Definition:ctsqJacobi

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The LaTeX DLMF and DRMF macro \ctsqJacobi represents the continuous ${\displaystyle{\displaystyle q}}$-Jacobi polynomial.

This macro is in the category of polynomials.

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

\ctsqJacobi{\alpha}{\beta}{m} produces ${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{m}}}}$

\ctsqJacobi{\alpha}{\beta}{m}@{x}{q} produces ${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{m}\!\left(x|q\right)}}}$

These are defined by
${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x|q\right):=\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}}\qHyperrphis{4}{3}@@{q^{-n},q^{n+\alpha+\beta+1},q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{i\theta}},q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{-i\theta}}}{q^{\alpha+1},-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{q}}}$

with ${\displaystyle{\displaystyle x=\cos\theta}}$.

Symbols List

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

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