Q-Charlier polynomials

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>

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- 66\,n{x}^{2}+11\,n{x}^{3}+36\,{n}^{2}-66\,{n}^{2}x+36\,{n}^{2}{x}^{2}- 6\,{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

Q-CHARLIER ABS COMPLEX 3D MAPLE PLOT

Q-CHARLIER IM COMPLEX 3D MAPLE PLOT

Q-CHARLIER RE COMPLEX 3D MAPLE PLOT

Q-CHARLIER ABS DENSITY MAPLE PLOT

Q-CHARLIER IM DENSITY MAPLE PLOT

Q-CHARLIER RE DENSITY MAPLE PLOT

References

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, 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