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Line 20: ==Relation to other polynomials==
==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==
==Gallery==
In mathematics, the q -Charlier polynomialsorthogonal 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(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a)</math>
Orthogonality
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Recurrence and difference relations
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Generating function
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Relation to other polynomials
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 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 Template:Dlmf 
