This file finds Verbose versions of the differntial equations for the generating functions of integrals, 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 ----------------------------------------------------------------------------\ ---------------- A Linear Differential Equation With Polynomial Coefficients Satisfied by The\ Generating Function of the Inegral of the, 1, 1 -th power of Certain C-finite Polynomial Sequence times, ------, From x=, 2 x + 1 -1, to x=, 1 By Shalosh B. Ekhad Proposition: Let , P[n](x), be the sequence of polynomial definied by the following recurrence P[n](x) = 2 x P[n - 1](x) - P[n - 2](x) Subject to the initial conditions P[0] = 1, P[1] = x Equivalently, in terms of generating function infinity ----- \ n -t x + 1 ) P[n](x) t = -------------- / 2 ----- t - 2 t x + 1 n = 0 Let's define the sequence of numbers 1 / | P_n(x) s(n) = | ------ dx | 2 / x + 1 -1 and let, f(t), be the ordinary generating function infinity ----- \ n f(t) = ) s(n) t / ----- n = 0 Then f(t) satisfies the following inhomogeneous linear differntial equation \ with polynomial coefficients 4 2 6 (t - 1) (t + 1) (t + 6 t + 1) f(t) 6 4 2 /d \ + 4 t (3 t + 9 t + 13 t - 9) |-- f(t)| \dt / / 2 \ 4 2 4 2 4 2 |d | 8 (9 t - 2 t + 1) + (3 t + 6 t - 1) (t + 6 t + 1) |--- f(t)| = ------------------- | 2 | (t - 1) (t + 1) \dt / and in Maple notation 4*t*(3*t^6+9*t^4+13*t^2-9)*diff(f(t),t)+(3*t^4+6*t^2-1)*(t^4+6*t^2+1)*diff(diff (f(t),t),t) = 8*(9*t^4-2*t^2+1)/(t-1)/(t+1) The proof, using the beautiful continuous Almkvist-Zeilberger algorithm is l\ eft to the readers. This ends this article, that took, 0.050, seconds to generate. ---------------------------------------------------- -------------------------------------- --------------------------------------- 1 / 2 | CHEBYSHEVfirstKind[n](x) For the sequence defined by, | ------------------------- dx, we have | 2 / x + 1 -1 ----------------------------------------------------------------------------\ ---------------- A Linear Differential Equation With Polynomial Coefficients Satisfied by The\ Generating Function of the Inegral of the, 2, 1 -th power of Certain C-finite Polynomial Sequence times, ------, From x=, 2 x + 1 -1, to x=, 1 By Shalosh B. Ekhad Proposition: Let , P[n](x), be the sequence of polynomial definied by the following recurrence P[n](x) = 2 x P[n - 1](x) - P[n - 2](x) Subject to the initial conditions P[0] = 1, P[1] = x Equivalently, in terms of generating function infinity ----- \ n -t x + 1 ) P[n](x) t = -------------- / 2 ----- t - 2 t x + 1 n = 0 Let's define the sequence of numbers 1 / 2 | P_n(x) s(n) = | ------- dx | 2 / x + 1 -1 and let, f(t), be the ordinary generating function infinity ----- \ n f(t) = ) s(n) t / ----- n = 0 Then f(t) satisfies the following inhomogeneous linear differntial equation \ with polynomial coefficients 2 2 2 (t + 1) (t - 10 t + 1) f(t) 6 5 4 3 2 /d \ + (t - 1) (5 t - 60 t - 195 t - 128 t + 159 t - 36 t - 1) |-- f(t)| \dt / / 2 \ 2 2 2 |d | + 2 t (t + 1) (t - 10 t + 1) (t + 6 t + 1) (t - 1) |--- f(t)| = | 2 | \dt / 4 3 2 -14 t - 40 t + 60 t - 8 t + 2 and in Maple notation (t-1)*(5*t^6-60*t^5-195*t^4-128*t^3+159*t^2-36*t-1)*diff(f(t),t)+2*t*(t+1)*(t^2 -10*t+1)*(t^2+6*t+1)*(t-1)^2*diff(diff(f(t),t),t) = -14*t^4-40*t^3+60*t^2-8*t+2 The proof, using the beautiful continuous Almkvist-Zeilberger algorithm is l\ eft to the readers. This ends this article, that took, 0.059, seconds to generate. ---------------------------------------------------- -------------------------------------- --------------------------------------- 1 / 3 | CHEBYSHEVfirstKind[n](x) For the sequence defined by, | ------------------------- dx, we have | 2 / x + 1 -1 ----------------------------------------------------------------------------\ ---------------- A Linear Differential Equation With Polynomial Coefficients Satisfied by The\ Generating Function of the Inegral of the, 3, 1 -th power of Certain C-finite Polynomial Sequence times, ------, From x=, 2 x + 1 -1, to x=, 1 By Shalosh B. Ekhad Proposition: Let , P[n](x), be the sequence of polynomial definied by the following recurrence P[n](x) = 2 x P[n - 1](x) - P[n - 2](x) Subject to the initial conditions P[0] = 1, P[1] = x Equivalently, in terms of generating function infinity ----- \ n -t x + 1 ) P[n](x) t = -------------- / 2 ----- t - 2 t x + 1 n = 0 Let's define the sequence of numbers 1 / 3 | P_n(x) s(n) = | ------- dx | 2 / x + 1 -1 and let, f(t), be the ordinary generating function infinity ----- \ n f(t) = ) s(n) t / ----- n = 0 Then f(t) satisfies the following inhomogeneous linear differntial equation \ with polynomial coefficients 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 - 3845859 t + 2214) (t - 1) (t + 1) f(t) + 6 t (272322 t 30 28 26 24 - 397453525 t - 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 /d \ 34 32 + 300810965 t + 11070) |-- f(t)| + (3447198 t - 4351752165 t \dt / 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 / 2 \ 6 4 2 |d | - 248933359005 t - 132554541 t + 22786293 t - 1230) |--- f(t)| + t ( | 2 | \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) / 3 \ |d | 2 34 32 30 |--- f(t)| + 18 t (19926 t - 17969286 t - 2098343517 t | 3 | \dt / 28 26 24 - 3536517948 t + 583661844653 t - 2609936419982 t 22 20 18 + 1300504656243 t + 12385506129104 t - 30551065749775 t 16 14 12 + 31504128471354 t - 15955375513823 t + 3539779009684 t 10 8 6 4 - 202781422449 t + 5215549358 t + 3320320073 t - 56878056 t / 4 \ 2 |d | 3 4 2 + 301949 t + 492) |--- f(t)| + 9 t (t - 1) (t + 1) (t + 6 t + 1) | 4 | \dt / 4 2 24 22 20 18 (t + 198 t + 1) (2214 t - 1975377 t - 9117744 t + 424919689 t 16 14 12 10 - 1856719650 t + 3489133446 t - 3408370560 t + 1761442786 t / 5 \ 8 6 4 2 |d | - 438292878 t + 44440731 t - 3791184 t + 49093 t - 246) |--- f(t)| = - | 5 | \dt / 34 32 30 28 8 (38017578 t + 8766020157 t - 2950066486179 t + 11649994029815 t 26 24 22 + 87325629266849 t - 199688246056073 t - 332913118104631 t 20 18 16 + 1263215873428411 t - 1153385510898319 t + 149037198401773 t 14 12 10 + 311661437034215 t - 151439746163819 t + 19621301936675 t 8 6 4 2 - 1571687047811 t - 105866919757 t - 3633237895 t + 58833633 t - 38622 / 3 2 3 2 ) / (3 (t - 3 t + 3 t - 1) (t + 3 t + 3 t + 1)) / and in Maple notation 6*t*(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)*diff(f(t),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)*diff(diff(f(t),t),t)+t*(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)*diff( diff(diff(f(t),t),t),t)+18*t^2*(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)*diff(diff(diff(diff(f(t),t),t),t),t)+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)*diff(diff(diff(diff(diff(f(t),t),t),t), t),t) = -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) The proof, using the beautiful continuous Almkvist-Zeilberger algorithm is l\ eft to the readers. This ends this article, that took, 0.467, seconds to generate. ---------------------------------------------------- -------------------------------------- --------------------------------------- 1 / 4 | CHEBYSHEVfirstKind[n](x) For the sequence defined by, | ------------------------- dx, we have | 2 / x + 1 -1 ----------------------------------------------------------------------------\ ---------------- A Linear Differential Equation With Polynomial Coefficients Satisfied by The\ Generating Function of the Inegral of the, 4, 1 -th power of Certain C-finite Polynomial Sequence times, ------, From x=, 2 x + 1 -1, to x=, 1 By Shalosh B. Ekhad Proposition: Let , P[n](x), be the sequence of polynomial definied by the following recurrence P[n](x) = 2 x P[n - 1](x) - P[n - 2](x) Subject to the initial conditions P[0] = 1, P[1] = x Equivalently, in terms of generating function infinity ----- \ n -t x + 1 ) P[n](x) t = -------------- / 2 ----- t - 2 t x + 1 n = 0 Let's define the sequence of numbers 1 / 4 | P_n(x) s(n) = | ------- dx | 2 / x + 1 -1 and let, f(t), be the ordinary generating function infinity ----- \ n f(t) = ) s(n) t / ----- n = 0 Then f(t) satisfies the following inhomogeneous linear differntial equation \ with polynomial coefficients 20 19 18 17 16 (21 t + 29820 t + 49345590 t - 1580645220 t + 12893742705 t 15 14 13 - 246400474704 t - 1365471135480 t + 2763583943280 t 12 11 10 - 1302150136710 t + 4668361454280 t + 5784834727812 t 9 8 7 + 4668361454280 t - 1302150136710 t + 2763583943280 t 6 5 4 3 - 1365471135480 t - 246400474704 t + 12893742705 t - 1580645220 t 2 21 20 + 49345590 t + 29820 t + 21) f(t) + (1449 t + 2093319 t 19 18 17 16 + 1311072210 t - 29560258170 t + 193151693625 t - 377627432001 t 15 14 13 + 225653071224 t + 23912487673320 t - 49956348787470 t 12 11 10 + 29414175337230 t - 30548824968372 t - 25558143106812 t 9 8 7 - 45457951818150 t + 20087866560630 t + 2427516079800 t 6 5 4 3 - 4452808166616 t + 1023025993029 t - 70816210485 t + 2023281330 t 2 /d \ 21 20 + 30601830 t + 89229 t - 21) |-- f(t)| + 6 t (623 t + 803901 t \dt / 19 18 17 16 + 368798846 t - 10375524190 t + 98797659455 t - 395243338267 t 15 14 13 - 324200092344 t + 9089897863288 t - 2453451611970 t 12 11 10 - 3170360104070 t + 14682377959316 t - 37393873787028 t 9 8 7 6 + 5590034127318 t + 475011395570 t + 1790389176840 t - 1533411836872 t 5 4 3 2 + 364748584691 t - 4662961943 t - 212458850 t + 25883970 t - 71477 t / 2 \ |d | 2 20 19 18 - 119) |--- f(t)| + 16 t (t - 1) (119 t + 136156 t + 51838490 t | 2 | \dt / 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 / 3 \ 2 |d | 3 2 - 9004966 t + 95004 t + 63) |--- f(t)| + 32 t (t - 34 t + 1) | 3 | \dt / 2 15 14 13 12 (t + 6 t + 1) (7 t + 7263 t + 2579559 t - 39304577 t 11 10 9 8 + 115159683 t + 13151995 t - 1791411853 t + 246925995 t 7 6 5 4 3 - 1932521707 t + 127067037 t + 18085461 t + 35501597 t - 17028831 t / 4 \ 2 2 |d | 19 18 + 576969 t - 14063 t - 7) (t - 1) |--- f(t)| = 4 (3423 t - 744749 t | 4 | \dt / 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 (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)*diff(f(t),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)*diff(diff(f(t),t),t)+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)*diff(diff(diff(f(t),t),t),t)+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*diff(diff(diff(diff(f(t),t),t),t),t) = 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) The proof, using the beautiful continuous Almkvist-Zeilberger algorithm is l\ eft to the readers. This ends this article, that took, 0.506, seconds to generate. ---------------------------------------------------- -------------------------------------- This took, 1.124, seconds.