Q-Charlier polynomials: Difference between revisions
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{{DISPLAYTITLE:''q''-Charlier polynomials }} | |||
In [[mathematics]], the '''''q''-Charlier polynomials'''<ref>There are similar named polynomials named alternative q-Charlier polynomials <math>K_n(x;a;q)</math> which is another name for q-Bessel polynomials.</ref> 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. | |||
The polynomials are given in terms of the [[basic hypergeometric function]] by | |||
:<math>C_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a).</math> | |||
==References== | |||
{{Reflist}} | |||
*{{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 | 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 | 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=Chapter 18: Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}} | |||
* {{cite thesis |last=Sadjang |first=Patrick Njionou |title=Moments of Classical Orthogonal Polynomials |type=Ph.D. thesis |year=2013 |publisher=[[University of Kassel]] |url=https://kobra.uni-kassel.de/items/663d9455-5e61-4a16-8f7b-b2bad71df4c6}} | |||
[[Category:Orthogonal polynomials]] | |||
[[Category:Q-analogs]] | |||
[[Category:Special hypergeometric functions]] | |||
{{polynomial-stub}} | |||
Latest revision as of 17:02, 29 September 2026
In mathematics, the q-Charlier polynomials[1] are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Template:Harvs give a detailed list of their properties.
The polynomials are given in terms of the basic hypergeometric function by
- <math>C_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a).</math>
References
- ↑ There are similar named polynomials named alternative q-Charlier polynomials <math>K_n(x;a;q)</math> which is another name for q-Bessel polynomials.
- Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, 96 (2nd ed.), Cambridge University Press, 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