commands:
Column[Table[ChebyshevT[i,x], {i, 1, 5, 1}]] (* returns T[1] ... T[5] *)

Chebyshev polynomials of even order have even symmetry and contain only even powers of x. Chebyshev polynomials of odd order have odd symmetry and contain only odd powers of x:

\[ T_n (-x) = (-1)^n T_n (x) , \qquad U_n (-x) = (-1)^n U_n (x) . \]
Some numerical values:
\[ T_n (1) = 1 , \qquad T_n (-1) = (-1)^n , \qquad T_n (0) = U_n (0) = \begin{cases} (-1)^k , & \quad \mbox{if $n=2k$ is even}, \\ 0 , & \quad \mbox{if $n=2k+1$ is odd}; \end{cases} \qquad U_n (1) = n+1, \qquad U_n (-1) = (-1)^{\lfloor n/2 \rfloor} \left( n+1 \right) . \]

Derivatives:

\[ \frac{{\text d}\,T_n (x)}{{\text d} x} = n\, U_{n-1} (x) , \qquad \frac{{\text d}\,U_n (x)}{{\text d} x} = \frac{1}{x^2 -1} \left[ (n+1)\, T_n (x) -x\,U_n (x) \right] . \]

Chebyshev polynomials of the second kind

Connections between Chebyshev polynomials

The Chebyshev polynomials of the first kind and second kind are connected via recurrence
\[ 2\,T_n (x) = V_n (x) + V_{n-1} (x) = W_n (x) - W_{n-1} (x) , \qquad n=2,3,4,\ldots . \]
Similar relations are valid for third and fourth kind polynomials:
\[ 2\,T_n (x) = U_n (x) - U_{n-2} (x) , \qquad n=1,2,3,\ldots . \]
Let us introduce new variables:
\begin{equation} u = \left[ \frac{1}{2} \left( 1 + x \right) \right]^{1/2} = \cos\frac{\theta}{2} , \qquad t = \left[ \frac{1}{2} \left( 1 - x \right) \right]^{1/2} = \sin\frac{\theta}{2} . \end{equation}

 

It is readily seen that
\[ W_n (x) = (-1)^n V_n (-x) . \]
Hence, it is normally sufficient to establish properties for third-kind polynomials, and then deduce analogous properties for fourth kind (by replacing x by −x). A key pair of formulae, for the third and fourth polynomials, establishes a strong link with first and second kinds:
\[ V_n (x) = u^{-1} T_{2n+1} (u) , \qquad W_n (x) = U_{2n} (u) , \]
where
\[ u = \frac{1}{\sqrt{2}} \,\sqrt{1+x} = \cos \left( \frac{\theta}{2} \right) \]
A further pair of formulae may be added
\[ T_n (x) = T_{2n} (u) , \qquad U_n (x) = \frac{1}{2}\, u^{-1} U_{2n+1} (u) . \]
Relation to the second kind polynomials:
\[ \begin{split} V_n (x) &= U_n (x) - U_{n-1} (x) , \\ W_n (x) &= U_n (x) + U_{n-1} (x) . \end{split} \]
These relations show that the Chebyshev polynomials of the third kind Vn(x) and forth kind Wn(x) are directly related, respectively, to the first- and second-kind Chebyshev polynomials.

From the relations above, follow two further consequences

\[ V_n (x) + V_{n-1} (x) = W_n (x) - W_{n-1} (x) = 2\,T_n (x) . \]

Note that Vn and Wn are neither even nor odd (unlike Tn and n). We have seen that the leading coefficient (a multiple of xn) is 2n in both Vn(x) and Wn(x), similar to Un(x), This leads to

\[ U_n (x) = \frac{1}{2} \left[ V_n (x) + W_n (x) \right] , \qquad n=1,2,\ldots . \]
This follows from the trigonometric identity:
\[ \frac{\sin \left( n+1 \right) \theta}{\sin\theta} = \frac{1}{2} \left[ \frac{\cos \left( n+\frac{1}{2} \right) \theta}{\cos \left( \frac{\theta}{2} \right)} + \frac{\sin \left( n+ \frac{1}{2} \right) \theta}{\sin\left( \frac{\theta}{2} \right)} \right] . \]

Another option using the recurrence:

CT[n_, x_] :=
CT[n, x] =
If[n == 0, 1,
If[n == 1, x,
Expand[2*x*ChebyshevT[n - 1, x] - ChebyshevT[n - 2, x]]]]

Extended Chebyshev polynomials

For many applications the range [0, 1] is more convenient to use than [−1, 1]. Thus, we map the independent variable x onto 2x −1 and label the shifted or extended Chebyshev polynomials by an additional star

\[ C_n^{\ast} (x) = C_n \left( 2x -1 \right) \qquad \mbox{with} \qquad C = \left\{ T, \ U, \ V, \ W \right\} . \]
The recurrence relation becomes
\begin{equation} C_n^{\ast} (x) = \left( 4x-2 \right) C_{n-1}^{\ast} (x) - C_{n-2}^{\ast} (x) , \end{equation}
and the initial polynomial values are
\[ T_0^{\ast} = U_0^{\ast} = V_0^{\ast} = W_0^{\ast} = 1, \qquad T_1^{\ast} (x) = 2x-1, \quad U_1^{\ast} (x) = 4x-2, \quad V_1^{\ast} (x) = 4x-3, \quad W_1^{\ast} (x) = 4x-1 . \]
Extended Chebyshev polynomials of the first kind are connected with the regular ones
\[ T_{2n} (x) = T_n^{\ast} \left( x^2 \right) \qquad \Longleftrightarrow \qquad T_n^{\ast} \left( x \right) = T_{2n} \left( \sqrt{x} \right) \]
Or they can be expressed through trigonometric functions:
\[ T_n^{\ast} \left( x \right) = \cos 2n\theta , \qquad\mbox{where} \quad x = \cos^2 \theta . \]
Similar formulas are valid for the second kind polynomials:
\[ 2x\,U_{n-1}^{\ast} \left( x^2 \right) = U_{2n-1} (x) . \]
Also
\[ x\,V_{n-1}^{\ast} \left( x^2 \right) = T_{2n-1} (x) , \qquad W_{n}^{\ast} \left( x^2 \right) = U_{2n} (x) . \]
  1. Clenshaw, C.W., Norton, H.J.: The solution of nonlinear ordinary differential equations in chebyshev series. The Computer Journal, 1963, {\bf 6}, Issue 1, 88–92; https://doi.org/10.1093

 

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