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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.

Orthogonality

The ordinary generating function for Legendre polynomials is
\begin{equation} \label{EqLege1.1} G(x,t) = \frac{1}{\sqrt{1 -2xt + t^2}} = \sum_{n\ge 0} P_n (x)\, t^n , \end{equation}
where Pn(x is the Legendre polynomial of degree n. Legendre's polynomials are orthogonal
\begin{equation} \label{Eqlegendre.4} \int_{-1}^1 P_n (x) \, P_m (x) \,{\text d} x = \begin{cases} 0 , & \ \mbox{for} \quad n\ne m, \\ \frac{2}{2n+1} , & \ \mbox{for} \quad n = m. \end{cases} \end{equation}
\[ \int_0^{\pi} P_n^m (\cos\theta )\,P_k^m (\cos\theta )\,\sin\theta\,{\text d}\theta = \frac{2\left( k+m \right)!}{\left( 2k+1 \right) \left( k-m \right)!}\,\delta_{n,k} \]
It turns out that the associated Legendre's polynomials are also orthogonal
\[ \int_{-1}^1 P_n^m (x) \, P_k^m (x) \,{\text d} x = \begin{cases} 0 , & \ \mbox{for} \quad n\ne k, \\ \frac{2}{2n+1} \cdot \frac{(n+m)!}{(n-m)!}, & \ \mbox{for} \quad n = k . \end{cases} \]
Also, they satisfy the orthogonality condition for fixed n with weight w = 1/(1 - x²):
\[ \int_{-1}^1 \frac{P_n^i (x) \, P_n^m (x)}{1 - x^2} \,{\text d} x = \begin{cases} 0 , & \ \mbox{for} \quad m\ne i, \\ \frac{(n+m)!}{2(n-m)!}, & \ \mbox{for} \quad m = i \ne 0 , \\ \infty , & \ \mbox{for} \quad m = i =0. \end{cases} \]
  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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