q-Charlier polynomials

In mathematics, the q-Charlier polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Definition

The q-Charlier polynomials are given in terms of the basic hypergeometric function by

\( \displaystyle c_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a) \)

Relation to other polynomials

Q-Charlier polynomials →Charlier polynomials

\( \lim_{q \to 1}C_{n}(q^{-n};a(1-q);q)=C_{n}(x;a)

Checking k=4 term of Q-Charlier polynomials

\( {\frac { \left( 1-{q}^{-n} \right) \left( 1-{q}^{-n}q \right) \left( 1-{q}^{-n}{q}^{2} \right) \left( 1-{q}^{-n}{q}^{3} \right) \left( 1-{q}^{-x} \right) \left( 1-{q}^{-x}q \right) \left( 1-{q}^{ -x}{q}^{2} \right) \left( 1-{q}^{-x}{q}^{3} \right) \left( {q}^{n} \right) ^{4}{q}^{4}}{{a}^{4} \left( 1-q \right) ^{5} \left( 1-{q}^{2} \right) \left( 1-{q}^{3} \right) \left( 1-{q}^{4} \right) }} \)

expand it:

\( \frac{1}{24}\,{\frac {36\,nx-66\,n{x}^{2}+36\,n{x}^{3}-6\,n{x}^{4}-66\,{n}^{2} x+121\,{n}^{2}{x}^{2}-66\,{n}^{2}{x}^{3}+11\,{n}^{2}{x}^{4}+36\,{n}^{3 }x-66\,{n}^{3}{x}^{2}+36\,{n}^{3}{x}^{3}-6\,{n}^{3}{x}^{4}-6\,{n}^{4}x +11\,{n}^{4}{x}^{2}-6\,{n}^{4}{x}^{3}+{n}^{4}{x}^{4}}{{a}^{4}}} \)

On the other hand

The k=4 term of Charlier polynomials is

\( \frac{1}{24}\,{\frac {{\it pochhammer} \left( -n,4 \right) {\it pochhammer} \left( -x,4 \right) }{{a}^{4}}} \)

expand it:

\( \frac{1}{24}\,{\frac {nx \left( 36-66\,x+36\,{x}^{2}-6\,{x}^{3}-66\,n+121\,nx- 66\,n{x}^{2}+11\,n{x}^{3}+36\,{n}^{2}-66\,{n}^{2}x+36\,{n}^{2}{x}^{2}- 6\,{n}^{2}{x}^{3}-6\,{n}^{3}+11\,{n}^{3}x-6\,{n}^{3}{x}^{2}+{n}^{3}{x} ^{3} \right) }{{a}^{4}}} \)

These two expresions are identical QED

References

Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8, MR 2128719
Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248

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