MuPAD - Demo

# ... Orthogonal polynomials

## Contents

1. The library package orthpoly
2. Index of algebraic examples

### The library package orthpoly

At first we load the orthpoly library package.

- `>> loadlib("orthpoly"): export(orthpoly):`
  ``

  Let's get some information about this library package.

- `>> info(orthpoly);`
  `Library 'orthpoly': Calculation of orthogonal polynomials
  Interface:
  laguerre
  gegenbauer
  legendre
  hermite
  jacobi
  chebyshev1
  chebyshev2`

  The function chebyshev1 generates the nth Chebyshev polynomial of the first kind.

- `>> chebyshev1(3, x);`
  `

           /            3       \
      poly \ - 3 x + 4 x  , [x] /

  `

  The function chebyshev2 generates the nth Chebyshev polynomial of the second kind.

- `>> chebyshev2(4, x);`
  `

           /       2       4           \
      poly \ - 12 x  + 16 x  + 1 , [x] /

  `

  The function gegenbauer generates the nth Gegenbauer polynomial.

- `>> gegenbauer(5, 2, x);`
  `

           /             3        5      \
      poly \ 24 x - 160 x  + 192 x , [x] /

  `

  To generate the nth Hermite polynomial type:

- `>> hermite(3, x);`
  `

           /             3       \
      poly \ - 12 x + 8 x  , [x] /

  `

  The function jacobi generates the nth Jacobi polynomial at a certain point.

- `>> jacobi(3, 1, 2, x);`
  `

           /            2       3             \
           |   7 x   7 x    21 x              |
      poly | - --- - ---- + ----- + 1/2 , [x] |
           \    2     2       2               /

  `

  To generate the nth generalized Laguerre polynomial type:

- `>> laguerre(3, 1/2, x);`
  `

           /             2    3               \
           |   35 x   7 x    x                |
      poly | - ---- + ---- - -- + 35/16 , [x] |
           \    8      4     6                /

  `

  And finally to generate the nth Legendre polynomial at a certain point type:

- `>> legendre(4, x);`
  `

           /       2       4             \
           |   15 x    35 x              |
      poly | - ----- + ----- + 3/8 , [x] |
           \     4       8               /

  `