319:
104:
314:{\displaystyle \displaystyle L_{n}^{(\alpha )}(x;q)={\frac {(q^{\alpha +1};q)_{n}}{(q;q)_{n}}}{}_{1}\phi _{1}(q^{-n};q^{\alpha +1};q,-q^{n+\alpha +1}x).}
493:
417:
426:
390:
352:
21:
483:
91:
25:
344:
17:
329:
Orthogonality is defined by the unimono nature of the polynomials' convergence at boundaries in integral form.
63:
95:
488:
422:
412:
386:
348:
453:
378:
467:
436:
400:
362:
463:
432:
396:
374:
358:
407:
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010),
477:
458:
67:
343:, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.),
74:). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (
382:
444:
Moak, Daniel S. (1981), "The q-analogue of the
Laguerre polynomials",
415:; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
369:
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010),
371:
Hypergeometric orthogonal polynomials and their q-analogues
408:
373:, Springer Monographs in Mathematics, Berlin, New York:
78:, 14) give a detailed list of their properties.
108:
107:
75:
313:
8:
90:-Laguerre polynomials are given in terms of
457:
283:
255:
239:
226:
216:
214:
204:
180:
158:
148:
118:
113:
106:
39:generalized Stieltjes–Wigert polynomials
418:NIST Handbook of Mathematical Functions
70:introduced by Daniel S. Moak (
62:) are a family of basic hypergeometric
339:Gasper, George; Rahman, Mizan (2004),
7:
409:"Chapter 18: Orthogonal Polynomials"
71:
14:
22:continuous q-Laguerre polynomials
494:Special hypergeometric functions
421:, Cambridge University Press,
304:
232:
201:
188:
177:
151:
142:
130:
125:
119:
92:basic hypergeometric functions
1:
26:little q-Laguerre polynomials
459:10.1016/0022-247X(81)90048-2
341:Basic hypergeometric series
510:
345:Cambridge University Press
18:big q-Laguerre polynomials
15:
383:10.1007/978-3-642-05014-5
484:Orthogonal polynomials
315:
64:orthogonal polynomials
316:
35:-Laguerre polynomials
446:J. Math. Anal. Appl.
105:
30:In mathematics, the
129:
96:q-Pochhammer symbol
413:Olver, Frank W. J.
311:
310:
109:
428:978-0-521-19225-5
392:978-3-642-05013-8
354:978-0-521-83357-8
211:
501:
470:
461:
439:
403:
365:
320:
318:
317:
312:
300:
299:
266:
265:
247:
246:
231:
230:
221:
220:
215:
212:
210:
209:
208:
186:
185:
184:
169:
168:
149:
128:
117:
53:
52:
509:
508:
504:
503:
502:
500:
499:
498:
474:
473:
443:
429:
406:
393:
375:Springer-Verlag
368:
355:
338:
335:
327:
279:
251:
235:
222:
213:
200:
187:
176:
154:
150:
103:
102:
84:
51:
46:
45:
44:
28:
12:
11:
5:
507:
505:
497:
496:
491:
486:
476:
475:
472:
471:
441:
427:
404:
391:
366:
353:
334:
331:
326:
323:
322:
321:
309:
306:
303:
298:
295:
292:
289:
286:
282:
278:
275:
272:
269:
264:
261:
258:
254:
250:
245:
242:
238:
234:
229:
225:
219:
207:
203:
199:
196:
193:
190:
183:
179:
175:
172:
167:
164:
161:
157:
153:
147:
144:
141:
138:
135:
132:
127:
124:
121:
116:
112:
83:
80:
47:
13:
10:
9:
6:
4:
3:
2:
506:
495:
492:
490:
487:
485:
482:
481:
479:
469:
465:
460:
455:
451:
447:
442:
438:
434:
430:
424:
420:
419:
414:
410:
405:
402:
398:
394:
388:
384:
380:
376:
372:
367:
364:
360:
356:
350:
346:
342:
337:
336:
332:
330:
325:Orthogonality
324:
307:
301:
296:
293:
290:
287:
284:
280:
276:
273:
270:
267:
262:
259:
256:
252:
248:
243:
240:
236:
227:
223:
217:
205:
197:
194:
191:
181:
173:
170:
165:
162:
159:
155:
145:
139:
136:
133:
122:
114:
110:
101:
100:
99:
97:
93:
89:
81:
79:
77:
73:
69:
66:in the basic
65:
61:
57:
50:
43:
40:
36:
34:
27:
23:
19:
452:(1): 20–47,
449:
445:
416:
370:
340:
328:
87:
85:
68:Askey scheme
59:
55:
48:
41:
38:
32:
31:
29:
478:Categories
333:References
82:Definition
16:See also:
489:Q-analogs
291:α
277:−
257:α
241:−
224:ϕ
160:α
123:α
94:and the
468:0618759
437:2723248
401:2656096
363:2128719
466:
435:
425:
399:
389:
361:
351:
24:, and
411:, in
37:, or
423:ISBN
387:ISBN
349:ISBN
98:by
86:The
76:2010
72:1981
454:doi
379:doi
480::
464:MR
462:,
450:81
448:,
433:MR
431:,
397:MR
395:,
385:,
377:,
359:MR
357:,
347:,
20:,
456::
440:.
381::
308:.
305:)
302:x
297:1
294:+
288:+
285:n
281:q
274:,
271:q
268:;
263:1
260:+
253:q
249:;
244:n
237:q
233:(
228:1
218:1
206:n
202:)
198:q
195:;
192:q
189:(
182:n
178:)
174:q
171:;
166:1
163:+
156:q
152:(
146:=
143:)
140:q
137:;
134:x
131:(
126:)
120:(
115:n
111:L
88:q
60:q
58:;
56:x
54:(
49:n
42:P
33:q
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.