This file finds differntial equations for the generating functions of integ\ rals, from -1 to 1 of powers of the Chebyshev polynomials times 1/(1+x^\ 2) for the first through fourth powers --------------------------------------- 1 / | CHEBYSHEVfirstKind[n](x) For the sequence defined by, | ------------------------ dx, we have | 2 / x + 1 -1 4 2 6 4 2 [6 (t - 1) (t + 1) (t + 6 t + 1) + 4 Dt t (3 t + 9 t + 13 t - 9) 4 2 4 2 4 2 2 8 (9 t - 2 t + 1) + (3 t + 6 t - 1) (t + 6 t + 1) Dt , -------------------] (t - 1) (t + 1) and in Maple notation [6*(t-1)*(t+1)*(t^4+6*t^2+1)+4*Dt*t*(3*t^6+9*t^4+13*t^2-9)+(3*t^4+6*t^2-1)*(t^4 +6*t^2+1)*Dt^2, 8*(9*t^4-2*t^2+1)/(t-1)/(t+1)] --------------------------------------- 1 / 2 | CHEBYSHEVfirstKind[n](x) For the sequence defined by, | ------------------------- dx, we have | 2 / x + 1 -1 2 2 2 [(t + 1) (t - 10 t + 1) 6 5 4 3 2 + (t - 1) (5 t - 60 t - 195 t - 128 t + 159 t - 36 t - 1) Dt 2 2 2 2 + 2 t (t + 1) (t - 10 t + 1) (t + 6 t + 1) (t - 1) Dt , 4 3 2 -14 t - 40 t + 60 t - 8 t + 2] and in Maple notation [(t+1)^2*(t^2-10*t+1)^2+(t-1)*(5*t^6-60*t^5-195*t^4-128*t^3+159*t^2-36*t-1)*Dt+ 2*t*(t+1)*(t^2-10*t+1)*(t^2+6*t+1)*(t-1)^2*Dt^2, -14*t^4-40*t^3+60*t^2-8*t+2] --------------------------------------- 1 / 3 | CHEBYSHEVfirstKind[n](x) For the sequence defined by, | ------------------------- dx, we have | 2 / x + 1 -1 2 24 22 20 18 [30 (t + 1) (2214 t - 3845859 t + 148467562 t + 16767551129 t 16 14 12 - 128901738398 t + 423467858250 t - 588795622116 t 10 8 6 4 + 423467858250 t - 128901738398 t + 16767551129 t + 148467562 t 2 3 3 32 30 - 3845859 t + 2214) (t - 1) (t + 1) + 6 Dt (272322 t - 397453525 t 28 26 24 - 1201592944 t + 257596746987 t - 1592468464084 t 22 20 18 + 4723270576143 t - 5805214788880 t - 91658235976433 t 16 14 12 + 273043063689984 t - 273022545252111 t + 102349971024400 t 10 8 6 4 - 9190976258575 t - 139022685228 t + 423174782229 t - 29715042960 t 2 34 32 + 300810965 t + 11070) t + (3447198 t - 4351752165 t 30 28 26 - 25773242211 t + 2218271952717 t + 13344967044609 t 24 22 20 - 235403962782051 t + 1197120262503609 t - 3046259853490935 t 18 16 14 + 3789070231123497 t - 1935346151040969 t - 45036686940921 t 12 10 8 + 308196185497623 t - 61180223912829 t + 9404455821471 t 6 4 2 2 3 - 248933359005 t - 132554541 t + 22786293 t - 1230) Dt + Dt ( 34 32 30 28 1923966 t - 2091502257 t - 82532558877 t + 438576142301 t 26 24 22 + 35940487209511 t - 287285299711739 t + 881082719849543 t 20 18 16 - 1120110502591271 t + 149668337718359 t + 1032871037995303 t 14 12 10 - 1004386418467783 t + 351592908660487 t - 44323191293227 t 8 6 4 2 + 4251359407735 t + 50028971741 t - 2717310589 t + 24259567 t + 1230) 4 34 32 30 28 t + 18 Dt (19926 t - 17969286 t - 2098343517 t - 3536517948 t 26 24 22 + 583661844653 t - 2609936419982 t + 1300504656243 t 20 18 16 + 12385506129104 t - 30551065749775 t + 31504128471354 t 14 12 10 - 15955375513823 t + 3539779009684 t - 202781422449 t 8 6 4 2 2 3 + 5215549358 t + 3320320073 t - 56878056 t + 301949 t + 492) t + 9 t 4 2 4 2 24 22 (t - 1) (t + 1) (t + 6 t + 1) (t + 198 t + 1) (2214 t - 1975377 t 20 18 16 14 - 9117744 t + 424919689 t - 1856719650 t + 3489133446 t 12 10 8 6 - 3408370560 t + 1761442786 t - 438292878 t + 44440731 t 4 2 5 34 32 - 3791184 t + 49093 t - 246) Dt , - 8 (38017578 t + 8766020157 t 30 28 26 - 2950066486179 t + 11649994029815 t + 87325629266849 t 24 22 20 - 199688246056073 t - 332913118104631 t + 1263215873428411 t 18 16 14 - 1153385510898319 t + 149037198401773 t + 311661437034215 t 12 10 8 - 151439746163819 t + 19621301936675 t - 1571687047811 t 6 4 2 / - 105866919757 t - 3633237895 t + 58833633 t - 38622) / (3 / 3 2 3 2 (t - 3 t + 3 t - 1) (t + 3 t + 3 t + 1))] and in Maple notation [30*(t^2+1)*(2214*t^24-3845859*t^22+148467562*t^20+16767551129*t^18-\ 128901738398*t^16+423467858250*t^14-588795622116*t^12+423467858250*t^10-\ 128901738398*t^8+16767551129*t^6+148467562*t^4-3845859*t^2+2214)*(t-1)^3*(t+1)^ 3+6*Dt*(272322*t^32-397453525*t^30-1201592944*t^28+257596746987*t^26-\ 1592468464084*t^24+4723270576143*t^22-5805214788880*t^20-91658235976433*t^18+ 273043063689984*t^16-273022545252111*t^14+102349971024400*t^12-9190976258575*t^ 10-139022685228*t^8+423174782229*t^6-29715042960*t^4+300810965*t^2+11070)*t+( 3447198*t^34-4351752165*t^32-25773242211*t^30+2218271952717*t^28+13344967044609 *t^26-235403962782051*t^24+1197120262503609*t^22-3046259853490935*t^20+ 3789070231123497*t^18-1935346151040969*t^16-45036686940921*t^14+308196185497623 *t^12-61180223912829*t^10+9404455821471*t^8-248933359005*t^6-132554541*t^4+ 22786293*t^2-1230)*Dt^2+Dt^3*(1923966*t^34-2091502257*t^32-82532558877*t^30+ 438576142301*t^28+35940487209511*t^26-287285299711739*t^24+881082719849543*t^22 -1120110502591271*t^20+149668337718359*t^18+1032871037995303*t^16-\ 1004386418467783*t^14+351592908660487*t^12-44323191293227*t^10+4251359407735*t^ 8+50028971741*t^6-2717310589*t^4+24259567*t^2+1230)*t+18*Dt^4*(19926*t^34-\ 17969286*t^32-2098343517*t^30-3536517948*t^28+583661844653*t^26-2609936419982*t ^24+1300504656243*t^22+12385506129104*t^20-30551065749775*t^18+31504128471354*t ^16-15955375513823*t^14+3539779009684*t^12-202781422449*t^10+5215549358*t^8+ 3320320073*t^6-56878056*t^4+301949*t^2+492)*t^2+9*t^3*(t-1)*(t+1)*(t^4+6*t^2+1) *(t^4+198*t^2+1)*(2214*t^24-1975377*t^22-9117744*t^20+424919689*t^18-1856719650 *t^16+3489133446*t^14-3408370560*t^12+1761442786*t^10-438292878*t^8+44440731*t^ 6-3791184*t^4+49093*t^2-246)*Dt^5, -8/3*(38017578*t^34+8766020157*t^32-\ 2950066486179*t^30+11649994029815*t^28+87325629266849*t^26-199688246056073*t^24 -332913118104631*t^22+1263215873428411*t^20-1153385510898319*t^18+ 149037198401773*t^16+311661437034215*t^14-151439746163819*t^12+19621301936675*t ^10-1571687047811*t^8-105866919757*t^6-3633237895*t^4+58833633*t^2-38622)/(t^3-\ 3*t^2+3*t-1)/(t^3+3*t^2+3*t+1)] --------------------------------------- 1 / 4 | CHEBYSHEVfirstKind[n](x) For the sequence defined by, | ------------------------- dx, we have | 2 / x + 1 -1 2 5 20 19 21 [21 + 49345590 t - 246400474704 t + 21 t + 29820 t + 6 t (623 t 20 19 18 17 + 803901 t + 368798846 t - 10375524190 t + 98797659455 t 16 15 14 - 395243338267 t - 324200092344 t + 9089897863288 t 13 12 11 - 2453451611970 t - 3170360104070 t + 14682377959316 t 10 9 8 - 37393873787028 t + 5590034127318 t + 475011395570 t 7 6 5 4 + 1790389176840 t - 1533411836872 t + 364748584691 t - 4662961943 t 3 2 2 21 - 212458850 t + 25883970 t - 71477 t - 119) Dt + (1449 t 20 19 18 17 + 2093319 t + 1311072210 t - 29560258170 t + 193151693625 t 16 15 14 - 377627432001 t + 225653071224 t + 23912487673320 t 13 12 11 - 49956348787470 t + 29414175337230 t - 30548824968372 t 10 9 8 - 25558143106812 t - 45457951818150 t + 20087866560630 t 7 6 5 4 + 2427516079800 t - 4452808166616 t + 1023025993029 t - 70816210485 t 3 2 17 + 2023281330 t + 30601830 t + 89229 t - 21) Dt - 1580645220 t 18 2 20 19 18 + 49345590 t + 16 t (t - 1) (119 t + 136156 t + 51838490 t 17 16 15 14 - 1885048388 t + 17951943987 t - 11283739984 t - 242833457032 t 13 12 11 + 1152434528880 t + 3717548546046 t - 1503190516344 t 10 9 8 7 + 5965291935004 t - 4243352819192 t + 34248918990 t - 67010026384 t 6 5 4 3 + 183552582136 t - 50701103440 t - 3222903733 t + 210419580 t 2 3 3 2 2 - 9004966 t + 95004 t + 63) Dt + 32 t (t - 34 t + 1) (t + 6 t + 1) ( 15 14 13 12 11 7 t + 7263 t + 2579559 t - 39304577 t + 115159683 t 10 9 8 7 + 13151995 t - 1791411853 t + 246925995 t - 1932521707 t 6 5 4 3 2 + 127067037 t + 18085461 t + 35501597 t - 17028831 t + 576969 t 2 4 3 4 - 14063 t - 7) (t - 1) Dt + 29820 t - 1580645220 t + 12893742705 t 10 8 9 + 5784834727812 t - 1302150136710 t + 4668361454280 t 11 13 12 + 4668361454280 t + 2763583943280 t - 1302150136710 t 7 6 14 + 2763583943280 t - 1365471135480 t - 1365471135480 t 15 16 19 18 - 246400474704 t + 12893742705 t , 4 (3423 t - 744749 t 17 16 15 14 + 73306845 t + 14879540673 t + 105075992604 t - 723333038676 t 13 12 11 + 1926330512868 t - 4620179155980 t + 1714586362434 t 10 9 8 7 + 819064523802 t - 3306480941226 t + 591237570318 t + 111717082860 t 6 5 4 3 - 205413995076 t + 69010809972 t - 1704115644 t + 464064087 t 2 - 21075285 t - 16779 t - 7)/(t - 1)] and in Maple notation [21-246400474704*t^15+49345590*t^2-246400474704*t^5+32*t^3*(t^2-34*t+1)*(t^2+6* t+1)*(7*t^15+7263*t^14+2579559*t^13-39304577*t^12+115159683*t^11+13151995*t^10-\ 1791411853*t^9+246925995*t^8-1932521707*t^7+127067037*t^6+18085461*t^5+35501597 *t^4-17028831*t^3+576969*t^2-14063*t-7)*(t-1)^2*Dt^4+16*t^2*(t-1)*(119*t^20+ 136156*t^19+51838490*t^18-1885048388*t^17+17951943987*t^16-11283739984*t^15-\ 242833457032*t^14+1152434528880*t^13+3717548546046*t^12-1503190516344*t^11+ 5965291935004*t^10-4243352819192*t^9+34248918990*t^8-67010026384*t^7+ 183552582136*t^6-50701103440*t^5-3222903733*t^4+210419580*t^3-9004966*t^2+95004 *t+63)*Dt^3-1365471135480*t^6+2763583943280*t^7+(1449*t^21+2093319*t^20+ 1311072210*t^19-29560258170*t^18+193151693625*t^17-377627432001*t^16+ 225653071224*t^15+23912487673320*t^14-49956348787470*t^13+29414175337230*t^12-\ 30548824968372*t^11-25558143106812*t^10-45457951818150*t^9+20087866560630*t^8+ 2427516079800*t^7-4452808166616*t^6+1023025993029*t^5-70816210485*t^4+ 2023281330*t^3+30601830*t^2+89229*t-21)*Dt+4668361454280*t^11+5784834727812*t^ 10-1580645220*t^3-1580645220*t^17+12893742705*t^16+21*t^20+29820*t^19+49345590* t^18+12893742705*t^4+2763583943280*t^13+29820*t+6*t*(623*t^21+803901*t^20+ 368798846*t^19-10375524190*t^18+98797659455*t^17-395243338267*t^16-324200092344 *t^15+9089897863288*t^14-2453451611970*t^13-3170360104070*t^12+14682377959316*t ^11-37393873787028*t^10+5590034127318*t^9+475011395570*t^8+1790389176840*t^7-\ 1533411836872*t^6+364748584691*t^5-4662961943*t^4-212458850*t^3+25883970*t^2-\ 71477*t-119)*Dt^2-1302150136710*t^8+4668361454280*t^9-1302150136710*t^12-\ 1365471135480*t^14, 4*(3423*t^19-744749*t^18+73306845*t^17+14879540673*t^16+ 105075992604*t^15-723333038676*t^14+1926330512868*t^13-4620179155980*t^12+ 1714586362434*t^11+819064523802*t^10-3306480941226*t^9+591237570318*t^8+ 111717082860*t^7-205413995076*t^6+69010809972*t^5-1704115644*t^4+464064087*t^3-\ 21075285*t^2-16779*t-7)/(t-1)] -------------------------------------- This took, 2.059, seconds.