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The Primus U(0,1)F
The series around U0:
| The primus is semi-symmetric with regard to U0. The other three of the fabfour can be derived from it by taking subsequent sums and differences of its terms (quartus and tertius) or subsequent differences between Un+1 and Un-1 (secundus). Its coefficients matrix is, like the sigma repeated matrix, another representation of Pascal's triangle.
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| The following theorems link terms around Un with terms at twice the index value. This is the series' development 'from the belly'. Theorem 2 is important as well as elegant: it shows that the primus on odd indices equals (tertius)*(quartus) while on even indices it equals (primus)*(secundus). Thus, factorization of the primus effectively boils down to the factorization of the other three. | ||
| U2n-1 = | (Un-Un-1)(Un+Un-1) | Theorem 2.1 |
| U2n = | Un(Un+1-Un-1) | Theorem 2.2 |
| U2n-1 = | Un(Un-Un-2)-1 | Theorem 3.1 |
| U2n = | (Un+1+Un)(Un-Un-1)-1 | Theorem 3.2 |
The primus coefficients matrix
The degree of a polynome is one less than its index.
Exponents decrease with steps of 2.
Note that each column displays the subsequent differences in the next one, or, to put it another way, each column displays the partial sums of the previous one.
Pascal's triangle appears starting at the top and looking right and down: 1 - 11 - 121 - 1331 - 14641 etc. To simplify the comparison, the main diagonal has been indicated.
Like Pascal's triangle, this matrix is the sigma repeated matrix revisited.
| U1: | 1 | ||||||||||||
| U2: | 1 | ||||||||||||
| U3: | 1 | -1 | |||||||||||
| U4: | 1 | -2 | |||||||||||
| U5: | 1 | -3 | 1 | ||||||||||
| U6: | 1 | -4 | 3 | ||||||||||
| U7: | 1 | -5 | 6 | -1 | |||||||||
| U8: | 1 | -6 | 10 | -4 | |||||||||
| U9: | 1 | -7 | 15 | -10 | 1 | ||||||||
| U10: | 1 | -8 | 21 | -20 | 5 | ||||||||
| U11: | 1 | -9 | 28 | -35 | 15 | -1 | |||||||
| U12: | 1 | -10 | 36 | -56 | 35 | -6 | |||||||
| U13: | 1 | -11 | 45 | -84 | 70 | -21 | 1 | ||||||
| U14: | 1 | -12 | 55 | -120 | 126 | -56 | 7 | ||||||
| U15: | 1 | -13 | 66 | -165 | 210 | -126 | 28 | -1 | |||||
| U16: | 1 | -14 | 78 | -220 | 330 | -252 | 84 | -8 | |||||
| U17: | 1 | -15 | 91 | -286 | 495 | -462 | 210 | -36 | 1 | ||||
| U18: | 1 | -16 | 105 | -364 | 715 | -792 | 462 | -120 | 9 | ||||
| U19: | 1 | -17 | 120 | -455 | 1001 | -1287 | 924 | -330 | 45 | -1 | |||
| U20: | 1 | -18 | 136 | -560 | 1365 | -2002 | 1716 | -792 | 165 | -10 | |||
| U21: | 1 | -19 | 153 | -680 | 1820 | -3003 | 3003 | -1716 | 495 | -55 | 1 | ||
| U22: | 1 | -20 | 171 | -816 | 2380 | -4368 | 5005 | -3432 | 1287 | -220 | 11 | ||
| U23: | 1 | -21 | 190 | -969 | 3060 | -6188 | 8008 | -6435 | 3003 | -715 | 66 | -1 | |
| U24: | 1 | -22 | 210 | -1140 | 3876 | -8568 | 12376 | -11440 | 6435 | -2002 | 286 | -12 | |
| U25: | 1 | -23 | 231 | -1330 | 4845 | -11628 | 18564 | -19448 | 12870 | -5005 | 1001 | -78 | 1 |
| U26: | 1 | -24 | 253 | -1540 | 5985 | -15504 | 27132 | -31824 | 24310 | -11440 | 3003 | -364 | 13 |