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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.

Recurrences

There are a large number of formulas involving Legendre and associated Legendre polynomials. We consider here only a few of the most useful.The following is a difference equation of order two follows from its ordinary generating function:
\begin{equation} \label{EqLege3.1} \left( n+1 \right) P_{n+1} (x) - \left( 2n+1 \right) P_n (x) + n\,P_{n-1} (x) = 0 , \qquad n=1,2,\ldots . \end{equation}
where Pn(x is the Legendre polynomial of degree n.

Upon differentiating of the generating function, we get the following recurrences:

\begin{equation} \label{EqLege3.2} P'_{n+1} (x) - P'_{n-1} (x) = \left( 2n+1 \right) P_n (x) , \end{equation}
\begin{equation} \label{EqLege3.3} P'_{n+1} (x) - x\,P'_{n} (x) = \left( n+1 \right) P_n (x) , \end{equation}
\begin{equation} \label{EqLege3.4} x\,P'_{n} (x) - P'_{n-1} (x) = n\, P_n (x) , \end{equation}
\begin{equation} \label{EqLege3.5} \left( x^2 -1 \right) P'_{n} (x) = nx\, P_n (x) - n\, P_{n-1} (x) . \end{equation}
Symmetry relation:
\begin{equation} \label{EqLege3.7} P_n (-x) = (-1)^n P_n (x) , \qquad n=1,2,\ldots . \end{equation}
Some numerical values:
\[ P_n (1) = 1 , \qquad P_n (-1) = (-1)^n . \] Also
\[ P_n (0) = \begin{cases} 0, & \ \mbox{when} \quad n \ \mbox{ is odd}, \\ (-1)^{n/2} \frac{(n-1)!!}{n!!} & \ \mbox{when} \quad n \ \mbox{ is even}. \end{cases} \]

Recurrences for Associated Legendre Polynomials

\begin{equation} \left( n-m+1 \right) P_{n+1}^m (x) = \left( 2n +1 \right) P_n^m (x) - \left( n+m \right) P_{n-1}^m (x) , \qquad n=1,2,\ldots . \end{equation}
\begin{equation} P_{n+1}^{m+2} (x) - \frac{\left( 2m+1 \right) x}{\left( 1 - x^2 \right)^{1/2}}\, P_n^{m+1} (x) + \left( n-m \right)\left( n+m+1 \right) P_n^m (x) = 0, n=1,2,\ldots . \end{equation}
\[ P_n^m (\cos \theta ) = (-1)^m \left( \sin\theta \right)^m \frac{{\text d}^m}{{\text d} (\cos \theta )^m} \,P_n (\cos\theta ). \]

 

Associated Legendre Functions of the Second Kind

\[ Q_n^m (x ) = \left( 1 - x^2 \right)^{m/2} \frac{{\text d}^m}{{\text d} x^m} \,Q_n (x ). \]
  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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