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| {{DISPLAYTITLE: ''q''-Laguerre polynomials}}
| | Частное предприятие «Илигран»<br>220073, г. [http://iligran.by/%d0%bd%d0%b0%d1%88%d0%b8-%d1%80%d0%b0%d0%b1%d0%be%d1%82%d1%8b/ разгрузка оборудования Минск], ул. Кальварийская, дом 25, офис 424<br>Телефоны:<br><br>+375 44 545-67-00<br><br>+375 29 379-91-88<br>+375 17 204 42 28 (факс)<br>+375 17 204 42 26 (факс)<br>+375 17 204 01 72<br>Email: 2044228@mail.ru<br><br>http://iligran.by |
| {{see also|big q-Laguerre polynomials|continuous q-Laguerre polynomials|little q-Laguerre polynomials}}
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| In mathematics, the '''''q''-Laguerre polynomials''', or '''generalized Stieltjes–Wigert polynomials''' ''P''{{su|b=''n''|p=(α)}}(''x'';''q'') are a family of basic hypergeometric [[orthogonal polynomials]] in the basic [[Askey scheme]] introduced by {{harvs|txt | last=Moak|first=Daniel S.|title=The q-analogue of the Laguerre polynomials|journal=. J. Math. Anal. Appl.| volume=81|issue=1|pages=20-47|year=1981}}. {{harvs|txt | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010|loc=14}} give a detailed list of their properties.
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| ==Definition==
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| The ''q''-Laguerre polynomials are given in terms of [[basic hypergeometric function]]s and the [[Pochhammer symbol]] by
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| :<math>\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) </math>
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| ==Orthogonality==
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| {{Empty section|date=September 2011}}
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| ==Recurrence and difference relations==
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| {{Empty section|date=September 2011}}
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| ==Rodrigues formula==
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| ==Generating function==
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| {{Empty section|date=September 2011}}
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| ==Relation to other polynomials==
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| {{Empty section|date=September 2011}}
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| ==References==
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| *{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 | mr=2128719 | year=2004 | volume=96}}
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| *{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}
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| *{{dlmf|id=18|title=|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
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| *{{citation | last=Moak|first=Daniel S.|title=The q-analogue of the Laguerre polynomials|journal=J. Math. Anal. Appl.| volume=81|issue=1|pages=20-47|year=1981| doi=10.1016/0022-247X(81)90048-2 |mr=618759}}
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| [[Category:Orthogonal polynomials]]
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| [[Category:Q-analogs]]
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| [[Category:Special hypergeometric functions]]
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Частное предприятие «Илигран»
220073, г. разгрузка оборудования Минск, ул. Кальварийская, дом 25, офис 424
Телефоны:
+375 44 545-67-00
+375 29 379-91-88
+375 17 204 42 28 (факс)
+375 17 204 42 26 (факс)
+375 17 204 01 72
Email: 2044228@mail.ru
http://iligran.by