Q-Charlier polynomials: Difference between revisions

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

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:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8
Template:Dlmf

Template:Cite thesis

Template:Polynomial-stub

[1] 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.

[1]

@@ Line 1: / Line 1: @@
 {{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 | 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
+:<math>C_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a).</math>
-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==
-{{Empty section|date=September 2011}}
-==Recurrence and difference relations==
-{{Empty section|date=September 2011}}
-==Rodrigues formula==
-{{Empty section|date=September 2011}}
-==Generating function==
-{{Empty section|date=September 2011}}
-==Relation to other polynomials==
-Q-Charlier polynomials →Charlier polynomials
-<math>\lim_{q \to 1}C_{n}(q^{-n};a(1-q);q)=C_{n}(x;a)</math>
-;Checking k=4 term of   Q-Charlier polynomials
-<math>{\frac { \left( 1-{q}^{-n} \right)  \left( 1-{q}^{-n}q \right)
- \left( 1-{q}^{-n}{q}^{2} \right)  \left( 1-{q}^{-n}{q}^{3} \right)
- \left( 1-{q}^{-x} \right)  \left( 1-{q}^{-x}q \right)  \left( 1-{q}^{
--x}{q}^{2} \right)  \left( 1-{q}^{-x}{q}^{3} \right)  \left( {q}^{n}
- \right) ^{4}{q}^{4}}{{a}^{4} \left( 1-q \right) ^{5} \left( 1-{q}^{2}
- \right)  \left( 1-{q}^{3} \right)  \left( 1-{q}^{4} \right) }}
-    </math>
-expand it:
-<math>\frac{1}{24}\,{\frac {36\,nx-66\,n{x}^{2}+36\,n{x}^{3}-6\,n{x}^{4}-66\,{n}^{2}
-x+121\,{n}^{2}{x}^{2}-66\,{n}^{2}{x}^{3}+11\,{n}^{2}{x}^{4}+36\,{n}^{3
-}x-66\,{n}^{3}{x}^{2}+36\,{n}^{3}{x}^{3}-6\,{n}^{3}{x}^{4}-6\,{n}^{4}x
-+11\,{n}^{4}{x}^{2}-6\,{n}^{4}{x}^{3}+{n}^{4}{x}^{4}}{{a}^{4}}}
-</math>
-On the other hand
-The k=4  term of  [[Charlier polynomials]] is
-<math>\frac{1}{24}\,{\frac {{\it pochhammer} \left( -n,4 \right) {\it pochhammer}
- \left( -x,4 \right) }{{a}^{4}}}</math>
-expand it:
-<math>\frac{1}{24}\,{\frac {nx \left( 36-66\,x+36\,{x}^{2}-6\,{x}^{3}-66\,n+121\,nx-
-\,n{x}^{2}+11\,n{x}^{3}+36\,{n}^{2}-66\,{n}^{2}x+36\,{n}^{2}{x}^{2}-
-\,{n}^{2}{x}^{3}-6\,{n}^{3}+11\,{n}^{3}x-6\,{n}^{3}{x}^{2}+{n}^{3}{x}
-^{3} \right) }{{a}^{4}}}
-</math>
-These two expresions are identical QED
-==Gallery==
-{|
-|[[File:Q-CHARLIER ABS COMPLEX 3D MAPLE PLOT.gif|thumb|Q-CHARLIER ABS COMPLEX 3D MAPLE PLOT]]
-|[[File:Q-CHARLIER IM COMPLEX 3D MAPLE PLOT.gif|thumb|Q-CHARLIER IM COMPLEX 3D MAPLE PLOT]]
-|[[File:Q-CHARLIER RE COMPLEX 3D MAPLE PLOT.gif|thumb|Q-CHARLIER RE COMPLEX 3D MAPLE PLOT]]
-|}
-{|
-|[[File:Q-CHARLIER ABS DENSITY MAPLE PLOT.gif|thumb|Q-CHARLIER ABS DENSITY MAPLE PLOT]]
-|[[File:Q-CHARLIER IM DENSITY MAPLE PLOT.gif|thumb|Q-CHARLIER IM DENSITY MAPLE PLOT]]
-|[[File:Q-CHARLIER RE DENSITY MAPLE PLOT.gif|thumb|Q-CHARLIER RE DENSITY MAPLE PLOT]]
-|}
 ==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 | 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=|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. 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}}

Waza (talk | contribs)

Latest revision as of 17:02, 29 September 2026

References