Method on @sym: laguerreL (n, x) ¶

Method on @sym: laguerreL (n, alpha, x) ¶

Symbolic Laguerre polynomials and associated Laguerre polynomials.

Example:

syms x n
laguerreL(5, x)
  ⇒ ans = (sym)
          5      4      3
         x    5⋅x    5⋅x       2
      - ─── + ──── - ──── + 5⋅x  - 5⋅x + 1
        120    24     3
laguerreL(n, x)
  ⇒ ans = (sym) laguerre(n, x)

When alpha is nonzero, we get generalized (associated) Laguerre polynomials:

laguerreL(n, 1, x)
  ⇒ ans = (sym) assoc_laguerre(n, 1, x)

The polynomials can be manipulated symbolically, for example:

L = laguerreL(n, x);
diff(L, x)
  ⇒ ans = (sym) -assoc_laguerre(n - 1, 1, x)

Laguerre Functions have non-positive integer N, such as

syms x
N = -3;
y = laguerreL(N, x)
  ⇒ y = (sym)
      ⎛ 2          ⎞
      ⎜x           ⎟  x
      ⎜── + 2⋅x + 1⎟⋅ℯ
      ⎝2           ⎠

These satisfy Laguerre’s differential equation:

x*diff(y, x, x) + (1 - x)*diff(y, x) + N*y == 0
  ⇒ (sym) ...
simplify(ans)
  ⇒ (sym) True

Note: the Generalized Laguerre Function is not implemented.

See also: laguerreL, @sym/chebychevT, @sym/chebychevU.

Package: symbolic