Difference between revisions of "Neumann polynomial"
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(Created page with "The Neumann polynomials are defined by the recurrence $$\left\{ \begin{array}{ll} O_0(s)&=\dfrac{1}{s} \\ O_1(s)&=\dfrac{1}{s^2}\\ O_{n}(s)&=O_{n-2}(s)-2O_{n-1}'(s);n \geq 2...") |
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They also obey the explicit formula | They also obey the explicit formula | ||
$$O_n(s) = \dfrac{n}{4} \displaystyle\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \dfrac{(n-1-k)!(\frac{2}{s})^{n+1-2k}}{k!}; n \geq 1.$$ | $$O_n(s) = \dfrac{n}{4} \displaystyle\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \dfrac{(n-1-k)!(\frac{2}{s})^{n+1-2k}}{k!}; n \geq 1.$$ | ||
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Latest revision as of 18:42, 24 May 2016
The Neumann polynomials are defined by the recurrence $$\left\{ \begin{array}{ll} O_0(s)&=\dfrac{1}{s} \\ O_1(s)&=\dfrac{1}{s^2}\\ O_{n}(s)&=O_{n-2}(s)-2O_{n-1}'(s);n \geq 2 \end{array} \right.$$ They also obey the explicit formula $$O_n(s) = \dfrac{n}{4} \displaystyle\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \dfrac{(n-1-k)!(\frac{2}{s})^{n+1-2k}}{k!}; n \geq 1.$$
