Q-Charlier polynomials: Difference between revisions
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{{DISPLAYTITLE:''q''-Charlier polynomials }} | {{DISPLAYTITLE:''q''-Charlier polynomials }} | ||
In mathematics, the '''''q''-Charlier polynomials''' are a family of basic hypergeometric [[orthogonal polynomials]] in the basic [[Askey scheme]]. {{harvs|txt | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | In mathematics, the '''''q''-Charlier polynomials''' are a family of basic hypergeometric [[orthogonal polynomials]] in the basic [[Askey scheme]]. {{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. | ||
==Definition== | ==Definition== | ||
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==Orthogonality== | ==Orthogonality== | ||
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==Recurrence and difference relations== | ==Recurrence and difference relations== | ||
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==Rodrigues formula== | ==Rodrigues formula== | ||
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==Generating function== | ==Generating function== | ||
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==Relation to other polynomials== | ==Relation to other polynomials== | ||
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==References== | ==References== | ||
*{{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 | | *{{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}} | ||
*{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | *{{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}} | ||
*{{dlmf|id=18|title=|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}} | *{{dlmf|id=18|title=|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}} | ||
[[Category:Orthogonal polynomials]] | [[Category:Orthogonal polynomials]] | ||
[[Category: | [[Category:Q-analogs]] | ||
[[Category:Special hypergeometric functions]] | [[Category:Special hypergeometric functions]] | ||
Revision as of 07:42, 5 September 2011
In mathematics, the q-Charlier polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Template:Harvs give a detailed list of their properties.
Definition
The q-Charlier polynomials are given in terms of the basic hypergeometric function by
- <math>\displaystyle c_n(x;a;q) = {}_2\phi_1(q^{-n},x;0;q,-q^{n+1}/a)</math>
Orthogonality
Recurrence and difference relations
Rodrigues formula
Generating function
Relation to other polynomials
References
- Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, 96 (2nd ed.), Cambridge University Press, doi:, ISBN 978-0-521-83357-8
- Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer-Verlag, doi:, ISBN 978-3-642-05013-8
- Template:Dlmf