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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 \( T_n (x) \) and second kind \( U_n (x) .\) Both Chebyshev polynomials are eigenfunctions of the corresponding singular Sturm--Liouville problems. Other two Chebyshev polynomials of the third kind and the fourth kind are not so popular in applications.

Generating functions

There are known four kinds of Chebyshev polynomials that could be defined through ordinary generating functions:

\begin{align} \frac{1-xt}{1-2xt + t^2} &= \sum_{n\ge 0} T_n (x)\, t^n , \label{EqGF.1} \\ \frac{1}{1-2xt + t^2} &= \sum_{n\ge 0} U_n (x)\, t^n , \label{EqGF.2} \\ \frac{1-t}{1-2xt + t^2} &= \sum_{n\ge 0} V_n (x)\, t^n , \label{EqGF.3} \\ \frac{1+t}{1-2xt + t^2} &= \sum_{n\ge 0} W_n (x)\, t^n . \label{EqGF.4} \end{align}
However, the first two, Tn and Un, are most popular because they are eigenfunctions of the following Sturm--Liouville problems, respectively:
\begin{align*} \left( 1 - x^2 \right) y'' - x\, y' + \lambda\,y &=0 , \qquad \lambda = n^2 , \\ \left( 1 - x^2 \right) y'' - 3x\, y' + \lambda\,y &=0 , \qquad \lambda = n(n+2) . \end{align}
Generating functions contain almost all information about Chebyshev polynomials, including explicit formulas and recurrence relations.

These polynomials could be defined "explicitly" either by the hypergeometric series (which becomes a finite sum)

\begin{align*} T_n (x) &= _2F_1 \left( -n, n, \frac{1}{2}; \frac{1-x}{2} \right) = \frac{n}{2} \sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \frac{(-1)^k}{n-k} \binom{n-k}{k} \left( 2x \right)^{n-2k} , \qquad n\ge 1, \end{align*} 

        cos ( t ) = x
        T(n,x) = cos ( n * t )
\begin{align*} U_n (x) &= \left( n+1 \right) _2F_1 \left( -n, n+2, \frac{3}{2}; \frac{1-x}{2} \right) = \sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} (-1)^k \binom{n-k}{k} \left( 2x \right)^{n-2k} , \qquad n\ge 0, \end{align*} 

        cos ( t ) = x
        U(n,x) = sin ( ( n + 1 ) t ) / sin ( t )
\begin{align*} V_n (x) &= _2F_1 \left( -n, n+1, \frac{1}{2}; \frac{1-x}{2} \right) = \sum_{k=0}^{n} (-1)^k \binom{2n-k}{k} 2^{n-k} \left( x-1 \right)^{n-k} , \qquad n\ge 0, \end{align*} 

        cos ( t ) = x
        V(n,x) = cos ( (2n+1)*t/2) / cos ( t/2)
\begin{align*} W_n (x) &= \left( 2n+1 \right) _2F_1 \left( -n, n+1, \frac{3}{2}; \frac{1-x}{2} \right) = \left( 2n+1 \right) \sum_{k=0}^{n} \frac{2^{n-k}}{2n-2k+1} \binom{2n-k}{k} \left( x-1 \right)^{n-k} , \qquad n\ge 0; \end{align*} 

        cos ( t ) = x
        W(n,x) = sin((2*n+1)*t/2)/sin(t/2)
or by Rodrigues' formulas:
\begin{align*} T_n (x) &= \frac{(-1)^n 2^n n!}{(2n)!}\,\frac{{\text d}^n}{{\text d} x^n} \left( 1- x^2 \right)^{n-1/2} , \\ U_n (x) &= \frac{(-1)^n 2^n (n+1)!}{(2n+1)!}\left( 1 - x^2 \right)^{-1} \frac{{\text d}^n}{{\text d} x^n} \left( 1- x^2 \right)^{n+1/2} , \\ \left( 1-x \right)^{-1/2} \left( 1+x \right)^{1/2} V_n (x) &= \frac{(-1)^n 2^n n!}{(2n)!}\,\frac{{\text d}^n}{{\text d} x^n} \left( 1- x \right)^{n-1/2} \left( 1+ x \right)^{n+1/2}, \\ \left( 1-x \right)^{1/2} \left( 1+x \right)^{-1/2} W_n (x) &= \frac{(-1)^n 2^n n!}{(2n)!}\,\frac{{\text d}^n}{{\text d} x^n} \left( 1- x \right)^{n+1/2} \left( 1+ x \right)^{n-1/2} . \end{align*}

 

Chebyshev Polynomials of the First kind

Using Mathematica, we find first Chebyshev polynomials of the first kind by expanding the generating function into Maclaurin series:
Series[(1 - x*t)/(1 - 2*x*t + t^2), {t, 0, 10}]
       n     Chebyshev polynomial of the first kind
    n = 0         T0(x) = 1
    n = 1         T1(x) = x
    n = 2         T2(x) = 2x² −1
    n = 3         T3(x) = 4x³ −3x
    n = 4         T4(x) = 8x4 −8x² + 1
    n = 5         T5(x) = 16x5 −20x³ + 5x
    n = 6         T6(x) = 32x6 −48x4 + 18x² −1
    n = 7         T7(x) = 64x7 −112x5 + 56x³ −7x
    n = 8         T8(x) = 27x8 −256x6 + 160x4 −32x² + 1
    n = 9         T9(x) = 28x9 −576x7 + 432x5 −120x³ + 9x
    n = 10         T10(x) = 29x10 −1280x8 + 1120x6 −400x4 + 50x² − 1

Using Mathematica, we define the Chebyshev polynomials of the first kind from its generating function:

CT[n_, x_] :=
Module[{F, z, Dn}, F = ((1 - z^2)/(1 - 2*x*z + z^2) + 1)/2;
Dn = D[F, {z, n}];
Expand[Dn/n! /. z -> 0]]

 

Chebyshev Polynomials of the Second kind

Using Mathematica, we find first Chebyshev polynomials of the second kind by expanding the generating function into Maclaurin series:
Series[(1)/(1 - 2*x*t + t^2), {t, 0, 10}]
       n     Chebyshev polynomial of the second kind
    n = 0         U0(x) = 1
    n = 1         U1(x) = 2x
    n = 2         U2(x) = 4x² −1
    n = 3         U3(x) = 8x³ −4x
    n = 4         U4(x) = 16x4 −12x² + 1
    n = 5         U5(x) = 25x5 −32x³ + 6x
    n = 6         U6(x) = 26x6 −80x4 + 24x² −1
    n = 7         U7(x) = 27x7 −192x5 + 80x³ −8x
    n = 8         U8(x) = 28x8 −448x6 + 240x4 −40x² + 1
    n = 9         U9(x) = 29x9 −1024x7 + 672x5 −160x³ + 10x
    n = 10         U10(x) = 210x10 −2304x8 + 1792x6 −560x4 + 60x² −1

 

Chebyshev Polynomials of the Third kind

Using Mathematica, we find first Chebyshev polynomials of the third kind:
Series[(1 - t)/(1 - 2*x*t + t^2), {t, 0, 10}]
       n     Chebyshev polynomial of the third kind
    n = 0         V0(x) = 1
    n = 1         V1(x) = 2x − 1
    n = 2         V2(x) = 4x² − 2x −1
    n = 3         V3(x) = 8x³ − 4x² −4x + 1
    n = 4         V4(x) = 16x4 − 8x³ −12x² + 4x + 1
    n = 5         V5(x) = 32x5 − 16x4 −32x³ + 12x² + 6x − 1
    n = 6         V6(x) = 64x6 − 32x5 − 80x4 + 32x³ + 24x² − 6x −1
    n = 7         V7(x) = 27x7 − 64x6 −192x5 + 80x4 + 80x³ − 24x² −8x + 1
    n = 8         W8(x) = 28x8 − 27x7 −448x6 + 192x5 + 240x4 − 80x³ −40x² + 8x + 1
    n = 9         V9(x) = 29x9 − 28x8 − 210x7 + 448x6 + 672x5 − 240x4 − 160x³ + 40x² + 10x − 1
    n = 10         V10(x) = 210x10 − 29x9 − 2304x8 + 210x7 + 1792x6 − 672x5 − 560x4 + 160x³ + 60x² − 10x − 1

 

Chebyshev Polynomials of the Fourth kind

Using Mathematica, we find first Chebyshev polynomials of the fourth kind:
Series[(1 + t)/(1 - 2*x*t + t^2), {t, 0, 10}]
       n     Chebyshev polynomial of the fourth kind
    n = 0         W0(x) = 1
    n = 1         W1(x) = 1 + 2x
    n = 2         W2(x) = 4x² + 2x −1
    n = 3         W3(x) = 8x³ +4x² −4x −1
    n = 4         W4(x) = 16x4 + 8x³ −12x² −4x + 1
    n = 5         W5(x) = 32x5 + 16x4 −32x³ −12x² + 6x + 1
    n = 6         W6(x) = 64x6 + 32x5 −80x4 −32x³ + 24x² + 6x −1
    n = 7         W7(x) = 27x7 + 64x6 −192x5 −80x4 + 80x³ + 24x² −8x −1
    n = 8         W8(x) = 28x8 + 27x7 −448x6 − 192x5 + 240x4 + 80x³ −40x² − 8x + 1
    n = 9         W9(x) = 29x9 + 28x8 − 210x7 −448x6 + 672x5 + 240x4 − 160x³ −40x² + 10x + 1
    n = 10         W10(x) = 210x10 + 29x9 − 2304x8 − 210x7 + 1792x6 + 672x5 − 560x4 − 160x³ + 60x² + 10x − 1

As we see from the table, the leading coefficient of Wn(x) is 2n.

  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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