Rodrigues' Formula for Hermite Polynomials

Theorem

$\map {H_n} x = \paren {-1}^n \map \exp {x^2} \map {\dfrac {\d^n} {\d x^n} } {\map \exp {-x^2} }$

where:

$n \in \N$ is a natural number

$H_n$ is the $n$th Hermite polynomial.

Proof

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Source of Name

This entry was named for Olinde Rodrigues.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 27$: Hermite Polynomials: Hermite Polynomials: $27.2$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hermite polynomial
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hermite polynomial
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 29$: Hermite Polynomials: Hermite Polynomials: $29.2.$