Little q-Jacobi: Special case

<< Little q-Jacobi

Little q-Jacobi: Special case

q-Meixner >>

Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x|q\right)=\qHyperrphis{2}{1}@@{q^{-n},q^{n+1}}{q}{q}{qx}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\int_{0}^{1}p_{m}\!\left(x|q\right)p_{n}\!\left(x|q\right)\,d_{q}x=(1-q)\sum_{k=0}^{\infty}q^{k}p_{m}\!\left(q^{k}|q\right)p_{n}\!\left(q^{k}|q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\int_{0}^{1}p_{m}\!\left(x|q\right)p_{n}\!\left(x|q\right)\,d_{q}x=\frac{(1-q)q^{n}}{(1-q^{2n+1})}\,\delta_{m,n}}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-xp_{n}\!\left(x|q\right)=A_{n}p_{n+1}\!\left(x|q\right)-\left(A_{n}+C_{n}\right)p_{n}\!\left(x|q\right)+C_{n}p_{n-1}\!\left(x|q\right)}}}$

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

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{p}}_{n}\!\left(x\right)={\widehat{p}}_{n+1}\!\left(x\right)+(A_{n}+C_{n}){\widehat{p}}_{n}\!\left(x\right)+A_{n-1}C_{n}{\widehat{p}}_{n-1}\!\left(x\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x|q\right)=\frac{(-1)^{n}q^{-\binomial{n}{2}}\left(q^{n+1};q\right)_{n}}{\left(q;q\right)_{n}}{\widehat{p}}_{n}\!\left(x\right)}}}$

q-Difference equation

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle D(x)=x-1}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle y(x)=p_{n}\!\left(x|q\right)}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x|q\right)=\frac{q^{\binomial{n}{2}}(1-q)^{n}}{\left(q;q\right)_{n}}\left(\mathcal{D}_{q^{-1}}\right)^{n}\left[\left(qx;q\right)_{n}x^{n}\right]}}}$

Generating function

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{0}{1}@@{-}{q}{q}{qxt}\,\qHyperrphis{2}{1}@@{x^{-1},0}{q}{q}{xt}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{n}{2}}}{\left(q,q;q\right)_{n}}p_{n}\!\left(x|q\right)t^{n}}}}$

Limit relation

Little q-Legendre polynomial to Legendre / Spherical polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}p_{n}\!\left(x|q\right)=\LegendrePoly{n}@{1-2x}}}}$

<< Little q-Jacobi

Little q-Jacobi: Special case

q-Meixner >>