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: | ||
| − | $$H_n(x)=\displaystyle\sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \dfrac{(-1)^k n! ( | + | $$H_n(x)=\displaystyle\sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \dfrac{(-1)^k n! (2x)^{n-2k}}{k! (n-2k)!},$$ |
where $H_n$ denotes the [[Hermite (physicist)|physicist's Hermite polynomials]], $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the [[floor]] function, and $k!$ denotes the [[factorial]]. | where $H_n$ denotes the [[Hermite (physicist)|physicist's Hermite polynomials]], $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the [[floor]] function, and $k!$ denotes the [[factorial]]. | ||
==Proof== | ==Proof== | ||
Revision as of 23:32, 8 July 2016
Theorem
The following formula holds: $$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
- 1960: Earl David Rainville: Special Functions ... (previous) ... (next): $103. (2)$
