Difference between revisions of "Closed formula for physicist's Hermite polynomials"

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==Theorem==
 
==Theorem==
 
The following formula holds:
 
The following formula holds:
$$H_n(x)=\displaystyle\sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \dfrac{(-1)^k n! (2k)^{n-2k}}{k! (n-2k)!},$$
+
$$H_n(x)=\displaystyle\sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \dfrac{(-1)^k n! (2x)^{n-2k}}{k! (n-2k)!},$$
 
where $H_n$ denotes the [[Hermite (physicist)|physicist's Hermite polynomials]], $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the [[floor]] function, and $k!$ denotes the [[factorial]].
 
where $H_n$ denotes the [[Hermite (physicist)|physicist's Hermite polynomials]], $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the [[floor]] function, and $k!$ denotes the [[factorial]].
 
==Proof==
 
==Proof==

Latest revision as of 23:33, 8 July 2016

Theorem

The following formula holds: $$H_n(x)=\displaystyle\sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \dfrac{(-1)^k n! (2x)^{n-2k}}{k! (n-2k)!},$$ where $H_n$ denotes the physicist's Hermite polynomials, $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the floor function, and $k!$ denotes the factorial.

Proof

References