25:
2328:
814:
be a recurrence equation with polynomial coefficients. There exist several algorithms which compute solutions of this equation. These algorithms can compute polynomial, rational, hypergeometric and d'Alembertian solutions. The solution of a homogeneous equation is given by the
4112:'s creative telescoping algorithm can transform such a hypergeometric sum into a recurrence equation with polynomial coefficients. This equation can then be solved to get for example a linear combination of hypergeometric solutions which is called a closed form solution of
2179:
1883:. This is the case if and only if the sequence is the solution of a first-order recurrence equation with polynomial coefficients. The set of hypergeometric sequences is not a subspace of the space of sequences as it is not closed under addition.
2184:
1914:
is the sum of hypergeometric sequences. The algorithm makes use of the Gosper-Petkovšek normal-form of a rational function. With this specific representation it is again sufficient to consider polynomial solutions of a transformed equation.
3209:
1994 Abramov and Petkovšek described an algorithm which computes the general d'Alembertian solution of a recurrence equation. This algorithm computes hypergeometric solutions and reduces the order of the recurrence equation recursively.
2004:
3934:
889:
1352:
2749:
2911:
420:
3424:
2376:
783:
3300:
2456:
1134:
976:
1080:
329:
292:
1881:
3750:
87:(or linear recurrence relations or linear difference equations) with polynomial coefficients. These equations play an important role in different areas of mathematics, specifically in
1518:
610:
2518:
3109:
164:
2590:
217:
1585:
4147:
If sequences are considered equal if they are equal in almost all terms, then this basis is finite. More on this can be found in the book A=B by Petkovšek, Wilf and
Zeilberger.
4071:
3204:
1631:
1458:
1412:
3532:
3155:
2957:
3971:
2323:{\textstyle c\in \mathbb {K} ,z\in {\overline {\mathbb {K} }},s\in \mathbb {N} ,r(n)\in {\overline {\mathbb {K} }}(n),\xi _{1},\dots ,\xi _{s}\in {\overline {\mathbb {K} }}}
1259:
684:
3480:
3600:
3251:
2544:
1192:
255:
130:
3015:
2485:
4157:
Abramov, Sergei A. (1989). "Problems in computer algebra that are connected with a search for polynomial solutions of linear differential and difference equations".
2986:
3035:
2402:
4106:
3797:
1002:
3323:
2671:
2644:
2617:
1970:
1943:
1725:
1698:
474:
447:
3640:
2846:
2818:
1999:
1912:
1812:
1788:
1745:
1523:
The other algorithms for finding more general solutions (e.g. rational or hypergeometric solutions) also rely on algorithms which compute polynomial solutions.
1221:
1163:
812:
546:
4130:
4011:
3991:
3660:
3552:
2789:
2769:
1671:
1651:
630:
514:
494:
3302:. A signed permutation matrix is a square matrix which has exactly one nonzero entry in every row and in every column. The nonzero entries can be
1363:
334:
1918:
A different and more efficient approach is due to Mark van Hoeij. Considering the roots of the first and the last coefficient polynomial
3328:
3802:
1368:
In the late 1980s Sergei A. Abramov described an algorithm which finds the general polynomial solution of a recurrence equation, i.e.
822:
422:
is called a linear recurrence equation with polynomial coefficients (all recurrence equations in this article are of this form). If
3665:
1264:
4307:
Abramov, Sergei A. (1989). "Rational solutions of linear differential and difference equations with polynomial coefficients".
2676:
2174:{\displaystyle y(n)=c\,r(n)\,z^{n}\,\Gamma (n-\xi _{1})^{e_{1}}\Gamma (n-\xi _{2})^{e_{2}}\cdots \Gamma (n-\xi _{s})^{e_{s}}}
1673:. Abramov showed how this universal denominator can be computed by only using the first and the last coefficient polynomial
2851:
1972:â called singularities â one can build a solution step by step making use of the fact that every hypergeometric sequence
1468:. In 1995 Abramov, Bronstein and Petkovšek showed that the polynomial case can be solved more efficiently by considering
4420:
Abramov, Sergei A.; Petkovšek, Marko (1994). "D'Alembertian solutions of linear differential and difference equations".
4256:
Abramov, Sergei A.; Bronstein, Manuel; Petkovšek, Marko (1995). "On polynomial solutions of linear operator equations".
4437:
4283:
4232:
2333:
697:
3756:
it is possible to see that there is no polynomial, rational or hypergeometric solution for this recurrence equation.
3485:
3259:
41:
2407:
1354:
is also a solution of the inhomogeneous problem and it is called the general solution of the inhomogeneous problem.
1085:
898:
98:
From the late 1980s, the first algorithms were developed to find solutions for these equations. Sergei A. Abramov,
1007:
297:
260:
102:
and Mark van Hoeij described algorithms to find polynomial, rational, hypergeometric and d'Alembertian solutions.
2820:
by
Abramov's algorithm. Considering all possibilities one gets the general solution of the recurrence equation.
1817:
3224:
4377:
Cluzeau, Thomas; van Hoeij, Mark (2006). "Computing
Hypergeometric Solutions of Linear Recurrence Equations".
1475:
551:
4342:
van Hoeij, Mark (1999). "Finite singularities and hypergeometric solutions of linear recurrence equations".
2494:
3040:
1464:) gave a degree bound for polynomial solutions. This way the problem can simply be solved by considering a
139:
133:
2549:
169:
3482:. Applying an algorithm to find hypergeometric solutions one can find the general hypergeometric solution
1544:
4179:
Petkovšek, Marko (1992). "Hypergeometric solutions of linear recurrences with polynomial coefficients".
3753:
1891:
1756:
1633:, can be found by using the notion of a universal denominator. A universal denominator is a polynomial
1465:
4016:
3160:
1590:
1417:
1371:
1747:
all rational solutions can be found by computing all polynomial solutions of a transformed equation.
3114:
2916:
4266:
3939:
3611:
4258:
Proceedings of the 1995 international symposium on
Symbolic and algebraic computation - ISSAC '95
1791:
1226:
816:
635:
4261:
3429:
3230:
1532:
892:
3557:
2527:
1894:
to get the general hypergeometric solution of a recurrence equation where the right-hand side
1168:
222:
113:
4422:
Proceedings of the international symposium on
Symbolic and algebraic computation - ISSAC '94
2991:
2461:
4490:
3325:. The sequence is determined by the linear recurrence equation with polynomial coefficients
1472:
solution of the recurrence equation in a specific power basis (i.e. not the ordinary basis
2962:
8:
3020:
2381:
1469:
84:
4076:
3767:
981:
4443:
4402:
4289:
3305:
2831:
2649:
2622:
2595:
1948:
1921:
1897:
1797:
1730:
1703:
1676:
452:
425:
92:
62:
4355:
4320:
3616:
2794:
1975:
1887:
1764:
1461:
1197:
1139:
788:
522:
99:
4433:
4394:
4359:
4324:
4279:
4238:
4228:
4196:
4192:
2521:
1538:
4293:
3111:. This is the case if and only if there are first-order linear recurrence operators
4447:
4425:
4406:
4386:
4351:
4316:
4271:
4188:
4115:
4109:
3996:
3976:
3645:
3537:
2774:
2754:
1656:
1636:
615:
499:
479:
516:
is zero the equation is called homogeneous, otherwise it is called inhomogeneous.
80:
72:
4222:
3254:
2488:
4390:
4484:
4398:
4363:
4328:
4200:
88:
4242:
4429:
4275:
1727:. Substituting this universal denominator for the unknown denominator of
76:
91:. The sequences which are solutions of these equations are called
3929:{\textstyle F(n,k+1)/F(n,k),F(n+1,k)/F(n,k)\in \mathbb {K} (n,k)}
884:{\textstyle \ker L=\{y\in \mathbb {K} ^{\mathbb {N} }\,:\,Ly=0\}}
612:
is a linear recurrence operator with polynomial coefficients and
1794:
if the ratio of two consecutive terms is a rational function in
4379:
Applicable
Algebra in Engineering, Communication and Computing
4221:
Petkovšek, Marko; Wilf, Herbert S.; Zeilberger, Doron (1996).
1653:
such that the denominator of every rational solution divides
2751:
it is possible to make an ansatz which gives candidates for
891:. As a subspace of the space of sequences this kernel has a
4461:
4159:
Moscow
University Computational Mathematics and Cybernetics
2791:
one can again make an ansatz to get the rational function
1347:{\textstyle c_{1}y^{(1)}+\dots +c_{m}y^{(m)}+{\tilde {y}}}
1136:
is called the general solution of the homogeneous problem
4255:
2744:{\textstyle \xi _{1},\dots ,\xi _{s},e_{1},\dots ,e_{s}}
2646:). Furthermore one can compute bounds for the exponents
2592:
have to be singularities of the equation (i.e. roots of
4309:
USSR Computational
Mathematics and Mathematical Physics
4220:
4118:
4079:
4019:
3999:
3979:
3942:
3805:
3770:
3648:
3619:
3560:
3540:
3432:
3308:
3262:
3163:
3117:
3043:
3023:
2994:
2965:
2919:
2854:
2797:
2777:
2757:
2679:
2652:
2625:
2598:
2552:
2530:
2497:
2464:
2410:
2384:
2336:
2187:
1978:
1951:
1924:
1820:
1767:
1706:
1679:
1659:
1639:
1593:
1547:
1478:
1420:
1374:
1267:
1229:
1200:
1171:
1142:
1088:
1010:
984:
901:
825:
791:
700:
638:
618:
554:
525:
502:
482:
455:
428:
300:
263:
225:
172:
142:
116:
79:
where the coefficient sequences can be represented as
3668:
3602:
describes the number of signed permutation matrices.
3488:
3331:
3233:
2834:
2007:
1900:
1800:
1733:
337:
3554:. Also considering the initial values, the sequence
2906:{\textstyle y=h_{1}\sum h_{2}\sum \cdots \sum h_{k}}
415:{\displaystyle \sum _{k=0}^{r}p_{k}(n)\,y(n+k)=f(n)}
4013:. A hypergeometric sum is a finite sum of the form
4124:
4100:
4065:
4005:
3985:
3965:
3928:
3791:
3744:
3654:
3634:
3594:
3546:
3526:
3474:
3418:
3317:
3294:
3245:
3198:
3149:
3103:
3029:
3009:
2980:
2951:
2905:
2840:
2812:
2783:
2763:
2743:
2665:
2638:
2611:
2584:
2538:
2512:
2479:
2450:
2396:
2370:
2322:
2173:
1993:
1964:
1937:
1906:
1875:
1806:
1782:
1739:
1719:
1692:
1665:
1645:
1625:
1579:
1512:
1452:
1406:
1346:
1253:
1215:
1186:
1157:
1128:
1074:
996:
970:
883:
806:
777:
678:
624:
604:
540:
508:
488:
468:
441:
414:
323:
286:
249:
211:
158:
124:
3419:{\displaystyle y(n)=4(n-1)^{2}\,y(n-2)+2\,y(n-1)}
2371:{\textstyle \xi _{i}-\xi _{j}\notin \mathbb {Z} }
778:{\textstyle \sum _{k=0}^{r}p_{k}(n)\,y(n+k)=f(n)}
4482:
4419:
4376:
4216:
4214:
4212:
4210:
4174:
4172:
3295:{\textstyle y(n)\in \mathbb {Q} ^{\mathbb {N} }}
1537:In 1989 Sergei A. Abramov showed that a general
2451:{\textstyle e_{1},\dots ,e_{s}\in \mathbb {Z} }
1129:{\textstyle c_{1},\dots ,c_{m}\in \mathbb {K} }
971:{\textstyle \{y^{(1)},y^{(2)},\dots ,y^{(m)}\}}
3218:
4207:
4169:
1364:Polynomial solutions of P-recursive equations
1075:{\textstyle c_{1}y^{(1)}+\dots +c_{m}y^{(m)}}
324:{\textstyle y\in \mathbb {K} ^{\mathbb {N} }}
287:{\textstyle f\in \mathbb {K} ^{\mathbb {N} }}
34:needs attention from an expert in mathematics
3662:elements is given by the recurrence equation
965:
902:
878:
838:
1876:{\textstyle y(n+1)/y(n)\in \mathbb {K} (n)}
3745:{\displaystyle y(n)=(n-1)\,y(n-2)+y(n-1).}
2823:
1750:
4341:
4265:
4178:
3944:
3907:
3699:
3507:
3397:
3372:
3286:
3280:
2532:
2501:
2444:
2364:
2311:
2255:
2230:
2211:
2195:
2050:
2039:
2026:
1860:
1610:
1564:
1504:
1437:
1391:
1122:
865:
861:
855:
849:
741:
642:
378:
315:
309:
278:
272:
196:
152:
144:
118:
1513:{\textstyle (x^{n})_{n\in \mathbb {N} }}
689:
605:{\textstyle L=\sum _{k=0}^{r}p_{k}N^{k}}
496:is called the order of the equation. If
4306:
4156:
2513:{\textstyle {\overline {\mathbb {K} }}}
1357:
4483:
3104:{\textstyle \Delta y=Ny-y=y(n+1)-y(n)}
3037:denotes the difference operator, i.e.
159:{\textstyle \mathbb {K} =\mathbb {Q} }
44:may be able to help recruit an expert.
3157:with rational coefficients such that
2585:{\textstyle \xi _{1},\dots ,\xi _{s}}
1526:
212:{\textstyle p_{k}(n)\in \mathbb {K} }
1580:{\textstyle y(n)\in \mathbb {K} (n)}
18:
4344:Journal of Pure and Applied Algebra
1414:, with a polynomial right-hand side
819:of the linear recurrence operator:
83:. P-recursive equations are linear
13:
3973:denotes the rational functions in
3044:
3024:
2995:
2913:for some hypergeometric sequences
2465:
2132:
2090:
2051:
1587:, with polynomial right-hand side
14:
4502:
4066:{\textstyle f(n)=\sum _{k}F(n,k)}
3199:{\textstyle L_{k}\cdots L_{1}y=0}
1626:{\textstyle f(n)\in \mathbb {K} }
1453:{\textstyle f(n)\in \mathbb {K} }
1407:{\textstyle y(n)\in \mathbb {K} }
331:an unknown sequence. The equation
2001:has a representation of the form
23:
4454:
4181:Journal of Symbolic Computation
3759:
3527:{\displaystyle y(n)=c\,2^{n}n!}
3150:{\textstyle L_{1},\dots ,L_{k}}
2952:{\textstyle h_{1},\dots ,h_{k}}
4413:
4370:
4335:
4300:
4249:
4150:
4141:
4095:
4083:
4060:
4048:
4029:
4023:
3966:{\textstyle \mathbb {K} (n,k)}
3960:
3948:
3923:
3911:
3900:
3888:
3877:
3859:
3850:
3838:
3827:
3809:
3786:
3774:
3736:
3724:
3715:
3703:
3696:
3684:
3678:
3672:
3629:
3623:
3605:
3570:
3564:
3498:
3492:
3463:
3457:
3442:
3436:
3413:
3401:
3388:
3376:
3363:
3350:
3341:
3335:
3272:
3266:
3098:
3092:
3083:
3071:
2807:
2801:
2474:
2468:
2270:
2264:
2246:
2240:
2155:
2135:
2113:
2093:
2074:
2054:
2036:
2030:
2017:
2011:
1988:
1982:
1870:
1864:
1853:
1847:
1836:
1824:
1777:
1771:
1620:
1614:
1603:
1597:
1574:
1568:
1557:
1551:
1493:
1479:
1447:
1441:
1430:
1424:
1401:
1395:
1384:
1378:
1338:
1324:
1318:
1289:
1283:
1239:
1178:
1067:
1061:
1032:
1026:
960:
954:
935:
929:
916:
910:
772:
766:
757:
745:
738:
732:
673:
661:
652:
646:
409:
403:
394:
382:
375:
369:
206:
200:
189:
183:
1:
4356:10.1016/s0022-4049(99)00008-0
4321:10.1016/s0041-5553(89)80002-3
4135:
105:
4193:10.1016/0747-7171(92)90038-6
3799:is called hypergeometric if
2505:
2315:
2259:
2215:
1460:. He (and a few years later
1254:{\textstyle L{\tilde {y}}=f}
1194:is a particular solution of
632:is the shift operator, i.e.
519:This can also be written as
134:field of characteristic zero
7:
3225:signed permutation matrices
3219:Signed permutation matrices
3213:
2848:is called d'Alembertian if
679:{\textstyle N\,y(n)=y(n+1)}
95:, P-recursive or D-finite.
36:. The specific problem is:
10:
4507:
3475:{\textstyle y(0)=1,y(1)=2}
1754:
1530:
1466:system of linear equations
1361:
60:
16:Linear recurrence equation
4424:. ACM. pp. 169â174.
4391:10.1007/s00200-005-0192-x
4260:. ACM. pp. 290â296.
3595:{\textstyle y(n)=2^{n}n!}
3246:{\displaystyle n\times n}
2539:{\textstyle \mathbb {K} }
1187:{\textstyle {\tilde {y}}}
250:{\textstyle k=0,\dots ,r}
125:{\textstyle \mathbb {K} }
3253:can be described by the
1082:for arbitrary constants
3426:and the initial values
3010:{\textstyle \Delta y=x}
2824:D'Alembertian solutions
2480:{\textstyle \Gamma (n)}
1751:Hypergeometric solution
476:are both nonzero, then
42:WikiProject Mathematics
4126:
4102:
4067:
4007:
3987:
3967:
3930:
3793:
3746:
3656:
3636:
3596:
3548:
3528:
3476:
3420:
3319:
3296:
3247:
3200:
3151:
3105:
3031:
3011:
2982:
2953:
2907:
2842:
2814:
2785:
2765:
2745:
2667:
2640:
2613:
2586:
2540:
2514:
2481:
2452:
2398:
2372:
2324:
2175:
1995:
1966:
1939:
1908:
1877:
1808:
1784:
1741:
1721:
1694:
1667:
1647:
1627:
1581:
1514:
1454:
1408:
1348:
1255:
1217:
1188:
1159:
1130:
1076:
1004:, then the formal sum
998:
972:
885:
808:
779:
721:
680:
626:
606:
581:
542:
510:
490:
470:
443:
416:
358:
325:
288:
251:
213:
160:
126:
38:to review the article.
4430:10.1145/190347.190412
4276:10.1145/220346.220384
4127:
4103:
4068:
4008:
3988:
3968:
3931:
3794:
3754:Petkovšek's algorithm
3752:Applying for example
3747:
3657:
3637:
3597:
3549:
3529:
3477:
3421:
3320:
3297:
3248:
3201:
3152:
3106:
3032:
3012:
2983:
2981:{\textstyle y=\sum x}
2954:
2908:
2843:
2815:
2786:
2766:
2746:
2668:
2641:
2614:
2587:
2541:
2515:
2482:
2453:
2399:
2373:
2325:
2176:
1996:
1967:
1940:
1909:
1878:
1809:
1785:
1757:Petkovšek's algorithm
1742:
1722:
1695:
1668:
1648:
1628:
1582:
1515:
1455:
1409:
1349:
1256:
1218:
1189:
1160:
1131:
1077:
999:
973:
886:
809:
780:
701:
690:Closed form solutions
681:
627:
607:
561:
543:
511:
491:
471:
444:
417:
338:
326:
289:
252:
214:
161:
127:
61:Further information:
4116:
4077:
4017:
3997:
3977:
3940:
3803:
3768:
3666:
3646:
3617:
3558:
3538:
3486:
3430:
3329:
3306:
3260:
3231:
3161:
3115:
3041:
3030:{\textstyle \Delta }
3021:
2992:
2963:
2917:
2852:
2832:
2795:
2775:
2755:
2677:
2650:
2623:
2596:
2550:
2528:
2495:
2462:
2408:
2397:{\textstyle i\neq j}
2382:
2334:
2185:
2005:
1976:
1949:
1922:
1898:
1818:
1798:
1765:
1731:
1704:
1677:
1657:
1637:
1591:
1545:
1476:
1418:
1372:
1358:Polynomial solutions
1265:
1227:
1198:
1169:
1140:
1086:
1008:
982:
899:
823:
789:
698:
636:
616:
552:
523:
500:
480:
453:
426:
335:
298:
261:
223:
170:
140:
114:
85:recurrence equations
69:P-recursive equation
4108:is hypergeometric.
4101:{\textstyle F(n,k)}
3792:{\textstyle F(n,k)}
2673:. For fixed values
1533:Abramov's algorithm
997:{\textstyle \ker L}
4122:
4098:
4063:
4044:
4003:
3983:
3963:
3926:
3789:
3742:
3652:
3632:
3592:
3544:
3534:for some constant
3524:
3472:
3416:
3318:{\textstyle \pm 1}
3315:
3292:
3243:
3196:
3147:
3101:
3027:
3007:
2978:
2949:
2903:
2838:
2810:
2781:
2761:
2741:
2666:{\textstyle e_{i}}
2663:
2639:{\textstyle p_{r}}
2636:
2612:{\textstyle p_{0}}
2609:
2582:
2536:
2510:
2477:
2448:
2394:
2368:
2320:
2171:
1991:
1965:{\textstyle p_{r}}
1962:
1938:{\textstyle p_{0}}
1935:
1904:
1873:
1804:
1780:
1737:
1720:{\textstyle p_{r}}
1717:
1693:{\textstyle p_{0}}
1690:
1663:
1643:
1623:
1577:
1527:Rational solutions
1510:
1450:
1404:
1344:
1251:
1213:
1184:
1155:
1126:
1072:
994:
968:
881:
804:
775:
676:
622:
602:
538:
506:
486:
469:{\textstyle p_{r}}
466:
442:{\textstyle p_{0}}
439:
412:
321:
284:
247:
209:
156:
122:
63:holonomic function
4035:
3635:{\textstyle y(n)}
2841:{\displaystyle y}
2813:{\textstyle r(n)}
2771:. For a specific
2522:algebraic closure
2508:
2318:
2262:
2218:
1994:{\textstyle y(n)}
1907:{\displaystyle f}
1807:{\displaystyle n}
1783:{\textstyle y(n)}
1740:{\displaystyle y}
1341:
1242:
1216:{\textstyle Ly=f}
1181:
1158:{\textstyle Ly=0}
807:{\textstyle Ly=f}
541:{\textstyle Ly=f}
67:In mathematics a
59:
58:
4498:
4476:
4475:
4473:
4472:
4462:"A000165 - OEIS"
4458:
4452:
4451:
4417:
4411:
4410:
4374:
4368:
4367:
4350:(1â3): 109â131.
4339:
4333:
4332:
4304:
4298:
4297:
4269:
4253:
4247:
4246:
4218:
4205:
4204:
4187:(2â3): 243â264.
4176:
4167:
4166:
4154:
4148:
4145:
4131:
4129:
4128:
4123:
4107:
4105:
4104:
4099:
4072:
4070:
4069:
4064:
4043:
4012:
4010:
4009:
4004:
3992:
3990:
3989:
3984:
3972:
3970:
3969:
3964:
3947:
3935:
3933:
3932:
3927:
3910:
3884:
3834:
3798:
3796:
3795:
3790:
3751:
3749:
3748:
3743:
3661:
3659:
3658:
3653:
3641:
3639:
3638:
3633:
3601:
3599:
3598:
3593:
3585:
3584:
3553:
3551:
3550:
3545:
3533:
3531:
3530:
3525:
3517:
3516:
3481:
3479:
3478:
3473:
3425:
3423:
3422:
3417:
3371:
3370:
3324:
3322:
3321:
3316:
3301:
3299:
3298:
3293:
3291:
3290:
3289:
3283:
3252:
3250:
3249:
3244:
3205:
3203:
3202:
3197:
3186:
3185:
3173:
3172:
3156:
3154:
3153:
3148:
3146:
3145:
3127:
3126:
3110:
3108:
3107:
3102:
3036:
3034:
3033:
3028:
3016:
3014:
3013:
3008:
2987:
2985:
2984:
2979:
2958:
2956:
2955:
2950:
2948:
2947:
2929:
2928:
2912:
2910:
2909:
2904:
2902:
2901:
2883:
2882:
2870:
2869:
2847:
2845:
2844:
2839:
2819:
2817:
2816:
2811:
2790:
2788:
2787:
2782:
2770:
2768:
2767:
2762:
2750:
2748:
2747:
2742:
2740:
2739:
2721:
2720:
2708:
2707:
2689:
2688:
2672:
2670:
2669:
2664:
2662:
2661:
2645:
2643:
2642:
2637:
2635:
2634:
2618:
2616:
2615:
2610:
2608:
2607:
2591:
2589:
2588:
2583:
2581:
2580:
2562:
2561:
2545:
2543:
2542:
2537:
2535:
2519:
2517:
2516:
2511:
2509:
2504:
2499:
2486:
2484:
2483:
2478:
2457:
2455:
2454:
2449:
2447:
2439:
2438:
2420:
2419:
2403:
2401:
2400:
2395:
2377:
2375:
2374:
2369:
2367:
2359:
2358:
2346:
2345:
2329:
2327:
2326:
2321:
2319:
2314:
2309:
2304:
2303:
2285:
2284:
2263:
2258:
2253:
2233:
2219:
2214:
2209:
2198:
2180:
2178:
2177:
2172:
2170:
2169:
2168:
2167:
2153:
2152:
2128:
2127:
2126:
2125:
2111:
2110:
2089:
2088:
2087:
2086:
2072:
2071:
2049:
2048:
2000:
1998:
1997:
1992:
1971:
1969:
1968:
1963:
1961:
1960:
1944:
1942:
1941:
1936:
1934:
1933:
1913:
1911:
1910:
1905:
1882:
1880:
1879:
1874:
1863:
1843:
1813:
1811:
1810:
1805:
1789:
1787:
1786:
1781:
1746:
1744:
1743:
1738:
1726:
1724:
1723:
1718:
1716:
1715:
1699:
1697:
1696:
1691:
1689:
1688:
1672:
1670:
1669:
1664:
1652:
1650:
1649:
1644:
1632:
1630:
1629:
1624:
1613:
1586:
1584:
1583:
1578:
1567:
1519:
1517:
1516:
1511:
1509:
1508:
1507:
1491:
1490:
1459:
1457:
1456:
1451:
1440:
1413:
1411:
1410:
1405:
1394:
1353:
1351:
1350:
1345:
1343:
1342:
1334:
1328:
1327:
1312:
1311:
1293:
1292:
1277:
1276:
1260:
1258:
1257:
1252:
1244:
1243:
1235:
1222:
1220:
1219:
1214:
1193:
1191:
1190:
1185:
1183:
1182:
1174:
1164:
1162:
1161:
1156:
1135:
1133:
1132:
1127:
1125:
1117:
1116:
1098:
1097:
1081:
1079:
1078:
1073:
1071:
1070:
1055:
1054:
1036:
1035:
1020:
1019:
1003:
1001:
1000:
995:
977:
975:
974:
969:
964:
963:
939:
938:
920:
919:
890:
888:
887:
882:
860:
859:
858:
852:
813:
811:
810:
805:
785:or equivalently
784:
782:
781:
776:
731:
730:
720:
715:
685:
683:
682:
677:
631:
629:
628:
623:
611:
609:
608:
603:
601:
600:
591:
590:
580:
575:
547:
545:
544:
539:
515:
513:
512:
507:
495:
493:
492:
487:
475:
473:
472:
467:
465:
464:
448:
446:
445:
440:
438:
437:
421:
419:
418:
413:
368:
367:
357:
352:
330:
328:
327:
322:
320:
319:
318:
312:
293:
291:
290:
285:
283:
282:
281:
275:
256:
254:
253:
248:
219:polynomials for
218:
216:
215:
210:
199:
182:
181:
165:
163:
162:
157:
155:
147:
131:
129:
128:
123:
121:
54:
51:
45:
27:
26:
19:
4506:
4505:
4501:
4500:
4499:
4497:
4496:
4495:
4481:
4480:
4479:
4470:
4468:
4460:
4459:
4455:
4440:
4418:
4414:
4375:
4371:
4340:
4336:
4305:
4301:
4286:
4254:
4250:
4235:
4219:
4208:
4177:
4170:
4155:
4151:
4146:
4142:
4138:
4117:
4114:
4113:
4078:
4075:
4074:
4039:
4018:
4015:
4014:
3998:
3995:
3994:
3978:
3975:
3974:
3943:
3941:
3938:
3937:
3906:
3880:
3830:
3804:
3801:
3800:
3769:
3766:
3765:
3762:
3667:
3664:
3663:
3647:
3644:
3643:
3618:
3615:
3614:
3608:
3580:
3576:
3559:
3556:
3555:
3539:
3536:
3535:
3512:
3508:
3487:
3484:
3483:
3431:
3428:
3427:
3366:
3362:
3330:
3327:
3326:
3307:
3304:
3303:
3285:
3284:
3279:
3278:
3261:
3258:
3257:
3232:
3229:
3228:
3221:
3216:
3181:
3177:
3168:
3164:
3162:
3159:
3158:
3141:
3137:
3122:
3118:
3116:
3113:
3112:
3042:
3039:
3038:
3022:
3019:
3018:
2993:
2990:
2989:
2964:
2961:
2960:
2943:
2939:
2924:
2920:
2918:
2915:
2914:
2897:
2893:
2878:
2874:
2865:
2861:
2853:
2850:
2849:
2833:
2830:
2829:
2826:
2796:
2793:
2792:
2776:
2773:
2772:
2756:
2753:
2752:
2735:
2731:
2716:
2712:
2703:
2699:
2684:
2680:
2678:
2675:
2674:
2657:
2653:
2651:
2648:
2647:
2630:
2626:
2624:
2621:
2620:
2603:
2599:
2597:
2594:
2593:
2576:
2572:
2557:
2553:
2551:
2548:
2547:
2531:
2529:
2526:
2525:
2500:
2498:
2496:
2493:
2492:
2463:
2460:
2459:
2443:
2434:
2430:
2415:
2411:
2409:
2406:
2405:
2383:
2380:
2379:
2363:
2354:
2350:
2341:
2337:
2335:
2332:
2331:
2310:
2308:
2299:
2295:
2280:
2276:
2254:
2252:
2229:
2210:
2208:
2194:
2186:
2183:
2182:
2163:
2159:
2158:
2154:
2148:
2144:
2121:
2117:
2116:
2112:
2106:
2102:
2082:
2078:
2077:
2073:
2067:
2063:
2044:
2040:
2006:
2003:
2002:
1977:
1974:
1973:
1956:
1952:
1950:
1947:
1946:
1929:
1925:
1923:
1920:
1919:
1899:
1896:
1895:
1888:Marko Petkovšek
1859:
1839:
1819:
1816:
1815:
1799:
1796:
1795:
1766:
1763:
1762:
1759:
1753:
1732:
1729:
1728:
1711:
1707:
1705:
1702:
1701:
1684:
1680:
1678:
1675:
1674:
1658:
1655:
1654:
1638:
1635:
1634:
1609:
1592:
1589:
1588:
1563:
1546:
1543:
1542:
1541:solution, i.e.
1535:
1529:
1503:
1496:
1492:
1486:
1482:
1477:
1474:
1473:
1462:Marko Petkovšek
1436:
1419:
1416:
1415:
1390:
1373:
1370:
1369:
1366:
1360:
1333:
1332:
1317:
1313:
1307:
1303:
1282:
1278:
1272:
1268:
1266:
1263:
1262:
1234:
1233:
1228:
1225:
1224:
1199:
1196:
1195:
1173:
1172:
1170:
1167:
1166:
1141:
1138:
1137:
1121:
1112:
1108:
1093:
1089:
1087:
1084:
1083:
1060:
1056:
1050:
1046:
1025:
1021:
1015:
1011:
1009:
1006:
1005:
983:
980:
979:
953:
949:
928:
924:
909:
905:
900:
897:
896:
854:
853:
848:
847:
824:
821:
820:
790:
787:
786:
726:
722:
716:
705:
699:
696:
695:
692:
637:
634:
633:
617:
614:
613:
596:
592:
586:
582:
576:
565:
553:
550:
549:
524:
521:
520:
501:
498:
497:
481:
478:
477:
460:
456:
454:
451:
450:
433:
429:
427:
424:
423:
363:
359:
353:
342:
336:
333:
332:
314:
313:
308:
307:
299:
296:
295:
294:a sequence and
277:
276:
271:
270:
262:
259:
258:
224:
221:
220:
195:
177:
173:
171:
168:
167:
151:
143:
141:
138:
137:
117:
115:
112:
111:
108:
100:Marko Petkovšek
73:linear equation
65:
55:
49:
46:
40:
28:
24:
17:
12:
11:
5:
4504:
4494:
4493:
4478:
4477:
4453:
4439:978-0897916387
4438:
4412:
4369:
4334:
4299:
4285:978-0897916998
4284:
4267:10.1.1.46.9373
4248:
4234:978-1568810638
4233:
4227:. A K Peters.
4206:
4168:
4149:
4139:
4137:
4134:
4125:{\textstyle f}
4121:
4097:
4094:
4091:
4088:
4085:
4082:
4062:
4059:
4056:
4053:
4050:
4047:
4042:
4038:
4034:
4031:
4028:
4025:
4022:
4006:{\textstyle k}
4002:
3986:{\textstyle n}
3982:
3962:
3959:
3956:
3953:
3950:
3946:
3925:
3922:
3919:
3916:
3913:
3909:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3883:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3858:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3833:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3788:
3785:
3782:
3779:
3776:
3773:
3761:
3758:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3655:{\textstyle n}
3651:
3642:of a set with
3631:
3628:
3625:
3622:
3610:The number of
3607:
3604:
3591:
3588:
3583:
3579:
3575:
3572:
3569:
3566:
3563:
3547:{\textstyle c}
3543:
3523:
3520:
3515:
3511:
3506:
3503:
3500:
3497:
3494:
3491:
3471:
3468:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3415:
3412:
3409:
3406:
3403:
3400:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3369:
3365:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3314:
3311:
3288:
3282:
3277:
3274:
3271:
3268:
3265:
3242:
3239:
3236:
3223:The number of
3220:
3217:
3215:
3212:
3195:
3192:
3189:
3184:
3180:
3176:
3171:
3167:
3144:
3140:
3136:
3133:
3130:
3125:
3121:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3026:
3006:
3003:
3000:
2997:
2977:
2974:
2971:
2968:
2946:
2942:
2938:
2935:
2932:
2927:
2923:
2900:
2896:
2892:
2889:
2886:
2881:
2877:
2873:
2868:
2864:
2860:
2857:
2837:
2825:
2822:
2809:
2806:
2803:
2800:
2784:{\textstyle z}
2780:
2764:{\textstyle z}
2760:
2738:
2734:
2730:
2727:
2724:
2719:
2715:
2711:
2706:
2702:
2698:
2695:
2692:
2687:
2683:
2660:
2656:
2633:
2629:
2606:
2602:
2579:
2575:
2571:
2568:
2565:
2560:
2556:
2534:
2507:
2503:
2489:Gamma function
2476:
2473:
2470:
2467:
2446:
2442:
2437:
2433:
2429:
2426:
2423:
2418:
2414:
2393:
2390:
2387:
2366:
2362:
2357:
2353:
2349:
2344:
2340:
2317:
2313:
2307:
2302:
2298:
2294:
2291:
2288:
2283:
2279:
2275:
2272:
2269:
2266:
2261:
2257:
2251:
2248:
2245:
2242:
2239:
2236:
2232:
2228:
2225:
2222:
2217:
2213:
2207:
2204:
2201:
2197:
2193:
2190:
2166:
2162:
2157:
2151:
2147:
2143:
2140:
2137:
2134:
2131:
2124:
2120:
2115:
2109:
2105:
2101:
2098:
2095:
2092:
2085:
2081:
2076:
2070:
2066:
2062:
2059:
2056:
2053:
2047:
2043:
2038:
2035:
2032:
2029:
2025:
2022:
2019:
2016:
2013:
2010:
1990:
1987:
1984:
1981:
1959:
1955:
1932:
1928:
1903:
1872:
1869:
1866:
1862:
1858:
1855:
1852:
1849:
1846:
1842:
1838:
1835:
1832:
1829:
1826:
1823:
1803:
1792:hypergeometric
1779:
1776:
1773:
1770:
1755:Main article:
1752:
1749:
1736:
1714:
1710:
1687:
1683:
1666:{\textstyle u}
1662:
1646:{\textstyle u}
1642:
1622:
1619:
1616:
1612:
1608:
1605:
1602:
1599:
1596:
1576:
1573:
1570:
1566:
1562:
1559:
1556:
1553:
1550:
1531:Main article:
1528:
1525:
1506:
1502:
1499:
1495:
1489:
1485:
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136:(for example
135:
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89:combinatorics
86:
82:
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70:
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53:
43:
39:
35:
32:This article
30:
21:
20:
4469:. Retrieved
4465:
4456:
4421:
4415:
4382:
4378:
4372:
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4308:
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4180:
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4158:
4152:
4143:
3763:
3760:Applications
3609:
3222:
3208:
2827:
2487:denotes the
1917:
1885:
1760:
1536:
1522:
1470:power series
1367:
693:
518:
109:
97:
68:
66:
50:October 2019
47:
37:
33:
4491:Polynomials
4315:(6): 7â12.
3764:A function
3612:involutions
3606:Involutions
2988:means that
2828:A sequence
2546:. Then the
1761:A sequence
81:polynomials
4471:2018-07-02
4136:References
4110:Zeilberger
1790:is called
106:Definition
4399:0938-1279
4364:0022-4049
4329:0041-5553
4262:CiteSeerX
4201:0747-7171
4037:∑
3904:∈
3731:−
3710:−
3691:−
3408:−
3383:−
3357:−
3310:±
3276:∈
3238:×
3175:⋯
3132:…
3087:−
3060:−
3045:Δ
3025:Δ
2996:Δ
2973:∑
2934:…
2891:∑
2888:⋯
2885:∑
2872:∑
2726:…
2701:ξ
2694:…
2682:ξ
2574:ξ
2567:…
2555:ξ
2506:¯
2466:Γ
2441:∈
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2389:≠
2361:∉
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2306:∈
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2260:¯
2250:∈
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2206:∈
2192:∈
2181:for some
2146:ξ
2142:−
2133:Γ
2130:⋯
2104:ξ
2100:−
2091:Γ
2065:ξ
2061:−
2052:Γ
1892:algorithm
1857:∈
1607:∈
1561:∈
1501:∈
1434:∈
1388:∈
1339:~
1298:⋯
1240:~
1179:~
1119:∈
1103:…
1041:⋯
989:
944:…
845:∈
830:
703:∑
563:∑
340:∑
305:∈
268:∈
239:…
193:∈
93:holonomic
77:sequences
4485:Category
4466:oeis.org
4294:14963237
4243:33898705
3255:sequence
3227:of size
3214:Examples
1890:gave an
1886:In 1992
1539:rational
4448:2802734
4407:7496623
2458:. Here
1814:, i.e.
1261:, then
1223:, i.e.
4446:
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4327:
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4264:
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4199:
4073:where
3936:where
3017:where
895:. Let
817:kernel
548:where
4444:S2CID
4403:S2CID
4290:S2CID
2330:with
1165:. If
893:basis
132:be a
71:is a
4434:ISBN
4395:ISSN
4360:ISSN
4325:ISSN
4280:ISBN
4239:OCLC
4229:ISBN
4197:ISSN
3993:and
2959:and
2520:the
2491:and
2404:and
2378:for
1945:and
1700:and
694:Let
449:and
110:Let
4426:doi
4387:doi
4352:doi
4348:139
4317:doi
4272:doi
4224:A=B
4189:doi
2619:or
1520:).
986:ker
827:ker
166:),
75:of
4487::
4464:.
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4209:^
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2015:n
2012:(
2009:y
1989:)
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1150:=
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1106:,
1100:,
1095:1
1091:c
1068:)
1065:m
1062:(
1058:y
1052:m
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1044:+
1038:+
1033:)
1030:1
1027:(
1023:y
1017:1
1013:c
992:L
966:}
961:)
958:m
955:(
951:y
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933:2
930:(
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914:1
911:(
907:y
903:{
879:}
876:0
873:=
870:y
867:L
863::
856:N
850:K
842:y
839:{
836:=
833:L
802:f
799:=
796:y
793:L
773:)
770:n
767:(
764:f
761:=
758:)
755:k
752:+
749:n
746:(
743:y
739:)
736:n
733:(
728:k
724:p
718:r
713:0
710:=
707:k
674:)
671:1
668:+
665:n
662:(
659:y
656:=
653:)
650:n
647:(
644:y
640:N
620:N
598:k
594:N
588:k
584:p
578:r
573:0
570:=
567:k
559:=
556:L
536:f
533:=
530:y
527:L
504:f
484:r
462:r
458:p
435:0
431:p
410:)
407:n
404:(
401:f
398:=
395:)
392:k
389:+
386:n
383:(
380:y
376:)
373:n
370:(
365:k
361:p
355:r
350:0
347:=
344:k
316:N
310:K
302:y
279:N
273:K
265:f
257:,
245:r
242:,
236:,
233:0
230:=
227:k
207:]
204:n
201:[
197:K
190:)
187:n
184:(
179:k
175:p
153:Q
149:=
145:K
119:K
52:)
48:(
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