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

$$O_0(x) = {1\over x}$$

$$O_1(x) = {1\over x^2}$$

$$O_2(x) = {1\over x}+{4\over x^3}.$$

See also Schläfli Polynomial

References

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