22:
1432:
on 4 points, so there are 192/24 = 8 = 2 × 4 essentially different solutions given by acting on the local Heun function by these symmetries, which give solutions for each of the 2 exponents for each of the 4 singular points. The complete list of 192 symmetries was given by
491:
and ∞ with exponents (0, 1 − γ), (0, 1 − δ), (0, 1 − ϵ), and (α, β). Every second-order linear ODE on the extended complex plane with at most four regular singular points, such as the
1372:
417:
736:
916:
1085:
1233:
1437:
using machine calculation. Several previous attempts by various authors to list these by hand contained many errors and omissions; for example, most of the 48 local solutions listed by Heun contain serious errors.
151:
470:
503:
Coalescence of various regular singularities of the Heun equation into irregular singularities give rise to several confluent forms of the equation, as shown in the table below.
1247:
218:
537:
750:
930:
1099:
1631:
1753:
1685:
1640:
1604:
1425:
497:
65:
43:
88:
1447:
425:
209:
1758:
1541:
472:
is taken so that the characteristic exponents for the regular singularity at infinity are α and β (see below).
36:
30:
1428:
obtained by Kummer. The symmetries fixing the local Heun function form a group of order 24 isomorphic to the
1709:
Hahn W.(1971) On linear geometric difference equations with accessory parameters.Funkcial. Ekvac., 14, 73–78
1367:{\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left(\gamma +z\right)z{\frac {dw}{dz}}+\left(\alpha z-q\right)w=0}
484:
412:{\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left{\frac {dw}{dz}}+{\frac {\alpha \beta z-q}{z(z-1)(z-a)}}w=0.}
47:
731:{\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left{\frac {dw}{dz}}+{\frac {\alpha \beta z-q}{z(z-1)(z-a)}}w=0}
1560:
1713:
Takemura, K. (2017), "Degenerations of
Ruijsenaars–van Diejen operator and q-Painlevé equations",
1722:
1699:
1663:
1584:
1550:
1528:
1492:
1681:
1636:
1626:
1600:
911:{\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left{\frac {dw}{dz}}+{\frac {\alpha z-q}{z(z-1)}}w=0}
1080:{\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left{\frac {dw}{dz}}+{\frac {\alpha z-q}{z^{2}}}w=0}
1732:
1673:
1568:
1520:
493:
1695:
1650:
1614:
1580:
1500:
1691:
1646:
1610:
1576:
1496:
1429:
1415:
206:
1228:{\displaystyle {\frac {d^{2}w}{dz^{2}}}-\left{\frac {dw}{dz}}+{\frac {\alpha z-q}{z}}w=0}
1564:
1509:"Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit vier Verzweigungspunkten"
1747:
1532:
1486:
1411:
1391:
1703:
1588:
1572:
79:
1677:
1736:
1599:, Oxford Science Publications, The Clarendon Press Oxford University Press,
154:
1410:
Heun's equation has a group of symmetries of order 192, isomorphic to the
1478:
1387:
1668:
1524:
1658:
Valent, Galliano (2007), "Heun functions versus elliptic functions",
1555:
1629:; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
1508:
1727:
1539:
Maier, Robert S. (2007), "The 192 solutions of the Heun equation",
1660:
Difference equations, special functions and orthogonal polynomials
500:, can be transformed into this equation by a change of variable.
1488:
Theory of differential equations. 4. Ordinary linear equations
1467:
15:
189:, if it is regular at all three finite singular points
146:{\displaystyle H\ell (a,q;\alpha ,\beta ,\gamma ,\delta ;z)}
1622:
465:{\displaystyle \epsilon =\alpha +\beta -\gamma -\delta +1}
1477:
A. Erdélyi, F. Oberhettinger, W. Magnus and F. Tricomi
1662:, World Sci. Publ., Hackensack, NJ, pp. 664–686,
1250:
1102:
933:
753:
540:
428:
221:
91:
169: = 0. The local Heun function is called a
1366:
1227:
1079:
910:
730:
464:
411:
145:
165:that is holomorphic and 1 at the singular point
1468:DLMF §31.12 Confluent Forms of Heun’s Equation
925:0 (irregular, rank 1), ∞ (irregular, rank 1)
8:
1390:of Heun's equation has been discovered by
1726:
1667:
1621:Sleeman, B. D.; Kuznetzov, V. B. (2010),
1554:
1424:, analogous to the 24 symmetries of the
1310:
1276:
1258:
1251:
1249:
1195:
1172:
1145:
1128:
1110:
1103:
1101:
1060:
1040:
1017:
996:
985:
976:
959:
941:
934:
932:
861:
838:
809:
796:
779:
761:
754:
752:
663:
640:
617:
596:
583:
566:
548:
541:
539:
427:
344:
321:
298:
277:
264:
247:
229:
222:
220:
90:
66:Learn how and when to remove this message
1399:
507:
29:This article includes a list of general
1632:NIST Handbook of Mathematical Functions
1460:
1450:, a generalization of Heun polynomials.
1479:Higher Transcendental functions vol. 3
1434:
1426:hypergeometric differential equations
7:
1395:
498:hypergeometric differential equation
158:
205:Heun's equation is a second-order
35:it lacks sufficient corresponding
14:
1485:Forsyth, Andrew Russell (1959) ,
181: = 1, and is called a
20:
1754:Ordinary differential equations
1635:, Cambridge University Press,
893:
881:
713:
701:
698:
686:
394:
382:
379:
367:
210:ordinary differential equation
140:
98:
1:
1715:Journal of Integrable Systems
1597:Heun's differential equations
1573:10.1090/S0025-5718-06-01939-9
193: = 0, 1,
745:0, 1, ∞ (irregular, rank 1)
163:Heun's differential equation
1448:Heine–Stieltjes polynomials
509:Forms of the Heun Equation
483:. Heun's equation has four
177:, if it is also regular at
1775:
1678:10.1142/9789812770752_0057
1595:Ronveaux, A., ed. (1995),
1542:Mathematics of Computation
1094:0, ∞ (irregular, rank 2)
1481:(McGraw Hill, NY, 1953).
200:
485:regular singular points
50:more precise citations.
1368:
1242:∞ (irregular, rank 3)
1229:
1081:
912:
732:
466:
413:
147:
1737:10.1093/integr/xyx008
1513:Mathematische Annalen
1369:
1230:
1082:
913:
733:
467:
414:
161:) is the solution of
148:
1248:
1100:
931:
751:
538:
426:
219:
89:
1565:2007MaCom..76..811M
1507:Heun, Karl (1889),
510:
481:accessory parameter
475:The complex number
155:Karl L. W. Heun
84:local Heun function
1627:Olver, Frank W. J.
1525:10.1007/bf01443849
1493:Dover Publications
1364:
1225:
1077:
908:
728:
508:
487:: 0, 1,
462:
409:
212:(ODE) of the form
143:
1759:Special functions
1687:978-981-270-643-0
1642:978-0-521-19225-5
1606:978-0-19-859695-0
1398:) and studied by
1377:
1376:
1328:
1283:
1214:
1190:
1153:
1135:
1066:
1035:
1004:
991:
966:
922:Doubly Confluent
897:
856:
825:
804:
786:
717:
658:
633:
612:
591:
573:
398:
339:
314:
293:
272:
254:
76:
75:
68:
1766:
1739:
1730:
1706:
1671:
1653:
1623:"Heun functions"
1617:
1591:
1558:
1549:(258): 811–843,
1535:
1503:
1470:
1465:
1373:
1371:
1370:
1365:
1354:
1350:
1329:
1327:
1319:
1311:
1306:
1302:
1284:
1282:
1281:
1280:
1267:
1263:
1262:
1252:
1234:
1232:
1231:
1226:
1215:
1210:
1196:
1191:
1189:
1181:
1173:
1171:
1167:
1154:
1146:
1136:
1134:
1133:
1132:
1119:
1115:
1114:
1104:
1086:
1084:
1083:
1078:
1067:
1065:
1064:
1055:
1041:
1036:
1034:
1026:
1018:
1016:
1012:
1005:
997:
992:
990:
989:
977:
967:
965:
964:
963:
950:
946:
945:
935:
917:
915:
914:
909:
898:
896:
876:
862:
857:
855:
847:
839:
837:
833:
826:
824:
810:
805:
797:
787:
785:
784:
783:
770:
766:
765:
755:
737:
735:
734:
729:
718:
716:
681:
664:
659:
657:
649:
641:
639:
635:
634:
632:
618:
613:
611:
597:
592:
584:
574:
572:
571:
570:
557:
553:
552:
542:
511:
471:
469:
468:
463:
418:
416:
415:
410:
399:
397:
362:
345:
340:
338:
330:
322:
320:
316:
315:
313:
299:
294:
292:
278:
273:
265:
255:
253:
252:
251:
238:
234:
233:
223:
152:
150:
149:
144:
71:
64:
60:
57:
51:
46:this article by
37:inline citations
24:
23:
16:
1774:
1773:
1769:
1768:
1767:
1765:
1764:
1763:
1744:
1743:
1712:
1688:
1669:math-ph/0512006
1657:
1643:
1620:
1607:
1594:
1538:
1506:
1495:, p. 158,
1484:
1474:
1473:
1466:
1462:
1457:
1444:
1430:symmetric group
1423:
1416:Coxeter diagram
1408:
1400:Takemura (2017)
1384:
1337:
1333:
1320:
1312:
1292:
1288:
1272:
1268:
1254:
1253:
1246:
1245:
1197:
1182:
1174:
1144:
1140:
1124:
1120:
1106:
1105:
1098:
1097:
1056:
1042:
1027:
1019:
981:
975:
971:
955:
951:
937:
936:
929:
928:
877:
863:
848:
840:
814:
795:
791:
775:
771:
757:
756:
749:
748:
682:
665:
650:
642:
622:
601:
582:
578:
562:
558:
544:
543:
536:
535:
424:
423:
363:
346:
331:
323:
303:
282:
263:
259:
243:
239:
225:
224:
217:
216:
203:
201:Heun's equation
183:Heun polynomial
87:
86:
72:
61:
55:
52:
42:Please help to
41:
25:
21:
12:
11:
5:
1772:
1770:
1762:
1761:
1756:
1746:
1745:
1742:
1741:
1710:
1707:
1686:
1655:
1641:
1618:
1605:
1592:
1536:
1504:
1482:
1472:
1471:
1459:
1458:
1456:
1453:
1452:
1451:
1443:
1440:
1421:
1407:
1404:
1383:
1380:
1379:
1378:
1375:
1374:
1363:
1360:
1357:
1353:
1349:
1346:
1343:
1340:
1336:
1332:
1326:
1323:
1318:
1315:
1309:
1305:
1301:
1298:
1295:
1291:
1287:
1279:
1275:
1271:
1266:
1261:
1257:
1243:
1240:
1236:
1235:
1224:
1221:
1218:
1213:
1209:
1206:
1203:
1200:
1194:
1188:
1185:
1180:
1177:
1170:
1166:
1163:
1160:
1157:
1152:
1149:
1143:
1139:
1131:
1127:
1123:
1118:
1113:
1109:
1095:
1092:
1088:
1087:
1076:
1073:
1070:
1063:
1059:
1054:
1051:
1048:
1045:
1039:
1033:
1030:
1025:
1022:
1015:
1011:
1008:
1003:
1000:
995:
988:
984:
980:
974:
970:
962:
958:
954:
949:
944:
940:
926:
923:
919:
918:
907:
904:
901:
895:
892:
889:
886:
883:
880:
875:
872:
869:
866:
860:
854:
851:
846:
843:
836:
832:
829:
823:
820:
817:
813:
808:
803:
800:
794:
790:
782:
778:
774:
769:
764:
760:
746:
743:
739:
738:
727:
724:
721:
715:
712:
709:
706:
703:
700:
697:
694:
691:
688:
685:
680:
677:
674:
671:
668:
662:
656:
653:
648:
645:
638:
631:
628:
625:
621:
616:
610:
607:
604:
600:
595:
590:
587:
581:
577:
569:
565:
561:
556:
551:
547:
533:
526:
522:
521:
518:
515:
479:is called the
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
422:The condition
420:
419:
408:
405:
402:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
361:
358:
355:
352:
349:
343:
337:
334:
329:
326:
319:
312:
309:
306:
302:
297:
291:
288:
285:
281:
276:
271:
268:
262:
258:
250:
246:
242:
237:
232:
228:
202:
199:
142:
139:
136:
133:
130:
127:
124:
121:
118:
115:
112:
109:
106:
103:
100:
97:
94:
74:
73:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
1771:
1760:
1757:
1755:
1752:
1751:
1749:
1738:
1734:
1729:
1724:
1720:
1716:
1711:
1708:
1705:
1701:
1697:
1693:
1689:
1683:
1679:
1675:
1670:
1665:
1661:
1656:
1652:
1648:
1644:
1638:
1634:
1633:
1628:
1624:
1619:
1616:
1612:
1608:
1602:
1598:
1593:
1590:
1586:
1582:
1578:
1574:
1570:
1566:
1562:
1557:
1552:
1548:
1544:
1543:
1537:
1534:
1530:
1526:
1522:
1518:
1514:
1510:
1505:
1502:
1498:
1494:
1490:
1489:
1483:
1480:
1476:
1475:
1469:
1464:
1461:
1454:
1449:
1446:
1445:
1441:
1439:
1436:
1431:
1427:
1420:
1417:
1413:
1412:Coxeter group
1405:
1403:
1401:
1397:
1393:
1389:
1381:
1361:
1358:
1355:
1351:
1347:
1344:
1341:
1338:
1334:
1330:
1324:
1321:
1316:
1313:
1307:
1303:
1299:
1296:
1293:
1289:
1285:
1277:
1273:
1269:
1264:
1259:
1255:
1244:
1241:
1239:Triconfluent
1238:
1237:
1222:
1219:
1216:
1211:
1207:
1204:
1201:
1198:
1192:
1186:
1183:
1178:
1175:
1168:
1164:
1161:
1158:
1155:
1150:
1147:
1141:
1137:
1129:
1125:
1121:
1116:
1111:
1107:
1096:
1093:
1090:
1089:
1074:
1071:
1068:
1061:
1057:
1052:
1049:
1046:
1043:
1037:
1031:
1028:
1023:
1020:
1013:
1009:
1006:
1001:
998:
993:
986:
982:
978:
972:
968:
960:
956:
952:
947:
942:
938:
927:
924:
921:
920:
905:
902:
899:
890:
887:
884:
878:
873:
870:
867:
864:
858:
852:
849:
844:
841:
834:
830:
827:
821:
818:
815:
811:
806:
801:
798:
792:
788:
780:
776:
772:
767:
762:
758:
747:
744:
741:
740:
725:
722:
719:
710:
707:
704:
695:
692:
689:
683:
678:
675:
672:
669:
666:
660:
654:
651:
646:
643:
636:
629:
626:
623:
619:
614:
608:
605:
602:
598:
593:
588:
585:
579:
575:
567:
563:
559:
554:
549:
545:
534:
531:
527:
524:
523:
519:
517:Singularities
516:
513:
512:
506:
505:
504:
501:
499:
495:
494:Lamé equation
490:
486:
482:
478:
473:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
406:
403:
400:
391:
388:
385:
376:
373:
370:
364:
359:
356:
353:
350:
347:
341:
335:
332:
327:
324:
317:
310:
307:
304:
300:
295:
289:
286:
283:
279:
274:
269:
266:
260:
256:
248:
244:
240:
235:
230:
226:
215:
214:
213:
211:
208:
198:
196:
192:
188:
184:
180:
176:
172:
171:Heun function
168:
164:
160:
156:
137:
134:
131:
128:
125:
122:
119:
116:
113:
110:
107:
104:
101:
95:
92:
85:
81:
70:
67:
59:
49:
45:
39:
38:
32:
27:
18:
17:
1718:
1714:
1659:
1630:
1596:
1556:math/0408317
1546:
1540:
1516:
1512:
1491:, New York:
1487:
1463:
1435:Maier (2007)
1418:
1409:
1385:
1091:Biconfluent
529:
502:
488:
480:
476:
474:
421:
204:
194:
190:
186:
182:
178:
174:
170:
166:
162:
83:
77:
62:
53:
34:
80:mathematics
48:introducing
1748:Categories
1728:1608.07265
1519:(2): 161,
1455:References
1406:Symmetries
742:Confluent
185:, denoted
173:, denoted
31:references
1533:120008459
1345:−
1339:α
1294:γ
1205:−
1199:α
1159:δ
1148:γ
1138:−
1050:−
1044:α
999:γ
979:δ
888:−
871:−
865:α
831:ϵ
819:−
812:δ
799:γ
708:−
693:−
676:−
670:β
667:α
627:−
620:ϵ
606:−
599:δ
586:γ
520:Equation
454:δ
451:−
448:γ
445:−
442:β
436:α
430:ϵ
389:−
374:−
357:−
351:β
348:α
308:−
301:ϵ
287:−
280:δ
267:γ
132:δ
126:γ
120:β
114:α
96:ℓ
56:June 2017
1442:See also
1388:q-analog
1382:q-analog
525:General
1704:8520520
1696:2451210
1651:2723248
1615:1392976
1581:2291838
1561:Bibcode
1501:0123757
1414:of the
1394: (
496:or the
44:improve
1702:
1694:
1684:
1649:
1639:
1613:
1603:
1589:749861
1587:
1579:
1531:
1499:
528:0, 1,
207:linear
157:
82:, the
33:, but
1723:arXiv
1721:(1),
1700:S2CID
1664:arXiv
1625:, in
1585:S2CID
1551:arXiv
1529:S2CID
1682:ISBN
1637:ISBN
1601:ISBN
1396:1971
1392:Hahn
1386:The
532:, ∞
514:Form
159:1889
1733:doi
1674:doi
1569:doi
1521:doi
78:In
1750::
1731:,
1717:,
1698:,
1692:MR
1690:,
1680:,
1672:,
1647:MR
1645:,
1611:MR
1609:,
1583:,
1577:MR
1575:,
1567:,
1559:,
1547:76
1545:,
1527:,
1517:33
1515:,
1511:,
1497:MR
1402:.
407:0.
197:.
187:Hp
175:Hf
1740:.
1735::
1725::
1719:2
1676::
1666::
1654:.
1571::
1563::
1553::
1523::
1422:4
1419:D
1362:0
1359:=
1356:w
1352:)
1348:q
1342:z
1335:(
1331:+
1325:z
1322:d
1317:w
1314:d
1308:z
1304:)
1300:z
1297:+
1290:(
1286:+
1278:2
1274:z
1270:d
1265:w
1260:2
1256:d
1223:0
1220:=
1217:w
1212:z
1208:q
1202:z
1193:+
1187:z
1184:d
1179:w
1176:d
1169:]
1165:z
1162:+
1156:+
1151:z
1142:[
1130:2
1126:z
1122:d
1117:w
1112:2
1108:d
1075:0
1072:=
1069:w
1062:2
1058:z
1053:q
1047:z
1038:+
1032:z
1029:d
1024:w
1021:d
1014:]
1010:1
1007:+
1002:z
994:+
987:2
983:z
973:[
969:+
961:2
957:z
953:d
948:w
943:2
939:d
906:0
903:=
900:w
894:)
891:1
885:z
882:(
879:z
874:q
868:z
859:+
853:z
850:d
845:w
842:d
835:]
828:+
822:1
816:z
807:+
802:z
793:[
789:+
781:2
777:z
773:d
768:w
763:2
759:d
726:0
723:=
720:w
714:)
711:a
705:z
702:(
699:)
696:1
690:z
687:(
684:z
679:q
673:z
661:+
655:z
652:d
647:w
644:d
637:]
630:a
624:z
615:+
609:1
603:z
594:+
589:z
580:[
576:+
568:2
564:z
560:d
555:w
550:2
546:d
530:a
489:a
477:q
460:1
457:+
439:+
433:=
404:=
401:w
395:)
392:a
386:z
383:(
380:)
377:1
371:z
368:(
365:z
360:q
354:z
342:+
336:z
333:d
328:w
325:d
318:]
311:a
305:z
296:+
290:1
284:z
275:+
270:z
261:[
257:+
249:2
245:z
241:d
236:w
231:2
227:d
195:a
191:z
179:z
167:z
153:(
141:)
138:z
135:;
129:,
123:,
117:,
111:;
108:q
105:,
102:a
99:(
93:H
69:)
63:(
58:)
54:(
40:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.