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orthpoly::laguerre -- the (generalized) Laguerre polynomials

Introduction

orthpoly::laguerre(n,a,x) computes the value of the generalized n-th degree Laguerre polynomial with parameter a at the point x.

Call(s)

orthpoly::laguerre(n, a, x)

Parameters

n - a nonnegative integer: the degree of the polynomial.
a - an arithmetical expression.
x - an indeterminate or an arithmetical expression. An indeterminate is either an identifier (of domain type DOM_IDENT) or an indexed identifier (of type "_index").

Returns

If x is an indeterminate, then a polynomial of domain type DOM_POLY is returned. If x is an arithmetical expression, then the value of the Laguerre polynomial at this point is returned as an arithmetical expression. If n is not a nonnegative integer, then orthpoly::laguerre returns itself symbolically.

Details

Example 1

Polynomials of domain type DOM_POLY are returned, if identifiers or indexed identifiers are specified:

>> orthpoly::laguerre(2, a, x)
               /                        /        2     \      \
               |      2                 | 3 a   a      |      |
           poly| 1/2 x  + (- a - 2) x + | --- + -- + 1 |, [x] |
               \                        \  2    2      /      /
>> orthpoly::laguerre(3, a, x[1])
          /
          |           3   / a       \     2
      poly| - 1/6 x[1]  + | - + 3/2 | x[1]  +
          \               \ 2       /
      
         /          2     \        /              3     \         \
         |   5 a   a      |        | 11 a    2   a      |         |
         | - --- - -- - 3 | x[1] + | ---- + a  + -- + 1 |, [x[1]] |
         \    2    2      /        \  6          6      /         /

However, using arithmetical expressions as input the ``values'' of these polynomials are returned:

>> orthpoly::laguerre(2, 4, 6*x)
                         (5 - 6 x) (7 - 6 x)
                         ------------------- - 5/2
                                  2
>> orthpoly::laguerre(2, 2/3*I, x[1] + 2)
        (- x[1] - (1 - 2/3 I)) ((1 + 2/3 I) - x[1])
        ------------------------------------------- - (1/2 + 1/3 I)
                             2

``Arithmetical expressions'' include numbers:

>> orthpoly::laguerre(2, a, sqrt(2)),
   orthpoly::laguerre(3, 0.4, 8 + I),
   orthpoly::laguerre(1000, 3, 0.3);
             2
      3 a   a       1/2      1/2
      --- + -- - 2 2    - a 2    + 2, - 4.969333333 - 8.713333334 I,
       2    2
      
         -15691.69498

If no integer degree is specified, then orthpoly::laguerre returns itself symbolically:

>> orthpoly::laguerre(n, a, x), orthpoly::laguerre(1/2, a, x)
        orthpoly::laguerre(n, a, x), orthpoly::laguerre(1/2, a, x)

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