883:
487:
878:{\displaystyle {\begin{aligned}\,_{q}!&=_{q}\cdot _{q}\cdots _{q}\cdot _{q}\\&={\frac {1-q}{1-q}}\cdot {\frac {1-q^{2}}{1-q}}\cdots {\frac {1-q^{n-1}}{1-q}}\cdot {\frac {1-q^{n}}{1-q}}\\&=1\cdot (1+q)\cdots (1+q+\cdots +q^{n-2})\cdot (1+q+\cdots +q^{n-1}).\end{aligned}}}
443:
1421:
492:
1310:
289:
1022:
1166:
1765:
1541:
1860:
An example from atomic physics is the model of molecular condensate creation from an ultra cold fermionic atomic gas during a sweep of an external magnetic field through the
1593:
1051:
1617:-generalization of the subsets of the set. This has been a fruitful point of view in finding interesting new theorems. For example, there are
452:-analog among the many possible options is unmotivated. However, it appears naturally in several contexts. For example, having decided to use
1968:, 98-17, Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics.
326:
1328:
2004:
1192:
222:
941:
911:
1996:
1988:
1941:
1907:
123:. The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of
1077:
2152:
1183:
2147:
71:
2092:
C. Sun; N. A. Sinitsyn (2016). "Landau-Zener extension of the Tavis-Cummings model: Structure of the solution".
128:
112:
2217:
1821:
1779:
1721:
1497:
1961:
1556:
2142:
2113:
2032:
1850:
limit usually corresponds to relatively simple dynamics, e.g., without nonlinear interactions, while
1622:
1957:
1030:
1061:
140:
48:
2222:
2103:
1861:
1700:
29:
Type of mathematical generalization such that the original version is the limit as q approaches 1
1915:
1824:, which recovers combinatorics as linear algebra over the field with one element: for example,
2000:
1992:
1984:
1937:
1881:
148:
132:
2121:
2040:
120:
85:
2078:
2195:
2074:
2066:
1890:
1829:
185:
2117:
2036:
1638:
189:
144:
124:
1868:-deformed version of the SU(2) algebra of operators, and its solution is described by
1843:-analogs are often found in exact solutions of many-body problems. In such cases, the
2211:
2017:
1947:
1925:
1911:
1895:
1626:
1316:
181:
169:
136:
81:
2094:
1929:
1462:
1454:
1446:
177:
152:
43:
of a theorem, identity or expression is a generalization involving a new parameter
2183:
897:
475:
100:
33:
2125:
2171:
1825:
2159:
2045:
2202:
2190:
2178:
2166:
1966:
The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
471:
17:
156:
107:-analogs find applications in a number of areas, including the study of
1027:
In particular, one recovers the usual factorial by taking the limit as
116:
108:
438:{\displaystyle _{q}={\frac {1-q^{n}}{1-q}}=1+q+q^{2}+\ldots +q^{n-1}.}
1792:-analogs as deformations, one can consider the combinatorial case of
1416:{\displaystyle e_{q}(x)=\sum _{n=0}^{\infty }{\frac {x^{n}}{_{q}!}}.}
62:-analogs that arise naturally, rather than in arbitrarily contriving
2108:
80:-analogs are most frequently studied in the mathematical fields of
47:
that returns the original theorem, identity or expression in the
1857:
gives insight into the complex nonlinear regime with feedbacks.
1305:{\displaystyle {\binom {n}{k}}_{q}={\frac {_{q}!}{_{q}!_{q}!}}.}
1182:, also known as Gaussian coefficients, Gaussian polynomials, or
892:-analog appears naturally in several contexts. Notably, while
1983:, New York: Halstead Press, Chichester: Ellis Horwood, 1983,
284:{\displaystyle \lim _{q\rightarrow 1}{\frac {1-q^{n}}{1-q}}=n}
2069:; Harper, L. H. (1971), "Matching theory, an introduction",
1952:
Classical and
Quantum Orthogonal Polynomials in One Variable
1017:{\displaystyle \sum _{w\in S_{n}}q^{{\text{inv}}(w)}=_{q}!.}
910:! counts permutations while keeping track of the number of
99:
is often discrete-valued (for example, it may represent a
1060:-factorial also has a concise definition in terms of the
1433:-Fourier transform, have been defined in this context.
1445:
The
Gaussian coefficients count subspaces of a finite
918:) denotes the number of inversions of the permutation
162:
themselves are closely related to elliptic integrals.
1724:
1559:
1500:
1331:
1195:
1161:{\displaystyle _{q}!={\frac {(q;q)_{n}}{(1-q)^{n}}}.}
1080:
1033:
944:
490:
329:
225:
216:-analogs of the nonnegative integers. The equality
1613:-generalization of a set, and the subspaces as the
2071:Advances in Probability and Related Topics, Vol. 1
1872:-deformed exponential and binomial distributions.
1817:in the formulae, hence the need to take a limit).
1759:
1587:
1535:
1415:
1304:
1160:
1045:
1016:
877:
437:
283:
180:. The connection here is similar, in that much of
1742:
1729:
1576:
1563:
1518:
1505:
1213:
1200:
1609:Thus, one can regard a finite vector space as a
1479:is especially appropriate.) Then the number of
227:
58:. Typically, mathematicians are interested in
1864:. This process is described by a model with a
139:in particular. The connection passes through
8:
1550:approach 1, we get the binomial coefficient
74:, which was introduced in the 19th century.
1981:q-Hypergeometric Functions and Applications
1175:-factorials, one can move on to define the
931:denotes the set of permutations of length
2107:
2044:
2025:Journal of Nonlinear Mathematical Physics
1748:
1741:
1728:
1726:
1723:
1575:
1562:
1560:
1558:
1524:
1517:
1504:
1502:
1499:
1398:
1381:
1375:
1369:
1358:
1336:
1330:
1287:
1268:
1241:
1228:
1219:
1212:
1199:
1197:
1194:
1146:
1122:
1103:
1091:
1079:
1032:
1002:
973:
972:
960:
949:
943:
853:
810:
731:
718:
689:
676:
653:
640:
611:
595:
576:
551:
532:
506:
495:
491:
489:
448:By itself, the choice of this particular
420:
401:
362:
349:
340:
328:
255:
242:
230:
224:
66:-analogs of known results. The earliest
1922:, Cambridge University Press, Cambridge.
1972:
1429:-trigonometric functions, along with a
2073:, New York: Dekker, pp. 169–215,
1707:). Then the number of fixed points of
1836:Applications in the physical sciences
168:-analogs also appear in the study of
7:
1760:{\displaystyle {\binom {n}{k}}_{q}.}
1536:{\displaystyle {\binom {n}{k}}_{q}.}
1487:-dimensional vector space over the
1733:
1567:
1509:
1370:
1204:
25:
1832:over the field with one element.
1598:or in other words, the number of
70:-analog studied in detail is the
1588:{\displaystyle {\binom {n}{k}},}
1067:, a basic building-block of all
88:. In these settings, the limit
1453:be the number of elements in a
1954:, Cambridge University Press.
1936:, Cambridge University Press,
1820:This can be formalized in the
1483:-dimensional subspaces of the
1395:
1388:
1348:
1342:
1284:
1277:
1265:
1252:
1238:
1231:
1184:Gaussian binomial coefficients
1143:
1130:
1119:
1106:
1088:
1081:
1046:{\displaystyle q\rightarrow 1}
1037:
999:
992:
984:
978:
865:
828:
822:
785:
779:
767:
592:
585:
573:
560:
548:
541:
529:
522:
503:
496:
337:
330:
234:
188:, resulting in connections to
1:
1810:(often one cannot simply let
127:in general (see, for example
294:suggests that we define the
2148:Encyclopedia of Mathematics
1934:Basic Hypergeometric Series
1663:be a cyclic group of order
155:play a prominent role; the
72:basic hypergeometric series
2239:
2126:10.1103/PhysRevA.94.033808
1777:
1691:has a canonical action on
1636:
192:, which in turn relate to
184:is set in the language of
115:, and expressions for the
2018:"A Method for q-calculus"
1683:-element set {1, 2, ...,
1655:-th power of a primitive
2046:10.2991/jnmp.2003.10.4.5
1679:-element subsets of the
1667:generated by an element
212:-theory begins with the
1784:Conversely, by letting
1659:-th root of unity. Let
1602:-element subsets of an
1491:-element field equals
896:! counts the number of
2016:Ernst, Thomas (2003).
1822:field with one element
1780:Field with one element
1761:
1589:
1537:
1475:, so using the letter
1417:
1374:
1306:
1180:-binomial coefficients
1162:
1047:
1018:
879:
439:
285:
113:multi-fractal measures
2199:-binomial coefficient
1762:
1590:
1538:
1461:is then a power of a
1418:
1354:
1307:
1163:
1048:
1019:
880:
466:, one may define the
440:
286:
1808: → 1
1722:
1557:
1498:
1329:
1193:
1078:
1031:
942:
488:
327:
302:, also known as the
223:
95:is often formal, as
2118:2016PhRvA..94c3808S
2037:2003JNMP...10..487E
914:. That is, if inv(
141:hyperbolic geometry
1979:Exton, H. (1983),
1862:Feshbach resonance
1757:
1701:cyclic permutation
1585:
1533:
1413:
1302:
1158:
1065:-Pochhammer symbol
1043:
1014:
967:
875:
873:
435:
281:
241:
149:elliptic integrals
2143:"Umbral calculus"
1920:Special Functions
1740:
1695:given by sending
1623:Sperner's theorem
1574:
1516:
1408:
1297:
1211:
1153:
976:
945:
749:
713:
671:
635:
380:
273:
226:
133:Apollonian gasket
121:dynamical systems
86:special functions
16:(Redirected from
2230:
2156:
2130:
2129:
2111:
2089:
2083:
2081:
2067:Rota, Gian-Carlo
2063:
2057:
2056:
2054:
2053:
2048:
2022:
2013:
2007:
1977:
1962:Swarttouw, R. F.
1948:Ismail, M. E. H.
1856:
1849:
1830:algebraic groups
1816:
1809:
1798:
1788:vary and seeing
1766:
1764:
1763:
1758:
1753:
1752:
1747:
1746:
1745:
1732:
1594:
1592:
1591:
1586:
1581:
1580:
1579:
1566:
1542:
1540:
1539:
1534:
1529:
1528:
1523:
1522:
1521:
1508:
1474:
1422:
1420:
1419:
1414:
1409:
1407:
1403:
1402:
1386:
1385:
1376:
1373:
1368:
1341:
1340:
1311:
1309:
1308:
1303:
1298:
1296:
1292:
1291:
1273:
1272:
1250:
1246:
1245:
1229:
1224:
1223:
1218:
1217:
1216:
1203:
1167:
1165:
1164:
1159:
1154:
1152:
1151:
1150:
1128:
1127:
1126:
1104:
1096:
1095:
1052:
1050:
1049:
1044:
1023:
1021:
1020:
1015:
1007:
1006:
988:
987:
977:
974:
966:
965:
964:
884:
882:
881:
876:
874:
864:
863:
821:
820:
754:
750:
748:
737:
736:
735:
719:
714:
712:
701:
700:
699:
677:
672:
670:
659:
658:
657:
641:
636:
634:
623:
612:
604:
600:
599:
581:
580:
556:
555:
537:
536:
511:
510:
444:
442:
441:
436:
431:
430:
406:
405:
381:
379:
368:
367:
366:
350:
345:
344:
290:
288:
287:
282:
274:
272:
261:
260:
259:
243:
240:
186:Riemann surfaces
98:
94:
57:
21:
2238:
2237:
2233:
2232:
2231:
2229:
2228:
2227:
2208:
2207:
2141:
2138:
2133:
2091:
2090:
2086:
2065:
2064:
2060:
2051:
2049:
2020:
2015:
2014:
2010:
1978:
1974:
1904:
1891:Stirling number
1878:
1851:
1844:
1838:
1811:
1804:
1793:
1782:
1776:
1727:
1725:
1720:
1719:
1641:
1635:
1561:
1555:
1554:
1503:
1501:
1496:
1495:
1466:
1457:. (The number
1443:
1394:
1387:
1377:
1332:
1327:
1326:
1322:is defined as:
1283:
1264:
1251:
1237:
1230:
1198:
1196:
1191:
1190:
1142:
1129:
1118:
1105:
1087:
1076:
1075:
1029:
1028:
998:
968:
956:
940:
939:
930:
909:
872:
871:
849:
806:
752:
751:
738:
727:
720:
702:
685:
678:
660:
649:
642:
624:
613:
602:
601:
591:
572:
547:
528:
515:
502:
486:
485:
474:, known as the
470:-analog of the
457:
416:
397:
369:
358:
351:
336:
325:
324:
262:
251:
244:
221:
220:
206:
190:elliptic curves
125:Fuchsian groups
96:
89:
52:
30:
23:
22:
15:
12:
11:
5:
2236:
2234:
2226:
2225:
2220:
2210:
2209:
2206:
2205:
2193:
2181:
2169:
2157:
2137:
2136:External links
2134:
2132:
2131:
2084:
2058:
2031:(4): 487–525.
2008:
2005:978-0470274538
1971:
1970:
1969:
1955:
1945:
1923:
1908:Andrews, G. E.
1903:
1900:
1899:
1898:
1893:
1888:
1877:
1874:
1837:
1834:
1815: = 1
1799:as a limit of
1797: = 1
1778:Main article:
1775:
1769:
1768:
1767:
1756:
1751:
1744:
1739:
1736:
1731:
1675:be the set of
1639:Cyclic sieving
1637:Main article:
1634:
1633:Cyclic sieving
1631:
1606:-element set.
1596:
1595:
1584:
1578:
1573:
1570:
1565:
1544:
1543:
1532:
1527:
1520:
1515:
1512:
1507:
1442:
1437:Combinatorial
1435:
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170:quantum groups
145:ergodic theory
129:Indra's pearls
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2235:
2224:
2221:
2219:
2218:Combinatorics
2216:
2215:
2213:
2204:
2200:
2198:
2194:
2192:
2188:
2186:
2182:
2180:
2176:
2174:
2170:
2168:
2164:
2162:
2158:
2154:
2150:
2149:
2144:
2140:
2139:
2135:
2127:
2123:
2119:
2115:
2110:
2105:
2102:(3): 033808.
2101:
2097:
2096:
2088:
2085:
2080:
2076:
2072:
2068:
2062:
2059:
2047:
2042:
2038:
2034:
2030:
2026:
2019:
2012:
2009:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1976:
1973:
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1963:
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1956:
1953:
1949:
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1943:
1939:
1935:
1931:
1927:
1924:
1921:
1917:
1913:
1909:
1906:
1905:
1901:
1897:
1896:Young tableau
1894:
1892:
1889:
1887:
1885:
1880:
1879:
1875:
1873:
1871:
1867:
1863:
1858:
1854:
1847:
1842:
1835:
1833:
1831:
1827:
1823:
1818:
1814:
1807:
1802:
1796:
1791:
1787:
1781:
1773:
1770:
1754:
1749:
1737:
1734:
1718:
1717:
1716:
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1687:}. The group
1686:
1682:
1678:
1674:
1670:
1666:
1662:
1658:
1654:
1650:
1646:
1640:
1632:
1630:
1628:
1627:Ramsey theory
1624:
1620:
1616:
1612:
1607:
1605:
1601:
1582:
1571:
1568:
1553:
1552:
1551:
1549:
1530:
1525:
1513:
1510:
1494:
1493:
1492:
1490:
1486:
1482:
1478:
1473:
1469:
1464:
1460:
1456:
1452:
1448:
1440:
1436:
1434:
1432:
1428:
1410:
1404:
1399:
1391:
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1365:
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1359:
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1345:
1337:
1333:
1325:
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1319:
1299:
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1288:
1280:
1274:
1269:
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1258:
1255:
1247:
1242:
1234:
1225:
1220:
1208:
1205:
1189:
1188:
1187:
1185:
1181:
1179:
1174:
1155:
1147:
1139:
1136:
1133:
1123:
1115:
1112:
1109:
1100:
1097:
1092:
1084:
1074:
1073:
1072:
1070:
1066:
1064:
1059:
1054:
1040:
1034:
1011:
1008:
1003:
995:
989:
981:
969:
961:
957:
953:
950:
946:
938:
937:
936:
934:
929:
925:
921:
917:
913:
908:
903:
899:
895:
891:
868:
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2095:Phys. Rev. A
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2050:. Retrieved
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1980:
1975:
1965:
1951:
1933:
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1912:Askey, R. A.
1883:
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1840:
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1803:-analogs as
1800:
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1463:prime number
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1447:vector space
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1958:Koekoek, R.
1828:are simple
1826:Weyl groups
1071:-theories:
462:-analog of
298:-analog of
119:of chaotic
101:prime power
34:mathematics
2212:Categories
2187:-factorial
2109:1606.08430
2052:2011-07-27
1997:0470274530
1989:0853124914
1942:0521833574
1930:Rahman, M.
1926:Gasper, G.
1902:References
935:, we have
912:inversions
900:of length
479:-factorial
208:Classical
176:-deformed
135:) and the
18:Q-analogue
2223:Q-analogs
2203:MathWorld
2191:MathWorld
2179:MathWorld
2167:MathWorld
2153:EMS Press
1651:) be the
1371:∞
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539:⋅
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425:−
411:…
374:−
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196:-series.
2175:-bracket
1964:(1998),
1950:(2005),
1932:(2004),
1918:(1999),
1886:-analogs
1882:List of
1876:See also
1546:Letting
1441:-analogs
320:, to be
307:-bracket
131:and the
109:fractals
2163:-analog
2155:, 2001
2114:Bibcode
2079:0282855
2033:Bibcode
1916:Roy, R.
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