Big q-Laguerre

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Big q-Laguerre

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Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b;q\right)=\qHyperrphis{3}{2}@@{q^{-n},0,x}{aq,bq}{q}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b;q\right)=\frac{1}{\left(b^{-1}q^{-n};q\right)_{n}}\,\qHyperrphis{2}{1}@@{q^{-n},aqx^{-1}}{aq}{q}{\frac{x}{b}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\int_{bq}^{aq}\frac{\left(a^{-1}x,b^{-1}x;q\right)_{\infty}}{\left(x;q\right)_{\infty}}P_{m}\!\left(x;a,b;q\right)P_{n}\!\left(x;a,b;q\right)\,d_{q}x{}=aq(1-q)\frac{\left(q,a^{-1}b,ab^{-1}q;q\right)_{\infty}}{\left(aq,bq;q\right)_{\infty}}\frac{\left(q;q\right)_{n}}{\left(aq,bq;q\right)_{n}}(-abq^{2})^{n}q^{\binomial{n}{2}}\,\delta_{m,n}}}}$

Recurrence relation

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

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

Monic recurrence relation

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

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

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b;q\right)=\frac{1}{\left(aq,bq;q\right)_{n}}{\widehat{P}}_{n}\!\left(x\right)}}}$

q-Difference equation

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

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

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

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

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b;q\right)-P_{n}\!\left(qx;a,b;q\right)=\frac{q^{-n+1}(1-q^{n})}{(1-aq)(1-bq)}xP_{n-1}\!\left(qx;aq,bq;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}P_{n}\!\left(x;a,b;q\right)=\frac{q^{-n+1}(1-q^{n})}{(1-q)(1-aq)(1-bq)}P_{n-1}\!\left(qx;aq,bq;q\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle(x-a)(x-b)P_{n}\!\left(x;a,b;q\right)-ab(1-x)P_{n}\!\left(qx;a,b;q\right){}=(1-a)(1-b)xP_{n+1}\!\left(x;aq^{-1},bq^{-1};q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}\left[w(x;a,b;q)P_{n}\!\left(x;a,b;q\right)\right]{}=\frac{(1-a)(1-b)}{ab(1-q)}w(x;aq^{-1},bq^{-1};q)P_{n+1}\!\left(x;aq^{-1},bq^{-1};q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;a,b;q)=\frac{\left(a^{-1}x,b^{-1}x;q\right)_{\infty}}{\left(x;q\right)_{\infty}}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;a,b;q)P_{n}\!\left(x;a,b;q\right){}=\frac{a^{n}b^{n}q^{n(n+1)}(1-q)^{n}}{\left(aq,bq;q\right)_{n}}\left(\mathcal{D}_{q}\right)^{n}\left[w(x;aq^{n},bq^{n};q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;a,b;q)=\frac{\left(a^{-1}x,b^{-1}x;q\right)_{\infty}}{\left(x;q\right)_{\infty}}}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\left(bqt;q\right)_{\infty}\cdot\qHyperrphis{2}{1}@@{aqx^{-1},0}{aq}{q}{xt}=\sum_{n=0}^{\infty}\frac{\left(bq;q\right)_{n}}{\left(q;q\right)_{n}}P_{n}\!\left(x;a,b;q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\left(aqt;q\right)_{\infty}\cdot\qHyperrphis{2}{1}@@{bqx^{-1},0}{bq}{q}{xt}=\sum_{n=0}^{\infty}\frac{\left(aq;q\right)_{n}}{\left(q;q\right)_{n}}P_{n}\!\left(x;a,b;q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\left(t;q\right)_{\infty}\cdot\qHyperrphis{3}{2}@@{0,0,x}{aq,bq}{q}{t}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{n}{2}}}{\left(q;q\right)_{n}}P_{n}\!\left(x;a,b;q\right)t^{n}}}}$

Limit relations

Big q-Jacobi polynomial to Big q-Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,0,c;q\right)=P_{n}\!\left(x;a,c;q\right)}}}$

Big q-Laguerre polynomial to Little q-Laguerre / Wall polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{b\rightarrow-\infty}P_{n}\!\left(bqx;a,b;q\right)=p_{n}\!\left(x;a;q\right)}}}$

Big q-Laguerre polynomial to Al-Salam-Carlitz I polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow 0}\frac{P_{n}\!\left(aqx;a,ab;q\right)}{a^{n}}=q^{n}U^{(b)}_{n}\!\left(x;q\right)}}}$

Big q-Laguerre polynomial to Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P_{n}\!\left(x;q^{\alpha},(1-q\right)^{-1}q^{\beta};q)=\frac{L^{\alpha}_{n}\left(x-1\right)}{L^{\alpha}_{n}\left(0\right)}}}}$

Remark

${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}\!\left(q^{-x};p,N;q\right)=P_{n}\!\left(q^{-x};p,q^{-N-1};q\right)}}}$

<< Continuous q-Jacobi: Special cases

Big q-Laguerre

Little q-Jacobi >>