This page contains all calculated cases for the Polynomials described in (0x67dbeaae) on the unit sphere \(\mathbb{S}^d\).
\(k=1\) Link to heading
We have
\[ P_1(t)=t. \]See (0x67dbefac) for a derivation.
\(k=2\) Link to heading
We have
\[ P_2(t)= t^{2} + (d-1)t. \]See (0x67dbf45f) for a derivation (which I should rewrite).
\(k=3\) Link to heading
We have
\[ P_3(t)= t^{3} + 3 (d-1) t^{2} + \left[2 (d-1)^{2} + 2 (d-1)\right] t. \]See (0x67dd1bc1) for a derivation (which I should rewrite).
\(k=4\) Link to heading
We have
\[ P_4(t) = t^{4} + 6 (d-1) t^{3} + \left[11 (d-1)^{2} + 12 (d-1)\right]t^{2} + \left[6 (d-1)^{3} + 14 (d-1)^{2} + 8 (d-1)\right]t. \]See (0x67dd6de8) for a derivation (which I should rewrite).
\(k=5\) Link to heading
We have
\[ P_5(t) = t^{5} + 10 (d-1) t^{4} + \left[35 (d-1)^{2} + 40 (d-1)\right]t^{3} + \left[50 (d-1)^{3} + 134 (d-1)^{2} + 96 (d-1)\right]t^{2} + \left[24 (d-1)^{4} + 100 (d-1)^{3} + 140 (d-1)^{2} + 64 (d-1)\right]t. \]This is calculated with a computer program. A different representation can be found in (0x67e68f92) .
\(k=6\) Link to heading
We have
\[ P_6(t) = t^{6} + 15 (d-1) t^{5} + \left[85 (d-1)^{2} + 100 (d-1)\right]t^{4} + \left[225 (d-1)^{3} + 654 (d-1)^{2} + 520 (d-1)\right]t^{3} + \left[274 (d-1)^{4} + 1310 (d-1)^{3} + 2192 (d-1)^{2} + 1264 (d-1)\right]t^{2} + \left[120 (d-1)^{5} + 780 (d-1)^{4} + 1924 (d-1)^{3} + 2096 (d-1)^{2} + 832 (d-1)\right]t. \]This is calculated with a computer program.
\(k=7\) Link to heading
We have
\[ P_7(t)=t^{7} + 21 (d-1)t^{6} + \left[175 (d-1)^{2} + 210 (d-1)\right]t^{5} + \left[735 (d-1)^{3} + 2254 (d-1)^{2} + 1904 (d-1)\right]t^{4} + \left[1624 (d-1)^{4} + 8484 (d-1)^{3} + 15764 (d-1)^{2} + 10224 (d-1)\right]t^{3} + \left[1764 (d-1)^{5} + 13040 (d-1)^{4} + 37540 (d-1)^{3} + 49136 (d-1)^{2} + 24384 (d-1)\right]t^{2} + \left[720 (d-1)^{6} + 6720 (d-1)^{5} + 25400 (d-1)^{4} + 47880 (d-1)^{3} + 44352 (d-1)^{2} + 15872 (d-1)\right]t. \]This is calculated with a computer program.
\(k=8\) Link to heading
We have
\[ P_8(t) = t^{8} + 28 (d-1) t^{7} + \left[322 (d-1)^{2} + 392 (d-1)\right]t^{6} + \left[1960 (d-1)^{3} + 6244 (d-1)^{2} + 5488 (d-1)\right]t^{5} + \left[6769 (d-1)^{4} + 37632 (d-1)^{3} + 74928 (d-1)^{2} + 52160 (d-1)\right]t^{4} + \left[13132 (d-1)^{5} + 106028 (d-1)^{4} + 337440 (d-1)^{3} + 493696 (d-1)^{2} + 276800 (d-1)\right]t^{3} + \left[13068 (d-1)^{6} + 137368 (d-1)^{5} + 595540 (d-1)^{4} + 1314000 (d-1)^{3} + 1459936 (d-1)^{2} + 648064 (d-1)\right]t^{2} + \left[5040 (d-1)^{7} + 63840 (d-1)^{6} + 340968 (d-1)^{5} + 970904 (d-1)^{4} + 1536480 (d-1)^{3} + 1266048 (d-1)^{2} + 418304 (d-1)\right]t. \]This is calculated with a computer program (in 22 seconds).
\(k=9\) Link to heading
We have
\[ P_9(t)=t^{9} + 36 (d-1) t^{8} + \left[546 (d-1)^{2} + 672 (d-1)\right]t^{7} + \left[4536 (d-1)^{3} + 14868 (d-1)^{2} + 13440 (d-1)\right]t^{6} + \left[22449 (d-1)^{4} + 130704 (d-1)^{3} + 273336 (d-1)^{2} + 199488 (d-1)\right]t^{5} + \left[67284 (d-1)^{5} + 579564 (d-1)^{4} + 1979856 (d-1)^{3} + 3122240 (d-1)^{2} + 1889600 (d-1)\right]t^{4} + \left[118124 (d-1)^{6} + 1352112 (d-1)^{5} + 6443724 (d-1)^{4} + 15773648 (d-1)^{3} + 19623648 (d-1)^{2} + 9845760 (d-1)\right]t^{3} + \left[109584 (d-1)^{7} + 1549296 (d-1)^{6} + 9363048 (d-1)^{5} + 30627672 (d-1)^{4} + 56668032 (d-1)^{3} + 55795200 (d-1)^{2} + 22695936 (d-1)\right]t^{2} + \left[40320 (d-1)^{8} + 665280 (d-1)^{7} + 4757472 (d-1)^{6} + 18919888 (d-1)^{5} + 44758576 (d-1)^{4} + 62391552 (d-1)^{3} + 46973440 (d-1)^{2} + 14553088 (d-1)\right]t. \]This is calculated with a computer program (in 2m 30s).
\(k=10\) Link to heading
We have
\[ t^{10} + 45 (d-1)t^{9} + \left[870 (d-1)^{2} + 1080 (d-1)\right]t^{8} + \left[9450 (d-1)^{3} + 31668 (d-1)^{2} + 29232 (d-1)\right]t^{7} + \left[63273 (d-1)^{4} + 381780 (d-1)^{3} + 828240 (d-1)^{2} + 625248 (d-1)\right]t^{6} + \left[269325 (d-1)^{5} + 2437980 (d-1)^{4} + 8780040 (d-1)^{3} + 14613920 (d-1)^{2} + 9318336 (d-1)\right]t^{5} + \left[723680 (d-1)^{6} + 8836020 (d-1)^{5} + 45155556 (d-1)^{4} + 119064368 (d-1)^{3} + 160100000 (d-1)^{2} + 86953856 (d-1)\right]t^{4} + \left[1172700 (d-1)^{7} + 17984832 (d-1)^{6} + 118802628 (d-1)^{5} + 428061480 (d-1)^{4} + 879340224 (d-1)^{3} + 969147264 (d-1)^{2} + 445052928 (d-1)\right]t^{3} + \left[1026576 (d-1)^{8} + 18755280 (d-1)^{7} + 150100680 (d-1)^{6} + 675980152 (d-1)^{5} + 1835493888 (d-1)^{4} + 2984074944 (d-1)^{3} + 2673288448 (d-1)^{2} + 1012829184 (d-1)\right]t^{2} + \left[362880 (d-1)^{9} + 7560000 (d-1)^{8} + 69631200 (d-1)^{7} + 367181040 (d-1)^{6} + 1202722960 (d-1)^{5} + 2486211520 (d-1)^{4} + 3141921024 (d-1)^{3} + 2199694336 (d-1)^{2} + 646008832 (d-1)\right]t . \]This is calculated with a computer program (in 14m 10s).
\(k=11\) Link to heading
We have
\[ t^{11} + 55 (d-1)t^{10} + \left[1320 (d-1)^{2} + 1650 (d-1)\right]t^{9} + \left[18150 (d-1)^{3} + 61908 (d-1)^{2} + 58080 (d-1)\right]t^{8} + \left[157773 (d-1)^{4} + 979440 (d-1)^{3} + 2186184 (d-1)^{2} + 1692768 (d-1)\right]t^{7} + \left[902055 (d-1)^{5} + 8493540 (d-1)^{4} + 31866648 (d-1)^{3} + 55243232 (d-1)^{2} + 36583360 (d-1)\right]t^{6} + \left[3416930 (d-1)^{6} + 43894950 (d-1)^{5} + 236734476 (d-1)^{4} + 660140800 (d-1)^{3} + 939517088 (d-1)^{2} + 539311104 (d-1)\right]t^{5} + \left[8409500 (d-1)^{7} + 137386392 (d-1)^{6} + 971176140 (d-1)^{5} + 3760527656 (d-1)^{4} + 8333431456 (d-1)^{3} + 9938223104 (d-1)^{2} + 4946323456 (d-1)\right]t^{4} + \left[12753576 (d-1)^{8} + 251751720 (d-1)^{7} + 2190617856 (d-1)^{6} + 10796147824 (d-1)^{5} + 32298819248 (d-1)^{4} + 58269363840 (d-1)^{3} + 58361091328 (d-1)^{2} + 24917015552 (d-1)\right]t^{3} + \left[10628640 (d-1)^{9} + 243462240 (d-1)^{8} + 2486821392 (d-1)^{7} + 14681347648 (d-1)^{6} + 54418355152 (d-1)^{5} + 128886805440 (d-1)^{4} + 189422834176 (d-1)^{3} + 157153183744 (d-1)^{2} + 56104587264 (d-1)\right]t^{2} + \left[3628800 (d-1)^{10} + 93139200 (d-1)^{9} + 1072906560 (d-1)^{8} + 7227644160 (d-1)^{7} + 31164265952 (d-1)^{6} + 88601259808 (d-1)^{5} + 164977916672 (d-1)^{4} + 192636495872 (d-1)^{3} + 126969352192 (d-1)^{2} + 35629531136 (d-1)\right]t . \]This is calculated with a computer program (in 1h 12m).