Difference between revisions of "Closed formula for physicist's Hermite polynomials"
From specialfunctionswiki
| (3 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
==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)!},$$ |
| − | where $H_n$ denotes the [[Hermite (physicist)|physicist's Hermite polynomials]] and $k!$ denotes the [[factorial]]. | + | where $H_n$ denotes the [[Hermite (physicist)|physicist's Hermite polynomials]], $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the [[floor]] function, and $k!$ denotes the [[factorial]]. |
==Proof== | ==Proof== | ||
Latest revision as of 23:33, 8 July 2016
Theorem
The following formula holds: $$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
- 1960: Earl David Rainville: Special Functions ... (previous) ... (next): $103. (2)$
