Definition:AlSalamIsmail: Difference between revisions

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Revision as of 00:32, 6 March 2017

The LaTeX DLMF and DRMF macro \AlSalamIsmail represents the Al-Salam Ismail q 𝑞 {\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 U n Al-Salam-Ismail-q-polynomial-U 𝑛 {\displaystyle{\displaystyle{\displaystyle U_{n}}}}
\AlSalamIsmail{n}@{x}{s}{q} produces U n ⁡ ( x ; s , q ) Al-Salam-Ismail-q-polynomial-U 𝑛 𝑥 𝑠 𝑞 {\displaystyle{\displaystyle{\displaystyle U_{n}\!\left(x;s,q\right)}}}

These are defined by U 2 ⁢ n ⁡ ( x 1 / 2 ; a , b ) := q n 2 - n ⁢ ( - b ) n ⁢ \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - n , - q 1 - n / a , - a ⁢ q n , q n + 1 - q , q 1 / 2 , - q 1 / 2 ⁢ q ⁢ x ⁢ a ⁢ q b assign Al-Salam-Ismail-q-polynomial-U 2 𝑛 superscript 𝑥 1 2 𝑎 𝑏 superscript 𝑞 superscript 𝑛 2 𝑛 superscript 𝑏 𝑛 \qHyperrphis 43 @ @ superscript 𝑞 𝑛 superscript 𝑞 1 𝑛 𝑎 𝑎 superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 𝑞 superscript 𝑞 1 2 superscript 𝑞 1 2 𝑞 𝑥 𝑎 𝑞 𝑏 {\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

U 2 ⁢ n + 1 ⁡ ( x 1 / 2 ; a , b ) = q n 2 - n ⁢ ( - b ) n ⁢ ( 1 + a ⁢ q n ) ⁢ ( 1 - q n + 1 ) ⁢ x 1 / 2 ( 1 - q ) ⁢ \qHyperrphis ⁢ 43 ⁢ @ ⁢ @ ⁢ q - n , - q 1 - n / a , - a ⁢ q n + 1 , q n + 2 - q , q 3 / 2 , - q 3 / 2 ⁢ q ⁢ x ⁢ a ⁢ q b . Al-Salam-Ismail-q-polynomial-U 2 𝑛 1 superscript 𝑥 1 2 𝑎 𝑏 superscript 𝑞 superscript 𝑛 2 𝑛 superscript 𝑏 𝑛 1 𝑎 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 superscript 𝑥 1 2 1 𝑞 \qHyperrphis 43 @ @ superscript 𝑞 𝑛 superscript 𝑞 1 𝑛 𝑎 𝑎 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 2 𝑞 superscript 𝑞 3 2 superscript 𝑞 3 2 𝑞 𝑥 𝑎 𝑞 𝑏 {\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

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


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