Q-Charlier polynomials - Revision history

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	~~<!-- Please do not remove or change this AfD message until the discussion has been closed. -->~~
	~~{{Article for deletion/dated\|page=Q-Charlier polynomials\|timestamp=20260922002037\|year=2026\|month=September\|day=22\|substed=yes\|help=off}}~~
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	{{DISPLAYTITLE:''q''-Charlier polynomials }}		{{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.		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.

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Nominated for deletion; see Wikipedia:Articles for deletion/Q-Charlier polynomials.

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			<!-- Please do not remove or change this AfD message until the discussion has been closed. -->
			{{Article for deletion/dated\|page=Q-Charlier polynomials\|timestamp=20260922002037\|year=2026\|month=September\|day=22\|substed=yes\|help=off}}
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	{{DISPLAYTITLE:''q''-Charlier polynomials }}		{{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.		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.

wikipedia>Barçaforlife: Designated as stub

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Designated as stub

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	{{DISPLAYTITLE:''q''-Charlier polynomials }}		{{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.		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.

	~~==Definition==~~

	The polynomials are given in terms of the [[basic hypergeometric function]] by		The polynomials are given in terms of the [[basic hypergeometric function]] by
	:<math>~~\displaystyle~~ C_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a).</math>		:<math>C_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a).</math>

	==References==		==References==
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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}}		*{{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}}		*{{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. \|publisher=~~Universität~~ Kassel \|~~citeseerx~~=~~10.1.1~~.~~643~~.~~3896~~ }}
			* {{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:Orthogonal polynomials]]
	[[Category:Q-analogs]]		[[Category:Q-analogs]]
	[[Category:Special hypergeometric functions]]		[[Category:Special hypergeometric functions]]

			{{polynomial-stub}}

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	*{{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}}		*{{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. \|publisher=Universität Kassel \|citeseerx=10.1.1.643.3896 ~~\|url=https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.643.3896&rep=rep1&type=pdf \|access-date=February 21, 2021~~}}		*{{cite thesis \|last=Sadjang \|first=Patrick Njionou \|title=Moments of Classical Orthogonal Polynomials \|type=Ph.D. \|publisher=Universität Kassel \|citeseerx=10.1.1.643.3896 }}

	[[Category:Orthogonal polynomials]]		[[Category:Orthogonal polynomials]]
	[[Category:Q-analogs]]		[[Category:Q-analogs]]
	[[Category:Special hypergeometric functions]]		[[Category:Special hypergeometric functions]]

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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}}		*{{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}}		*{{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}}		*{{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}}

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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}}		*{{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}}		*{{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. \|publisher=Universität Kassel \|url=https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.643.3896&rep=rep1&type=pdf \|access-date=February 21, 2021}}		*{{cite thesis \|last=Sadjang \|first=Patrick Njionou \|title=Moments of Classical Orthogonal Polynomials \|type=Ph.D. \|publisher=Universität Kassel \|citeseerx=10.1.1.643.3896 \|url=https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.643.3896&rep=rep1&type=pdf \|access-date=February 21, 2021}}

	[[Category:Orthogonal polynomials]]		[[Category:Orthogonal polynomials]]
	[[Category:Q-analogs]]		[[Category:Q-analogs]]
	[[Category:Special hypergeometric functions]]		[[Category:Special hypergeometric functions]]

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 The polynomials are given in terms of the [[basic hypergeometric function]] by
 :<math>\displaystyle C_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a).</math>
 ==References==

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	==References==		==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 \| doi=10.2277/0521833574 \| mr=2128719 \| year=2004 \| volume=96}}		*{{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=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=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 \| 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}}		*{{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=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. \|publisher=Universität Kassel \|url=https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.643.3896&rep=rep1&type=pdf \|access-date=February 21, 2021}}		*{{cite thesis \|last=Sadjang \|first=Patrick Njionou \|title=Moments of Classical Orthogonal Polynomials \|type=Ph.D. \|publisher=Universität Kassel \|url=https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.643.3896&rep=rep1&type=pdf \|access-date=February 21, 2021}}

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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}}		*{{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 \| 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}}		*{{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=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. \|publisher=Universität Kassel \|url=https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.643.3896&rep=rep1&type=pdf \|access-date=February 21, 2021}}		*{{cite thesis \|last=Sadjang \|first=Patrick Njionou \|title=Moments of Classical Orthogonal Polynomials \|type=Ph.D. \|publisher=Universität Kassel \|url=https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.643.3896&rep=rep1&type=pdf \|access-date=February 21, 2021}}