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Neumann Polynomial

Polynomials which obey the Recurrence Relation

O_{n+1}(x)=(n+1){2\over x}O_n(x)-{n+1\over n-1}O_{n-1}(x)+{2n\over x}\sin^2({\textstyle{1\over 2}}n\pi).

The first few are
$\displaystyle O_0(x)$ $\textstyle =$ $\displaystyle {1\over x}$  
$\displaystyle O_1(x)$ $\textstyle =$ $\displaystyle {1\over x^2}$  
$\displaystyle O_2(x)$ $\textstyle =$ $\displaystyle {1\over x}+{4\over x^3}.$  

See also Schläfli Polynomial


von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 196, 1993.

© 1996-9 Eric W. Weisstein