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Splitting U(a,b,c)F
| U(a,b,c)F | Primus splitvalues | Secundus splitvalues | Tertius splitvalues | Quartus splitvalues |
| [1] | -Fa+2b-c | a+ c/(F-2) | (a+b)+ 2c/(F-2) | -(a-b) |
| [2] | Fb-2a+c | b+ c/(F-2) | Fb-(a-b)+ Fc/(F-2) | Fb-(a+b)+c |
| [3] | a | -a- 2c/(F-2) | -b- 2c/(F-2) | b |
| [4] | Fa-b+c | -Fa+b- Fc/(F-2) | -a- 2c/(F-2) | a |
| Splitting U(a,b,c)F the general series | Primus doubled | U(-Fa+2b-c, Fb-2a+c)F doubled | Secundus doubled | U[a+ c/(F-2), b+ c/(F-2)]F doubled | Tertius doubled | U[a+b+ 2c/(F-2), Fb-(a-b)+ Fc/(F-2)]F doubled | Quartus doubled | U[-(a-b), Fb-(a+b)+c]F doubled | |||||||||||||||||
| U-4 | F4a-F3(b-c)-F2(3a-c)+F(2b-c)+a | = | -F | * | -F3a+F2(b-c)+F(3a-c)-2b+c | + | a | = | F2-2 | * | F2a-Fb-a+ c(F2-F-1)/(F-2) | + | -a- 2c/(F-2) | = | F2-F-1 | * | F2a+F(a-b)-(a+b) +c(F2-2)/(F-2) | + | -b -2c/(F-2) | = | -F2-F+1 | * | -F2a+F(a+b-c)+(a-b) | + | b |
| U-3 | F3a-F2(b-c)-F(2a-c)+b | = | -1 | * | -F3a+F2(b-c)+F(3a-c)-2b+c | + | Fa-b+c | = | F | * | F2a-Fb-a+ c(F2-F-1)/(F-2) | + | -Fa+b -Fc/(F-2) | = | F-1 | * | F2a+F(a-b)-(a+b) +c(F2-2)/(F-2) | + | -a -2c/(F-2) | = | -F-1 | * | -F2a+F(a+b-c)+(a-b) | + | a |
| U-2 | F2a-F(b-c)-(a-c) | = | -1 | * | -F2a+F(b-c)+2a-c | + | a | = | F | * | Fa-b+ c(F-1)/(F-2) | + | -a- 2c/(F-2) | = | F-1 | * | Fa+(a-b) +Fc/(F-2) | + | -b -2c/(F-2) | = | -F-1 | * | -Fa+(a+b)-c | + | b |
| U-1 | Fa-(b-c) | = | 0 | * | -F2a+F(b-c)+2a-c | + | Fa-b+c | = | 2 | * | Fa-b+ c(F-1)/(F-2) | + | -Fa+b -Fc/(F-2) | = | 1 | * | Fa+(a-b) +Fc/(F-2) | + | -a -2c/(F-2) | = | -1 | * | -Fa+(a+b)-c | + | a |
| U0 | a | = | 0 | * | -Fa+2b-c | + | a | = | 2 | * | a+ c/(F-2) | + | -a- 2c/(F-2) | = | 1 | * | (a+b)+ 2c/(F-2) | + | -b -2c/(F-2) | = | -1 | * | -(a-b) | + | b |
| U1 | b | = | 1 | * | -Fa+2b-c | + | Fa-b+c | = | F | * | a+ c/(F-2) | + | -Fa+b -Fc/(F-2) | = | 1 | * | (a+b)+ 2c/(F-2) | + | -a -2c/(F-2) | = | 1 | * | -(a-b) | + | a |
| U2 | Fb-(a-c) | = | 1 | * | Fb-2a+c | + | a | = | F | * | b+ c/(F-2) | + | -a- 2c/(F-2) | = | 1 | * | Fb-(a-b)+ Fc/(F-2) | + | -b -2c/(F-2) | = | 1 | * | Fb-(a+b)+c | + | b |
| U3 | F2b-F(a-c)-(b-c) | = | F | * | Fb-2a+c | + | Fa-b+c | = | F2-2 | * | b+ c/(F-2) | + | -Fa+b -Fc/(F-2) | = | F-1 | * | Fb-(a-b)+ Fc/(F-2) | + | -a -2c/(F-2) | = | F+1 | * | Fb-(a+b)+c | + | a |
| U4 | F3b-F2(a-c)-F(2b-c)+a | = | F | * | F2b-F(a-c)-2b+c | + | a | = | F2-2 | * | Fb-a+ c(F-1)/(F-2) | + | -a- 2c/(F-2) | = | F-1 | * | F2b-F(a-b)-(a+b)+ c(F2-2)/(F-2) | + | -b -2c/(F-2) | = | F+1 | * | F2b-F(a+b-c)+(a-b) | + | b |
| U5 | F4b-F3(a-c)-F2(3b-c)+F(2a-c)+b | = | F2-1 | * | F2b-F(a-c)-2b+c | + | Fa-b+c | = | F3-3F | * | Fb-a+ c(F-1)/(F-2) | + | -Fa+b -Fc/(F-2) | = | F2-F-1 | * | F2b-F(a-b)-(a+b)+ c(F2-2)/(F-2) | + | -a -2c/(F-2) | = | F2+F-1 | * | F2b-F(a+b-c)+(a-b) | + | a |
This is the alpha and omega of splitting in the U(a,b,c)F domain: the split of the general series.
Secundus- and tertius-based splits go clearly into the realm of rational numbers. Clearly too, the secundus loses the 'unity role' it has for zero series.