Definition:AlSalamIsmail

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The LaTeX DLMF and DRMF macro \AlSalamIsmail represents the Al-Salam Ismail ${\displaystyle{\displaystyle q}}$ polynomial.

This macro is in the category of polynomials.

In math mode, this macro can be called in the following ways:

\AlSalamIsmail{n} produces ${\displaystyle{\displaystyle{\displaystyle U_{n}}}}$

\AlSalamIsmail{n}@{x}{s}{q} produces ${\displaystyle{\displaystyle{\displaystyle U_{n}\!\left(x;s,q\right)}}}$

These are defined by ${\displaystyle{\displaystyle U_{2n}\!\left(x^{1/2};a,b\right):=q^{n^{2}-n}(-b)^{n}\qHyperrphis{4}{3}@@{q^{-n},-q^{1-n}/a,-aq^{n},q^{n+1}}{-q,q^{1/2},-q^{1/2}}{q}{\frac{xaq}{b}}}}$

and

${\displaystyle{\displaystyle U_{2n+1}\!\left(x^{1/2};a,b\right)=q^{n^{2}-n}\frac{(-b)^{n}(1+aq^{n})(1-q^{n+1})x^{1/2}}{(1-q)}\qHyperrphis{4}{3}@@{q^{-n},-q^{1-n}/a,-aq^{n+1},q^{n+2}}{-q,q^{3/2},-q^{3/2}}{q}{\frac{xaq}{b}}.}}$

Symbols List

${\displaystyle{\displaystyle{\displaystyle U_{n}}}}$ : Al-Salam Ismail polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamIsmail
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

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