Big q-Jacobi

<< Continuous q-Hahn

Big q-Jacobi

Big q-Jacobi: Special case >>

Basic hypergeometric representation

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

Orthogonality relation(s)

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

Recurrence relation

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=-acq^{n+1}\frac{(1-q^{n})(1-abc^{-1}q^{n})(1-bq^{n})}{(1-abq^{2n})(1-abq^{2n+1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-aq^{n+1})(1-abq^{n+1})(1-cq^{n+1})}{(1-abq^{2n+1})(1-abq^{2n+2})}}}}$

Monic recurrence relation

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=-acq^{n+1}\frac{(1-q^{n})(1-abc^{-1}q^{n})(1-bq^{n})}{(1-abq^{2n})(1-abq^{2n+1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-aq^{n+1})(1-abq^{n+1})(1-cq^{n+1})}{(1-abq^{2n+1})(1-abq^{2n+2})}}}}$

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

q-Difference equation

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

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

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

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

Forward shift operator

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

Backward shift operator

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

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

Rodrigues-type formula

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

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

Generating functions

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{aqx^{-1},0}{aq}{q}{xt}\,\qHyperrphis{1}{1}@@{bc^{-1}x}{bq}{q}{cqt}{}=\sum_{n=0}^{\infty}\frac{\left(cq;q\right)_{n}}{\left(bq,q;q\right)_{n}}P_{n}\!\left(x;a,b,c;q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{cqx^{-1},0}{cq}{q}{xt}\,\qHyperrphis{1}{1}@@{bc^{-1}x}{abc^{-1}q}{q}{aqt}{}=\sum_{n=0}^{\infty}\frac{\left(aq;q\right)_{n}}{\left(abc^{-1}q,q;q\right)_{n}}P_{n}\!\left(x;a,b,c;q\right)t^{n}}}}$

Limit relations

Askey-Wilson polynomial to Big q-Jacobi polynomial

${\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}\!\left(x;a,b,c,d\,|\,q\right)=\frac{a^{n}p_{n}\!\left(x;a,b,c,d\,|\,q\right)}{\left(ab,ac,ad;q\right)_{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow 0}{\tilde{p}}_{n}\!\left(\textstyle\frac{1}{2}a^{-1}x;a,a^{-1}\alpha q,a^{-1}\gamma q,a\beta\gamma^{-1}\,|\,q\right)=P_{n}\!\left(x;\alpha,\beta,\gamma;q\right)}}}$

q-Racah polynomial to Big q-Jacobi polynomial

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);a,b,c,0\,|\,q\right)=P_{n}\!\left(q^{-x};a,b,c;q\right)}}}$

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-Jacobi polynomial to Little q-Jacobi polynomial

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

Big q-Jacobi polynomial to q-Meixner polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow-\infty}P_{n}\!\left(q^{-x};a,-a^{-1}cd^{-1},c;q\right)=M_{n}\!\left(q^{-x};a,d;q\right)}}}$

Big q-Jacobi polynomial to Jacobi polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P_{n}\!\left(x;q^{\alpha},q^{\beta},0;q\right)=\frac{P^{(\alpha,\beta)}_{n}\left(2x-1\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P_{n}\!\left(x;q^{\alpha},q^{\beta},-q^{\gamma};q\right)=\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,0;q\right)=\frac{\left(bq;q\right)_{n}}{\left(aq;q\right)_{n}}(-1)^{n}a^{n}q^{n+\binomial{n}{2}}p_{n}\!\left(a^{-1}q^{-1}x;b,a;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,c,d;q\right)=\qHyperrphis{3}{2}@@{q^{-n},abq^{n+1},ac^{-1}qx}{aq,-ac^{-1}dq}{q}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\left(c^{-1}qx,-d^{-1}qx;q\right)_{\infty}}{\left(ac^{-1}qx,-bd^{-1}qx;q\right)_{\infty}}d_{q}x}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,c,d;q\right)=P_{n}\!\left(ac^{-1}qx;a,b,-ac^{-1}d;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,c;q\right)=P_{n}\!\left(x;a,b,aq,-cq;q\right)}}}$

Koornwinder Addendum: Big q-Jacobi

Different notation

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,c,d;q\right):=P_{n}\!\left(qac^{-1}x;a,b,-ac^{-1}d;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,c,d;q\right)=\qHyperrphis{3}{2}@@{q^{-n},q^{n+1}ab,qac^{-1}x}{qa,-qac^{-1}d}{q}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,c,d;q\right)=P_{n}\!\left(\lambda x;a,b,\lambda c,\lambda d;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,c;q\right)=P_{n}\!\left(-q^{-1}c^{-1}x;a,b,-ac^{-1},1;q\right)}}}$

Orthogonality relation

${\displaystyle{\displaystyle{\displaystyle\int_{-d}^{c}P_{m}\!\left(x;a,b,c,d;q\right)P_{n}\!\left(x;a,b,c,d;q\right)\frac{\left(qx/c,-qx/d;q\right)_{\infty}}{\left(qax/c,-qbx/d;q\right)_{\infty}}d_{q}x=h_{n}\delta_{m,n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)c\frac{\left(q,-d/c,-qc/d,q^{2}ab;q\right)_{\infty}}{\left(qa,qb,-qbc/d,-qad/c;q\right)_{\infty}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)z_{+}\frac{\left(q,a/c,a/d,b/c,b/d;q\right)_{\infty}}{\left(ab/(qcd);q\right)_{\infty}}\frac{\theta(z_{-}/z_{+},cdz_{-}z_{+};q)}{\theta(cz_{-},dz_{-},cz_{+},dz_{+};q)}}}}$

${\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=q^{\frac{1}{2}n(n-1)}\left(\frac{q^{2}a^{2}d}{c}\right)^{n}\frac{1-qab}{1-q^{2n+1}ab}\frac{\left(q,qb,-qbc/d;q\right)_{n}}{\left(qa,qab,-qad/c;q\right)_{n}}=(-1)^{n}\left(\frac{c^{2}}{ab}\right)^{n}q^{\frac{1}{2}n(n-1)}q^{2n}\frac{\left(q,qd/a,qd/b;q\right)_{n}}{\left(qcd/(ab),qc/a,qc/b;q\right)_{n}}\frac{1-qcd/(ab)}{1-q^{2n+1}cd/(ab)}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)c\frac{\left(q,-d/c,-qc/d,q^{2}ab;q\right)_{\infty}}{\left(qa,qb,-qbc/d,-qad/c;q\right)_{\infty}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)z_{+}\frac{\left(q,a/c,a/d,b/c,b/d;q\right)_{\infty}}{\left(ab/(qcd);q\right)_{\infty}}\frac{\theta(z_{-}/z_{+},cdz_{-}z_{+};q)}{\theta(cz_{-},dz_{-},cz_{+},dz_{+};q)}}}}$

Other hypergeometric representation and asymptotics

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,c,d;q\right)=\frac{\left(-qbd^{-1}x;q\right)_{n}}{\left(-q^{-n}a^{-1}cd^{-1};q\right)_{n}}\qHyperrphis{3}{2}@@{q^{-n},q^{-n}b^{-1},cx^{-1}}{qa,-q^{-n}b^{-1}dx^{-1}}{q}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,c,d;q\right)=(qac^{-1}x)^{n}\frac{\left(qb,cx^{-1};q\right)_{n}}{\left(qa,-qac^{-1}d;q\right)_{n}}\qHyperrphis{3}{2}@@{q^{-n},q^{-n}a^{-1},-qbd^{-1}x}{qb,q^{1-n}c^{-1}x}{q}{-q^{n+1}ac^{-1}d}}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,c,d;q\right)=(qac^{-1}x)^{n}\frac{\left(qb,q;q\right)_{n}}{\left(-qac^{-1}d;q\right)_{n}}\sum_{k=0}^{n}\frac{\left(cx^{-1};q\right)_{n-k}}{\left(q,qa;q\right)_{n-k}}\frac{\left(-qbd^{-1}x;q\right)_{k}}{\left(qb,q;q\right)_{k}}(-1)^{k}q^{\frac{1}{2}k(k-1)}(-dx^{-1})^{k}}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{n\to\infty}(qac^{-1}x)^{-n}P_{n}\!\left(x;a,b,c,d;q\right)=\frac{\left(cx^{-1},-dx^{-1};q\right)_{\infty}}{\left(-qac^{-1}d,qa;q\right)_{\infty}}}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{n\to\infty}(qac^{-1}x)^{-n}P_{n}\!\left(x;a,b,c,d;q\right)=\frac{\left(qb,cx^{-1};q\right)_{n}}{\left(qa,-qac^{-1}d;q\right)_{n}}\qHyperrphis{1}{1}@@{-qbd^{-1}x}{qb}{q}{-dx^{-1}}}}}$

Symmetry

${\displaystyle{\displaystyle{\displaystyle\frac{P_{n}\!\left(-x;a,b,c,d;q\right)}{P_{n}\!\left(-d/(qb);a,b,c,d;q\right)}=P_{n}\!\left(x;b,a,d,c;q\right)}}}$

Big q-Jacobi: Special values

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(c/(qa);a,b,c,d;q\right)=1}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(-d/(qb);a,b,c,d;q\right)=\left(-\frac{ad}{bc}\right)^{n}\frac{\left(qb,-qbc/d;q\right)_{n}}{\left(qa,-qad/c;q\right)_{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(c;a,b,c,d;q\right)=q^{\frac{1}{2}n(n+1)}\left(\frac{ad}{c}\right)^{n}\frac{\left(-qbc/d;q\right)_{n}}{\left(-qad/c;q\right)_{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(-d;a,b,c,d;q\right)=q^{\frac{1}{2}n(n+1)}(-a)^{n}\frac{\left(qb;q\right)_{n}}{\left(qa;q\right)_{n}}}}}$

Quadratic transformations

${\displaystyle{\displaystyle{\displaystyle P_{2n}\!\left(x;a,a,1,1;q\right)=\frac{p_{n}\!\left(x^{2};q^{-1},a^{2};q^{2}\right)}{p_{n}\!\left((qa)^{-2};q^{-1},a^{2};q^{2}\right)}}}}$
${\displaystyle{\displaystyle{\displaystyle P_{2n+1}\!\left(x;a,a,1,1;q\right)=\frac{qaxp_{n}\!\left(x^{2};q,a^{2};q^{2}\right)}{p_{n}\!\left((qa)^{-2};q,a^{2};q^{2}\right)}}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,a,1,1;q\right)=\frac{\left(qa^{2};q^{2}\right)_{n}}{\left(qa^{2};q\right)_{n}}(qax)^{n}\qHyperrphis{2}{1}@@{q^{-n},q^{-n+1}}{q^{-2n+1}a^{-2}}{q^{2}}{(ax)^{-2}}}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,a,1,1;q\right)=\frac{\left(q;q\right)_{n}}{\left(qa^{2};q\right)_{n}}(qa)^{n}\sum_{k=0}^{[\frac{1}{2}n]}(-1)^{k}q^{k(k-1)}\frac{\left(qa^{2};q^{2}\right)_{n-k}}{\left(q^{2};q^{2}\right)_{k}\left(q;q\right)_{n-2k}}x^{n-2k}}}}$

q-Chebyshev polynomials

${\displaystyle{\displaystyle{\displaystyle U_{n}\!\left(x,q,b\right)=(q^{-3}b)^{\frac{1}{2}n}\frac{1-q^{n+1}}{1-q}P_{n}\!\left(b^{-\frac{1}{2}}x;q^{\frac{1}{2}},q^{\frac{1}{2}},1,1;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle T_{n}\!\left(x,s,q\right)=(-s)^{\frac{1}{2}n}P_{n}\!\left((-qs)^{-\frac{1}{2}}x;q^{-\frac{1}{2}},q^{-\frac{1}{2}},1,1;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle U_{n}\!\left(x,s,q\right)=(-q^{-2}s)^{\frac{1}{2}n}\frac{1-q^{n+1}}{1-q}P_{n}\!\left((-qs)^{-\frac{1}{2}}x;q^{\frac{1}{2}},q^{\frac{1}{2}},1,1;q\right)}}}$

Limit to Discrete q-Hermite I

${\displaystyle{\displaystyle{\displaystyle\lim_{a\to 0}a^{-n}P_{n}\!\left(x;a,a,1,1;q\right)=q^{n}h_{n}\!\left(x;q\right)}}}$

Pseudo big q-Jacobi polynomials

${\displaystyle{\displaystyle{\displaystyle\int_{z_{-}q^{\mathbb{Z}}\cup z_{+}q^{\mathbb{Z}}}P_{m}\!\left(cx;c/b,d/a,c/a;q\right)P_{n}\!\left(cx;c/b,d/a,c/a;q\right)\frac{\left(ax,bx;q\right)_{\infty}}{\left(cx,dx;q\right)_{\infty}}d_{q}x=h_{n}\delta_{m,n}(m,n=0,1,\ldots,N)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)c\frac{\left(q,-d/c,-qc/d,q^{2}ab;q\right)_{\infty}}{\left(qa,qb,-qbc/d,-qad/c;q\right)_{\infty}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)z_{+}\frac{\left(q,a/c,a/d,b/c,b/d;q\right)_{\infty}}{\left(ab/(qcd);q\right)_{\infty}}\frac{\theta(z_{-}/z_{+},cdz_{-}z_{+};q)}{\theta(cz_{-},dz_{-},cz_{+},dz_{+};q)}}}}$

${\displaystyle{\displaystyle{\displaystyle h_{0}=\int_{z_{-}q^{\mathbb{Z}}\cup z_{+}q^{\mathbb{Z}}}\frac{\left(ax,bx;q\right)_{\infty}}{\left(cx,dx;q\right)_{\infty}}d_{q}x}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)c\frac{\left(q,-d/c,-qc/d,q^{2}ab;q\right)_{\infty}}{\left(qa,qb,-qbc/d,-qad/c;q\right)_{\infty}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)z_{+}\frac{\left(q,a/c,a/d,b/c,b/d;q\right)_{\infty}}{\left(ab/(qcd);q\right)_{\infty}}\frac{\theta(z_{-}/z_{+},cdz_{-}z_{+};q)}{\theta(cz_{-},dz_{-},cz_{+},dz_{+};q)}}}}$

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(cx;c/b,d/a,c/a;q\right)=P_{n}\!\left(-q^{-1}ax;c/b,d/a,-a/b,1;q\right)}}}$

Pseudo big q-Jacobi \longrightarrow Discrete Hermite II

${\displaystyle{\displaystyle{\displaystyle\lim_{a\to\infty}{\mathrm{i}^{n}}q^{\frac{1}{2}n(n-1)}P_{n}\!\left(q^{-1}a^{-1}\mathrm{i}x;a,a,1,1;q\right)=\tilde{h}_{n}\!\left(x;q\right)}}}$

Pseudo big q-Jacobi \longrightarrow Pseudo Jacobi

${\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}P_{n}\!\left(\mathrm{i}q^{\frac{1}{2}(-N-1+\mathrm{i}\nu)}x;-q^{-N-1},-q^{-N-1},q^{-N+\mathrm{i}\nu-1};q\right)=\frac{P_{n}\!\left(x;\nu,N\right)}{P_{n}\!\left(-\mathrm{i};\nu,N\right)}}}}$

<< Continuous q-Hahn

Big q-Jacobi

Big q-Jacobi: Special case >>