Page 3 of 9

**Splitting U(2,F)F**

U(2,F)F | Primus splitvalues | Secundus splitvalues | Tertius splitvalues | Quartus splitvalues |

[1] | 0 | 2 | F+2 | F-2 |

[2] | F^{2}-4 | F | F^{2}+F-2 | F^{2}-F-2 |

[3] | 2 | -2 | -F | F |

[4] | F | -F | -2 | 2 |

Splitting U(2,F)Fthe secundus | Primus doubled | Primus*(F^{2}-4)doubled | Secundus doubled | Secundus doubled | Tertius doubled | Tertius*(F+2) doubled | Quartus doubled | Quartus*(F-2) doubled | |||||||||||||||||

U_{-5} | F^{5}-5F^{3}+5F | = | -F | * | -(F^{2}-1)*(F^{2}-4) | + | F | = | F^{2}-2 | * | F^{3}-3F | + | -F | = | F^{2}-F-1 | * | (F^{2}-F-1)*(F+2) | + | -2 | = | -F^{2}-F+1 | * | -(F^{2}+F-1)*(F-2) | + | 2 |

U_{-4} | F^{4}-4F^{2}+2 | = | -F | * | -F*(F^{2}-4) | + | 2 | = | F^{2}-2 | * | F^{2}-2 | + | -2 | = | F^{2}-F-1 | * | (F-1)*(F+2) | + | -F | = | -F^{2}-F+1 | * | -(F+1)*(F-2) | + | F |

U_{-3} | F^{3}-3F | = | -1 | * | -F*(F^{2}-4) | + | F | = | F | * | F^{2}-2 | + | -F | = | F-1 | * | (F-1)*(F+2) | + | -2 | = | -F-1 | * | -(F+1)*(F-2) | + | 2 |

U_{-2} | F^{2}-2 | = | -1 | * | -(F^{2}-4) | + | 2 | = | F | * | F | + | -2 | = | F-1 | * | F+2 | + | -F | = | -F-1 | * | -(F-2) | + | F |

U_{-1} | F | = | 0 | * | -(F^{2}-4) | + | F | = | 2 | * | F | + | -F | = | 1 | * | F+2 | + | -2 | = | -1 | * | -(F-2) | + | 2 |

U_{0} | 2 | = | 0 | * | 0 | + | 2 | = | 2 | * | 2 | + | -2 | = | 1 | * | F+2 | + | -F | = | -1 | * | F-2 | + | F |

U_{1} | F | = | 1 | * | 0 | + | F | = | F | * | 2 | + | -F | = | 1 | * | F+2 | + | -2 | = | 1 | * | F-2 | + | 2 |

U_{2} | F^{2}-2 | = | 1 | * | F^{2}-4 | + | 2 | = | F | * | F | + | -2 | = | 1 | * | (F-1)*(F+2) | + | -F | = | 1 | * | (F+1)*(F-2) | + | F |

U_{3} | F^{3}-3F | = | F | * | F^{2}-4 | + | F | = | F^{2}-2 | * | F | + | -F | = | F-1 | * | (F-1)*(F+2) | + | -2 | = | F+1 | * | (F+1)*(F-2) | + | 2 |

U_{4} | F^{4}-4F^{2}+2 | = | F | * | F*(F^{2}-4) | + | 2 | = | F^{2}-2 | * | F^{2}-2 | + | -2 | = | F-1 | * | (F^{2}-F-1)*(F+2) | + | -F | = | F+1 | * | (F^{2}+F-1)*(F-2) | + | F |

U_{5} | F^{5}-5F^{3}+5F | = | F^{2}-1 | * | F*(F^{2}-4) | + | F | = | F^{3}-3F | * | F^{2}-2 | + | -F | = | F^{2}-F-1 | * | (F^{2}-F-1)*(F+2) | + | -2 | = | F^{2}+F-1 | * | (F^{2}+F-1)*(F-2) | + | 2 |

Note that the secundus-based split of a zero-series is

*neutral*: what goes in comes out - in this case the secundus itself.

Note also that a series operating on itself - in this case also the secundus - renders the secundus as its output series.

All other splits of the secundus give a multiple of the operator as output.