📂Functions

Rodrigues' Formula for Multiple Polynomials

Description

Rodrigues’ formula originally referred to an explicit form of the Legendre polynomials, but later became the general term for formulae representing the explicit forms of special functions expressed in polynomials.

Formulas

• Legendre Polynomials:

  $$ P_{l}(x)=\dfrac{1}{2^{l} l!} \dfrac{d^{l}}{dx^{l}}(x^{2}-1)^{l} $$

• Laguerre Polynomials:

  $$ L_{n}(x) = \frac{1}{n!}e^{x}\frac{ d ^{n}}{ dx^{n} }(x^{n}e^{-x}) $$

• Hermite Polynomials:

  $$ H_{n} = (-1)^{n} e^{x^2} {{d^{n}} \over {dx^{n}}} e^{-x^2} $$