Big q-Jacobi: Special case

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Big q-Jacobi: Special case

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Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;c;q\right)=\qHyperrphis{3}{2}@@{q^{-n},q^{n+1},x}{q,cq}{q}{q}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\int_{cq}^{q}P_{m}\!\left(x;c;q\right)P_{n}\!\left(x;c;q\right)\,d_{q}x{}=q(1-c)\frac{(1-q)}{(1-q^{2n+1})}\frac{\left(c^{-1}q;q\right)_{n}}{\left(cq;q\right)_{n}}(-cq^{2})^{n}q^{\binomial{n}{2}}\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle c<0}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle(x-1)P_{n}\!\left(x;c;q\right)=A_{n}P_{n+1}\!\left(x;c;q\right)-\left(A_{n}+C_{n}\right)P_{n}\!\left(x;c;q\right){}+C_{n}P_{n-1}\!\left(x;c;q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=-cq^{n+1}\frac{(1-q^{n})(1-c^{-1}q^{n})}{(1+q^{n})(1-q^{2n+1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+1})(1-cq^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}_{n}\!\left(x;c;q\right)={\widehat{P}}_{n+1}\!\left(x;c;q\right)+\left[1-(A_{n}+C_{n})\right]{\widehat{P}}_{n}\!\left(x;c;q\right)+A_{n-1}C_{n}{\widehat{P}}_{n-1}\!\left(x;c;q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=-cq^{n+1}\frac{(1-q^{n})(1-c^{-1}q^{n})}{(1+q^{n})(1-q^{2n+1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+1})(1-cq^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}}}}$

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;c;q\right)=\frac{\left(q^{n+1};q\right)_{n}}{\left(q,cq;q\right)_{n}}{\widehat{P}}_{n}\!\left(x;c;q\right)}}}$

q-Difference equation

${\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n})(1-q^{n+1})x^{2}y(x){}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle D(x)=(x-q)(x-cq)}}}$ &

${\displaystyle{\displaystyle{\displaystyle B(x)=q(x-1)(x-c)}}}$ &

${\displaystyle{\displaystyle{\displaystyle y(x)=P_{n}\!\left(x;c;q\right)}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;c;q\right)=\frac{c^{n}q^{n(n+1)}(1-q)^{n}}{\left(q,cq;q\right)_{n}}\left(\mathcal{D}_{q}\right)^{n}\left[\left(q^{-n}x,c^{-1}q^{-n}x;q\right)_{n}\right]}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;c;q\right)=\frac{(1-q)^{n}}{\left(q,cq;q\right)_{n}}\left(\mathcal{D}_{q}\right)^{n}\left[\left(qx^{-1},cqx^{-1};q\right)_{n}x^{2n}\right]}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{qx^{-1},0}{q}{q}{xt}\,\qHyperrphis{1}{1}@@{c^{-1}x}{q}{q}{cqt}=\sum_{n=0}^{\infty}\frac{\left(cq;q\right)_{n}}{\left(q,q;q\right)_{n}}P_{n}\!\left(x;c;q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{cqx^{-1},0}{cq}{q}{xt}\,\qHyperrphis{1}{1}@@{c^{-1}x}{c^{-1}q}{q}{qt}=\sum_{n=0}^{\infty}\frac{P_{n}\!\left(x;c;q\right)}{\left(c^{-1}q;q\right)_{n}}t^{n}}}}$

Limit relations

Big q-Legendre polynomial to Legendre / Spherical polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P_{n}\!\left(x;0;q\right)=\LegendrePoly{n}@{2x-1}}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P_{n}\!\left(x;-q^{\gamma};q\right)=\LegendrePoly{n}@{x}}}}$

<< Big q-Jacobi

Big q-Jacobi: Special case

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