In[1]:= << HolonomicFunctions.m HolonomicFunctions package by Christoph Koutschan, RISC-Linz, Version 1.5.1 (09.08.2011) --> Type ?HolonomicFunctions for help In[2]:= {L, f, g, h} = << certEx8.m; In[3]:= ApplyOreOperator[L, f] - D[g, x] - D[h, y] // Together Out[3]= 0See cert.m for the Mathematica code which was used to compute these certificates. (This file also uses the Singular interface.)

**A:**Chyzak's algorithm (the command`CreativeTelescoping`in Koutschan's package) applied to the rational function, yielding an ideal of telescopers in*x*and*t*. Then Chyzak's algorithm again applied to this ideal, yielding the desired telescoper in*t*only.**A':**same as**A**, but with*x*and*y*exchanged.**B:**Like in A, except that for the second application, Koutschan's ansatz (the command`FindCreativeTelescoping`in his package) was employed.**C:**Koutschan's command`FindCreativeTelescoping`was used for computing the telescoper in one shot.**D:**Equivalence from Section 2 of our paper, then Chyzak's algorithm (viz. the`CreativeTelescoping`command) for integrating the resulting algebraic functions.**E:**Equivalence from Section 2 of our paper, then step 1 of the algorithm from Section 3 of our paper, then Chyzak's algorithm (viz. the`CreativeTelescoping`command) for integrating the resulting algebraic functions.**F:**Equivalence from Section 2 of our paper, then the full algorithm from Section 3 of our paper, using Koutschan's implementation`SolveCoupledSystem`.

in | out | A | A' | B | C | D | E | F | ord | deg | bytecount | meaning |

in-1.m | out-1.m | 0.71 | 7.45 | 1.41 | 16.11 | 6.56 | 5.88 | 4.98 | 3 | 11 | 3600 | aztec diamond edge gfun, [x^0 y^0] |

in-2.m | out-2.m | 5.90 | 11.03 | 7.89 | 51.75 | 11.06 | 10.90 | 9.30 | 4 | 13 | 4344 | aztec diamond edge gfun, [x^1 y^0] |

in-3.m | out-3.m | 1.16 | 34.96 | 7.30 | 61.89 | 12.21 | 12.48 | 11.42 | 4 | 15 | 5368 | aztec diamond edge gfun, [x^0 y^1] |

in-4.m | out-4.m | 5.96 | 11.87 | 7.82 | 52.97 | 11.38 | 11.15 | 9.09 | 4 | 13 | 4344 | aztec diamond edge gfun, [x^(-1) y^0] |

in-5.m | out-5.m | 0.73 | 6.32 | 1.66 | 21.57 | 6.19 | 5.63 | 4.79 | 3 | 10 | 3376 | aztec diamond edge gfun, [x^0 y^(-1)] |

in-6.m | out-6.m | 6.04 | 11.47 | 11.32 | 77.44 | 11.54 | 11.28 | 9.27 | 4 | 14 | 5824 | aztec diamond edge gfun, [x^(-1) y^(-1)] |

in-7.m | out-7.m | 6.04 | 32.20 | 11.24 | 52.98 | 11.90 | 11.59 | 11.49 | 4 | 14 | 5160 | aztec diamond edge gfun, [x^(-1) y^1] |

in-8.m | out-8.m | 5.98 | 11.06 | 11.54 | 60.53 | 11.11 | 11.04 | 9.39 | 4 | 14 | 5824 | aztec diamond edge gfun, [x^1 y^(-1)] |

in-9.m | out-9.m | 5.94 | 31.59 | 11.50 | 47.14 | 12.09 | 12.54 | 11.71 | 4 | 14 | 5160 | aztec diamond edge gfun, [x^1 y^1] |

in-10.m | out-10.m | 34.02 | 495.75 | 23.72 | 465.36 | 69.87 | 87.31 | 60.35 | 7 | 11 | 10400 | cores of a planar graph, [x^(2n)y^n] |

in-11.m | out-11.m | 2.89 | 147.87 | 2.33 | 32.67 | 1.63 | 1.68 | 1.76 | 4 | 6 | 3920 | cores of a planar graph, [x^n y^(2n)] |

in-12.m | out-12.m | 18.82 | 409.63 | 20.45 | 199.60 | 54.72 | 63.09 | 44.66 | 6 | 9 | 7424 | cores of a planar graph, [x^(2n)y^(2n)] |

in-13.m | out-13.m | 314.34 | 2079.81 | 565.10 | 6022.78 | 6098.31 | 630.98 | 287.44 | 11 | 17 | 20608 | cores of a planar graph, [x^(3n)y^n] |

in-14.m | out-14.m | 1.82 | 55.74 | 1.57 | 16.63 | 17.88 | 1.57 | 1.84 | 3 | 4 | 2032 | cores of a planar graph, [x^n y^(3n)] |

in-15.m | out-15.m | 1385.74 | 5198.98 | 2434.02 | 44390.07 | 848.92 | 2491.66 | 1332.63 | 13 | 21 | 31920 | cores of a planar graph, [x^(4n)y^n] |

in-16.m | out-16.m | 252.61 | 2052.27 | 454.48 | 3623.02 | 337.07 | 414.81 | 298.00 | 10 | 16 | 19368 | cores of a planar graph, [x^(2n) y^(3n)] |

in-17.m | out-17.m | 13.47 | 194.59 | 9.54 | 98.62 | 29.84 | 33.49 | 25.17 | 5 | 8 | 6128 | cores of a planar graph, [x^(3n) y^(2n)] |

in-18.m | out-18.m | 3.12 | 1751.68 | 2.38 | 56.55 | 1.10 | 1.17 | 1.69 | 4 | 6 | 4016 | cores of a planar graph, [x^n y^(4n)] |

in-19.m | out-19.m | >30h | 2.17 | 4431.06 | 399.11 | 21.43 | 22.10 | 23.40 | 3 | 6 | 3992 | artificial example from the paper |

in-20.m | out-20.m | 19.79 | 19.98 | 9.50 | 104.79 | 6.93 | 9.49 | 6.85 | 4 | 9 | 5872 | random groves, [x^n y^n] |

in-21.m | out-21.m | 10712.13 | 125.60 | 421.60 | 5062.14 | 37.60 | 52.26 | 33.18 | 7 | 20 | 21784 | random groves, [x^(2n) y^n] |

in-22.m | out-22.m | 117.96 | 11099.43 | 87.98 | 6060.53 | 37.11 | 54.08 | 33.48 | 7 | 20 | 21784 | random groves, [x^n y^(2n)] |

in-23.m | out-23.m | >30h | 529.80 | 10946.79 | >30h | 166.86 | 403.41 | 173.14 | 10 | 32 | 50560 | random groves, [x^(3n) y^n] |

in-24.m | out-24.m | 16773.93 | 14784.08 | 765.81 | 15679.77 | 759.26 | 758.02 | 1997.19 | 7 | 27 | 31168 | random groves, [x^(2n) y^(2n)] |

in-25.m | out-25.m | 873.93 | >33h | 1125.10 | >30h | 168.78 | 244.90 | 158.93 | 10 | 32 | 50560 | random groves, [x^n y^(3n)] |

in-26.m | out-26.m | 126.55 | 125.61 | 275.13 | 2655.64 | 2666.71 | 280.68 | 127.92 | 6 | 23 | 13576 | knight walks ending at (0,0) |

in-27.m | out-27.m | 4294.85 | 2227.09 | 31336.71 | >30h | 2178.55 | 3116.89 | 4378.15 | 12 | 76 | 88128 | knight walks ending at (1,0) |

in-28.m | out-28.m | 2164.31 | 4283.18 | 21465.62 | >30h | 4264.47 | 5166.89 | 65365.06 | 12 | 76 | 87728 | knight walks ending at (0,1) |

in-29.m | out-29.m | 4271.47 | 2303.72 | 37185.64 | >30h | 2287.83 | 3273.19 | 2376.63 | 12 | 76 | 88176 | knight walks ending at (-1,0) |

in-30.m | out-30.m | 2441.47 | 4247.27 | 43244.79 | >30h | 5386.17 | 6190.89 | 64508.09 | 12 | 76 | 87728 | knight walks ending at (0,-1) |

in-31.m | out-31.m | 8333.90 | 7494.64 | 39162.78 | >30h | 7419.98 | 7915.04 | 72994.38 | 12 | 74 | 85536 | knight walks ending at (-1,-1) |

in-32.m | out-32.m | 8392.92 | 7465.50 | 36957.55 | >30h | 7276.75 | 7938.35 | 74181.14 | 12 | 74 | 85536 | knight walks ending at (-1,1) |

in-33.m | out-33.m | 7754.31 | 8209.88 | 34464.58 | running | 8528.18 | 7808.53 | 72216.13 | 12 | 74 | 84928 | knight walks ending at (1,-1) |

in-34.m | out-34.m | 7436.03 | 10732.05 | 34075.48 | running | 10663.64 | 7756.55 | 71008.12 | 12 | 74 | 84928 | knight walks ending at (1,1) |

in-35.m | out-35.m | 9.75 | 9.86 | 12.99 | 28.70 | 9.71 | 9.85 | 7.47 | 4 | 9 | 3384 | quantum random walks ending at (0,0) |

in-36.m | out-36.m | 5.53 | 5.81 | 3.74 | 11.70 | 5.55 | 5.46 | 3.99 | 3 | 9 | 3392 | quantum random walks ending at (1,0) |

in-37.m | out-37.m | 5.85 | 5.62 | 3.88 | 10.57 | 5.41 | 5.36 | 4.01 | 3 | 9 | 3312 | quantum random walks ending at (0,1) |

in-38.m | out-38.m | 6.89 | 6.22 | 6.93 | 19.09 | 5.77 | 5.35 | 3.89 | 3 | 11 | 3968 | quantum random walks ending at (-1,0) |

in-39.m | out-39.m | 6.00 | 6.62 | 4.92 | 13.98 | 6.61 | 6.33 | 5.44 | 3 | 11 | 3824 | quantum random walks ending at (0,-1) |

in-40.m | out-40.m | 11.41 | 11.16 | 13.95 | 53.75 | 11.06 | 11.25 | 9.25 | 4 | 13 | 4864 | quantum random walks ending at (-1,-1) |

in-41.m | out-41.m | 6.64 | 5.98 | 7.18 | 12.90 | 5.67 | 5.22 | 3.82 | 3 | 12 | 4200 | quantum random walks ending at (-1,1) |

in-42.m | out-42.m | 5.97 | 6.33 | 6.91 | 13.03 | 6.16 | 6.35 | 5.41 | 3 | 12 | 4344 | quantum random walks ending at (1,-1) |

in-43.m | out-43.m | 10.26 | 10.37 | 9.80 | 35.48 | 10.05 | 10.56 | 7.52 | 4 | 9 | 3384 | quantum random walks ending at (1,1) |

in-44.m | out-44.m | 3.48 | 2.61 | 2.59 | 24.55 | 0.36 | 0.40 | 0.59 | 3 | 5 | 2880 | diagonal of 3d-rooks |

in-45.m | out-45.m | 27.89 | 30.74 | 65.50 | 206.58 | 1.51 | 1.59 | 2.50 | 4 | 21 | 16840 | diagonal of 3d-rooks (variation) |

in-46.m | out-46.m | 4838.00 | 4986.07 | 8019.37 | >30h | 543.69 | 580.05 | 479.52 | 6 | 45 | 48200 | diagonal of 3d-rooks (variation) |

in-47.m | out-47.m | 3.05 | 5.72 | 1.69 | 29.06 | 0.55 | 0.61 | 0.89 | 3 | 10 | 3216 | diagonal of 3d-rooks (variation) |

in-48.m | out-48.m | 44.63 | 62.06 | 104.79 | 498.45 | 2.02 | 2.12 | 3.38 | 4 | 30 | 24712 | diagonal of 3d-rooks (variation) |

in-49.m | out-49.m | 12109.41 | 12318.27 | 39640.75 | >30h | 1252.31 | 1452.92 | 1186.72 | 6 | 71 | 90320 | diagonal of 3d-queens |

in-50.m | out-50.m | 2.87 | 3.48 | 1.59 | 19.93 | 0.42 | 0.46 | 0.67 | 3 | 5 | 2880 | diagonal of 3d-rooks |

in-51.m | out-51.m | 31.38 | 28.82 | 42.98 | 182.97 | 1.40 | 1.53 | 2.39 | 4 | 21 | 16840 | diagonal of 3d-rooks (variation) |

in-52.m | out-52.m | 5226.95 | 4912.46 | 8242.97 | >30h | 536.11 | 602.07 | 479.23 | 6 | 45 | 48200 | diagonal of 3d-rooks (variation) |

in-53.m | out-53.m | 5.76 | 3.03 | 6.69 | 24.42 | 0.39 | 0.44 | 0.64 | 3 | 10 | 3216 | diagonal of 3d-rooks (variation) |

in-54.m | out-54.m | 60.18 | 44.90 | 211.31 | 420.31 | 1.82 | 1.91 | 3.02 | 4 | 30 | 24712 | diagonal of 3d-rooks (variation) |

in-55.m | out-55.m | 11797.54 | 11598.18 | 38827.70 | >30h | 1260.61 | 1453.87 | 1201.81 | 6 | 71 | 90320 | diagonal of 3d-queens |

in-56.m | out-56.m | 2.80 | 2.77 | 1.60 | 22.96 | 22.69 | 1.55 | 2.58 | 3 | 5 | 2768 | diagonal of 3d-rooks |

in-57.m | out-57.m | 31.48 | 32.27 | 42.25 | 198.90 | 201.68 | 42.73 | 31.40 | 4 | 21 | 17048 | diagonal of 3d-rooks (variation) |

in-58.m | out-58.m | 5195.78 | 5082.99 | 8270.46 | >30h | 571.67 | 8262.68 | 5086.82 | 6 | 45 | 48552 | diagonal of 3d-rooks (variation) |

in-59.m | out-59.m | 6.47 | 5.91 | 7.53 | 28.93 | 0.54 | 0.71 | 0.88 | 3 | 10 | 3216 | diagonal of 3d-rooks (variation) |

in-60.m | out-60.m | 65.59 | 61.88 | 208.60 | 1270.58 | 2.06 | 2.14 | 3.70 | 4 | 30 | 24712 | diagonal of 3d-rooks (variation) |

in-61.m | out-61.m | 11805.01 | 11975.58 | 39649.67 | >30h | 1268.54 | 1416.92 | 1203.11 | 6 | 71 | 90320 | diagonal of 3d-queens |

in-62.m | out-62.m | 95.64 | 8.89 | 42.94 | 400.03 | 2.47 | 2.52 | 94.70 | 5 | 14 | 11392 | 2-1-1 diagonal of 3d-rooks |

in-63.m | out-63.m | >30h | 477.99 | 48682.20 | 86435.48 | 99.49 | 97.23 | 151.46 | 8 | 60 | 85752 | 2-1-1 diagonal of 3d-rooks (variation) |

in-65.m | out-65.m | 709.68 | 3.91 | 30.30 | 57.40 | 0.54 | 0.58 | 0.87 | 3 | 4 | 2536 | 2-1-1 diagonal of 3d-rooks (variation) |

in-66.m | out-66.m | >30h | 1166.03 | running | running | 183.22 | 183.68 | 242.61 | 9 | 93 | 163536 | 2-1-1 diagonal of 3d-rooks (variation) |

in-68.m | out-68.m | 225.47 | 271.87 | 515.36 | 1517.94 | 259.35 | 204.20 | 1704.81 | 4 | 40 | 24192 | randomly generated input |

in-69.m | out-69.m | 216.96 | 243.07 | 466.36 | 1374.09 | 249.16 | 202.67 | 989.00 | 4 | 40 | 23680 | randomly generated input |

in-70.m | out-70.m | 221.35 | 45.30 | 436.84 | 1232.80 | 43.09 | 38.34 | 26.76 | 4 | 40 | 23528 | randomly generated input |

in-71.m | out-71.m | 37.21 | 238.57 | 447.21 | 3060.84 | 48.05 | 32.82 | 27.33 | 4 | 40 | 24016 | randomly generated input |

in-72.m | out-72.m | 227.78 | 257.81 | 494.57 | 1357.72 | 263.46 | 210.24 | 1012.61 | 4 | 40 | 23920 | randomly generated input |

in-73.m | out-73.m | 36.77 | 231.64 | 351.79 | 2540.32 | 44.68 | 49.84 | 26.82 | 4 | 40 | 23296 | randomly generated input |

in-74.m | out-74.m | 225.23 | 242.18 | 438.31 | 1319.19 | 235.85 | 191.66 | 1185.38 | 4 | 40 | 23488 | randomly generated input |

in-75.m | out-75.m | 2.01 | 1.78 | 1.60 | 6.98 | 1.68 | 1.90 | 1.35 | 3 | 4 | 2592 | restricted 2D lattice walks with small steps |

in-76.m | out-76.m | 1.75 | 1.73 | 1.60 | 11.84 | 1.72 | 2.19 | 1.30 | 3 | 5 | 3192 | restricted 2D lattice walks with small steps |

in-77.m | out-77.m | 2.06 | 2.05 | 2.05 | 12.81 | 1.95 | 2.71 | 1.66 | 3 | 8 | 4448 | restricted 2D lattice walks with small steps |

in-78.m | out-78.m | 2.72 | 2.34 | 1.79 | 12.43 | 2.32 | 3.31 | 1.83 | 3 | 6 | 3776 | restricted 2D lattice walks with small steps |

in-79.m | out-79.m | 38.02 | 31.55 | 18.15 | 330.90 | 13.45 | 21.02 | 11.49 | 5 | 16 | 12880 | restricted 2D lattice walks with small steps |

in-80.m | out-80.m | 64.94 | 66.18 | 46.34 | 505.83 | 25.49 | 37.43 | 20.48 | 5 | 20 | 17136 | restricted 2D lattice walks with small steps |

in-81.m | out-81.m | 42.14 | 32.90 | 19.45 | 334.40 | 14.15 | 21.84 | 11.90 | 5 | 15 | 12656 | restricted 2D lattice walks with small steps |

in-82.m | out-82.m | 69.64 | 72.50 | 50.74 | 474.22 | 25.45 | 37.29 | 20.01 | 5 | 18 | 13424 | restricted 2D lattice walks with small steps |

in-83.m | out-83.m | 99.65 | 68.21 | 64.35 | 751.13 | 27.78 | 39.20 | 22.80 | 5 | 24 | 22200 | restricted 2D lattice walks with small steps |

in-84.m | out-84.m | 114.51 | 81.38 | 99.00 | 866.63 | 26.04 | 35.92 | 21.49 | 5 | 24 | 22656 | restricted 2D lattice walks with small steps |

in-85.m | out-85.m | 36.02 | 23.29 | 35.90 | 192.23 | 6.38 | 6.97 | 4.94 | 5 | 15 | 12464 | restricted 2D lattice walks with small steps |

in-86.m | out-86.m | 57.41 | 45.51 | 26.76 | 250.46 | 18.21 | 19.23 | 13.63 | 5 | 18 | 15072 | restricted 2D lattice walks with small steps |

in-87.m | out-87.m | 109.70 | 65.87 | 71.00 | 775.62 | 29.81 | 42.23 | 24.04 | 5 | 24 | 22160 | restricted 2D lattice walks with small steps |

in-88.m | out-88.m | 136.23 | 79.23 | 101.06 | 863.12 | 27.99 | 39.03 | 23.40 | 5 | 24 | 22584 | restricted 2D lattice walks with small steps |

in-89.m | out-89.m | 30.51 | 22.46 | 30.68 | 162.53 | 6.03 | 6.76 | 4.85 | 5 | 16 | 13368 | restricted 2D lattice walks with small steps |

in-90.m | out-90.m | 51.72 | 43.52 | 25.63 | 228.03 | 17.87 | 19.34 | 13.61 | 5 | 19 | 15944 | restricted 2D lattice walks with small steps |

in-91.m | out-91.m | 396.82 | 363.07 | 11.36 | 66.08 | 6.29 | 6.85 | 3.91 | 3 | 4 | 2736 | restricted 2D lattice walks with small steps |

in-92.m | out-92.m | 0.21 | 0.27 | 0.46 | 1.68 | 0.13 | 0.21 | 0.38 | 2 | 3 | 1992 | diagonal of Szego's ratfun |

in-93.m | out-93.m | 8.99 | 0.81 | 9.03 | 40.33 | 0.83 | 0.85 | 1.06 | 4 | 10 | 7976 | 2-1-1 diagonal of Szego's ratfun |

in-94.m | out-94.m | 0.76 | 11.96 | 2.97 | 33.38 | 0.91 | 0.73 | 0.91 | 4 | 10 | 7976 | 1-2-1 diagonal of Szego's ratfun |

in-95.m | out-95.m | 200.42 | 7.82 | 103.93 | 510.25 | 9.78 | 7.54 | 10.41 | 6 | 15 | 15528 | 3-1-1 diagonal of Szego's ratfun |

in-96.m | out-96.m | 12.37 | 13.81 | 16.10 | 81.62 | 17.77 | 14.17 | 51.66 | 4 | 11 | 9128 | 2-2-1 diagonal of Szego's ratfun |

in-97.m | out-97.m | 6.01 | 206.22 | 17.49 | 579.80 | 10.30 | 10.10 | 10.62 | 6 | 15 | 15528 | 1-3-1 diagonal of Szego's ratfun |