Big q-Jacobi
Big q-Jacobi
Basic hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobi{n}@{x}{a}{b}{c}{q}=\qHyperrphis{3}{2}@@{q^{-n},abq^{n+1},x}{aq,cq}{q}{q} }}
Orthogonality relation(s)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{cq}^{aq}\frac{\qPochhammer{a^{-1}x,c^{-1}x}{q}{\infty}}{\qPochhammer{x,bc^{-1}x}{q}{\infty}} \bigqJacobi{m}@{x}{a}{b}{c}{q}\bigqJacobi{n}@{x}{a}{b}{c}{q}\,d_qx {}=aq(1-q)\frac{\qPochhammer{q,abq^2,a^{-1}c,ac^{-1}q}{q}{\infty}}{\qPochhammer{aq,bq,cq,abc^{-1}q}{q}{\infty}} {}\frac{(1-abq)}{(1-abq^{2n+1})} \frac{\qPochhammer{q,bq,abc^{-1}q}{q}{n}}{\qPochhammer{aq,abq,cq}{q}{n}}(-acq^2)^nq^{\binomial{n}{2}}\,\Kronecker{m}{n} }}
Recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (x-1)\bigqJacobi{n}@{x}{a}{b}{c}{q}=A_n\bigqJacobi{n+1}@{x}{a}{b}{c}{q}-\left(A_n+C_n\right)\bigqJacobi{n}@{x}{a}{b}{c}{q} {}+C_n\bigqJacobi{n-1}@{x}{a}{b}{c}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicbigqJacobi{n}@@{x}{a}{b}{c}{q}=\monicbigqJacobi{n+1}@@{x}{a}{b}{c}{q}+\left[1-(A_n+C_n)\right]\monicbigqJacobi{n}@@{x}{a}{b}{c}{q}+A_{n-1}C_n\monicbigqJacobi{n-1}@@{x}{a}{b}{c}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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})}}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobi{n}@{x}{a}{b}{c}{q}=\frac{\qPochhammer{abq^{n+1}}{q}{n}}{\qPochhammer{aq,cq}{q}{n}}\monicbigqJacobi{n}@@{x}{a}{b}{c}{q} }}
q-Difference equation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q^{-n}(1-q^n)(1-abq^{n+1})x^2y(x) {}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=aq(x-1)(bx-c)}}
&
Forward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobi{n}@{x}{a}{b}{c}{q}-\bigqJacobi{n}@{qx}{a}{b}{c}{q} {}=\frac{q^{-n+1}(1-q^n)(1-abq^{n+1})}{(1-aq)(1-cq)}x\bigqJacobi{n-1}@{qx}{aq}{bq}{cq}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\bigqJacobi{n}@{x}{a}{b}{c}{q}=\frac{q^{-n+1}(1-q^n)(1-abq^{n+1})} {(1-q)(1-aq)(1-cq)}\bigqJacobi{n-1}@{qx}{aq}{bq}{cq}{q} }}
Backward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (x-a)(x-c)\bigqJacobi{n}@{x}{a}{b}{c}{q}-a(x-1)(bx-c)\bigqJacobi{n}@{qx}{a}{b}{c}{q} {}=(1-a)(1-c)x\bigqJacobi{n+1}@{x}{aq^{-1}}{bq^{-1}}{cq^{-1}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\left[w(x;a,b,c;q)\bigqJacobi{n}@{x}{a}{b}{c}{q}\right] {}=\frac{(1-a)(1-c)}{ac(1-q)}w(x;aq^{-1},bq^{-1},cq^{-1};q) {} \bigqJacobi{n+1}@{x}{aq^{-1}}{bq^{-1}}{cq^{-1}}{q} }}
Rodrigues-type formula
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;a,b,c;q)\bigqJacobi{n}@{x}{a}{b}{c}{q} {}=\frac{a^nc^nq^{n(n+1)}(1-q)^n} {\qPochhammer{aq,cq}{q}{n}}\left(\qderiv{q}\right)^n\left[w(x;aq^n,bq^n,cq^n;q)\right] }}
Generating functions
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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{\qPochhammer{cq}{q}{n}}{\qPochhammer{bq,q}{q}{n}}\bigqJacobi{n}@{x}{a}{b}{c}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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{\qPochhammer{aq}{q}{n}}{\qPochhammer{abc^{-1}q,q}{q}{n}}\bigqJacobi{n}@{x}{a}{b}{c}{q}t^n }}
Limit relations
Askey-Wilson polynomial to Big q-Jacobi polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \normAskeyWilsonptilde{n}@{x}{a}{b}{c}{d}{q}=\frac{a^n\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab,ac,ad}{q}{n}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{a\rightarrow 0}\normAskeyWilsonptilde{n}@{\textstyle\frac{1}{2}a^{-1}x}{a}{a^{-1}\alpha q}{ a^{-1}\gamma q}{a\beta \gamma^{-1}}{q}=\bigqJacobi{n}@{x}{\alpha}{\beta}{\gamma}{q} }}
q-Racah polynomial to Big q-Jacobi polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{a}{b}{c}{0}{q}=\bigqJacobi{n}@{q^{-x}}{a}{b}{c}{q} }}
Big q-Jacobi polynomial to Big q-Laguerre polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobi{n}@{x}{a}{0}{c}{q}=\bigqLaguerre{n}@{x}{a}{c}{q} }}
Big q-Jacobi polynomial to Little q-Jacobi polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{c\rightarrow -\infty}\bigqJacobi{n}@{cqx}{a}{b}{c}{q}=\littleqJacobi{n}@{x}{a}{b}{q} }}
Big q-Jacobi polynomial to q-Meixner polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{c\rightarrow -\infty}\bigqJacobi{n}@{q^{-x}}{a}{-a^{-1}cd^{-1}}{c}{q}=\qMeixner{n}@{q^{-x}}{a}{d}{q} }}
Big q-Jacobi polynomial to Jacobi polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\bigqJacobi{n}@{x}{q^{\alpha}}{q^{\beta}}{0}{q}=\frac{\Jacobi{\alpha}{\beta}{n}@{2x-1}}{\Jacobi{\alpha}{\beta}{n}@{1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\bigqJacobi{n}@{x}{q^{\alpha}}{q^{\beta}}{-q^{\gamma}}{q}=\frac{\Jacobi{\alpha}{\beta}{n}@{x}}{\Jacobi{\alpha}{\beta}{n}@{1}} }}
Remarks
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobi{n}@{x}{a}{b}{0}{q}=\frac{\qPochhammer{bq}{q}{n}}{\qPochhammer{aq}{q}{n}}(-1)^na^nq^{n+\binomial{n}{2}}\littleqJacobi{n}@{a^{-1}q^{-1}x}{b}{a}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q}=\qHyperrphis{3}{2}@@{q^{-n},abq^{n+1},ac^{-1}qx}{aq,-ac^{-1}dq}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{c^{-1}qx,-d^{-1}qx}{q}{\infty}}{\qPochhammer{ac^{-1}qx,-bd^{-1}qx}{q}{\infty}}d_qx }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q}=\bigqJacobi{n}@{ac^{-1}qx}{a}{b}{-ac^{-1}d}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobi{n}@{x}{a}{b}{c}{q}=\bigqJacobiIVparam{n}@{x}{a}{b}{aq}{-cq}{q} }}
Koornwinder Addendum: Big q-Jacobi
Different notation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q}:=\bigqJacobi{n}@{qac^{-1}x}{a}{b}{-ac^{-1}d}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q} =\qHyperrphis{3}{2}@@{q^{-n},q^{n+1}ab,qac^{-1}x}{qa,-qac^{-1}d}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q}=\bigqJacobiIVparam{n}@{\lambda x}{a}{b}{\lambda c}{\lambda d}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobi{n}@{x}{a}{b}{c}{q}=\bigqJacobiIVparam{n}@{-q^{-1}c^{-1}x}{a}{b}{-ac^{-1}}{1}{q} }}
Orthogonality relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-d}^c \bigqJacobiIVparam{m}@{x}{a}{b}{c}{d}{q} \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q} \frac{\qPochhammer{qx/c,-qx/d}{q}{\infty}}{\qPochhammer{qax/c,-qbx/d}{q}{\infty}} d_qx=h_n \Kronecker{m}{n} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h_0 =(1-q)z_+ \frac{\qPochhammer{q,a/c,a/d,b/c,b/d}{q}{\infty}}{\qPochhammer{ab/(qcd)}{q}{\infty}} \frac{\theta(z_-/z_+,cdz_-z_+;q)}{\theta(cz_-,dz_-,cz_+,dz_+;q)}}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{h_n}{h_0}=q^{\frac12 n(n-1)}\left(\frac{q^2a^2d}c\right)^n \frac{1-qab}{1-q^{2n+1}ab} \frac{\qPochhammer{q,qb,-qbc/d}{q}{n}}{\qPochhammer{qa,qab,-qad/c}{q}{n}} =(-1)^n\left(\frac{c^2}{ab}\right)^n q^{\frac12 n(n-1)} q^{2n} \frac{\qPochhammer{q,qd/a,qd/b}{q}{n}}{\qPochhammer{qcd/(ab),qc/a,qc/b}{q}{n}} \frac{1-qcd/(ab)}{1-q^{2n+1}cd/(ab)} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h_0 =(1-q)z_+ \frac{\qPochhammer{q,a/c,a/d,b/c,b/d}{q}{\infty}}{\qPochhammer{ab/(qcd)}{q}{\infty}} \frac{\theta(z_-/z_+,cdz_-z_+;q)}{\theta(cz_-,dz_-,cz_+,dz_+;q)}}}
Other hypergeometric representation and asymptotics
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q} =\frac{\qPochhammer{-qbd^{-1}x}{q}{n}}{\qPochhammer{-q^{-n}a^{-1}cd^{-1}}{q}{n}} \qHyperrphis{3}{2}@@{q^{-n},q^{-n}b^{-1},cx^{-1}}{qa,-q^{-n}b^{-1}dx^{-1}}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q} =(qac^{-1}x)^n \frac{\qPochhammer{qb,cx^{-1}}{q}{n}}{\qPochhammer{qa,-qac^{-1}d}{q}{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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q} =(qac^{-1}x)^n \frac{\qPochhammer{qb,q}{q}{n}}{\qPochhammer{-qac^{-1}d}{q}{n}} \sum_{k=0}^n\frac{\qPochhammer{cx^{-1}}{q}{n-k}}{\qPochhammer{q,qa}{q}{n-k}} \frac{\qPochhammer{-qbd^{-1}x}{q}{k}}{\qPochhammer{qb,q}{q}{k}} (-1)^k q^{\frac12 k(k-1)}(-dx^{-1})^k }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{n\to\infty}(qac^{-1}x)^{-n} \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q} =\frac{\qPochhammer{cx^{-1},-dx^{-1}}{q}{\infty}}{\qPochhammer{-qac^{-1}d,qa}{q}{\infty}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{n\to\infty}(qac^{-1}x)^{-n} \bigqJacobiIVparam{n}@{x}{a}{b}{c}{d}{q} =\frac{\qPochhammer{qb,cx^{-1}}{q}{n}}{\qPochhammer{qa,-qac^{-1}d}{q}{n}} \qHyperrphis{1}{1}@@{-qbd^{-1}x}{qb}{q}{-dx^{-1}} }}
Symmetry
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\bigqJacobiIVparam{n}@{-x}{a}{b}{c}{d}{q}}{\bigqJacobiIVparam{n}@{-d/(qb)}{a}{b}{c}{d}{q}} =\bigqJacobiIVparam{n}@{x}{b}{a}{d}{c}{q} }}
Big q-Jacobi: Special values
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{c/(qa)}{a}{b}{c}{d}{q}=1 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{-d/(qb)}{a}{b}{c}{d}{q}=\left(- \frac{ad}{bc}\right)^n \frac{\qPochhammer{qb,-qbc/d}{q}{n}}{\qPochhammer{qa,-qad/c}{q}{n}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{c}{a}{b}{c}{d}{q}= q^{\frac12 n(n+1)}\left(\frac{ad}c\right)^n \frac{\qPochhammer{-qbc/d}{q}{n}}{\qPochhammer{-qad/c}{q}{n}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{-d}{a}{b}{c}{d}{q}=q^{\frac12 n(n+1)} (-a)^n \frac{\qPochhammer{qb}{q}{n}}{\qPochhammer{qa}{q}{n}} }}
Quadratic transformations
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{2n}@{x}{a}{a}{1}{1}{q}=\frac{\littleqJacobi{n}@{x^2}{q^{-1}}{a^2}{q^2}}{\littleqJacobi{n}@{(qa)^{-2}}{q^{-1}}{a^2}{q^2}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{2n+1}@{x}{a}{a}{1}{1}{q}=\frac{qax \littleqJacobi{n}@{x^2}{q}{a^2}{q^2}}{\littleqJacobi{n}@{(qa)^{-2}}{q}{a^2}{q^2}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{x}{a}{a}{1}{1}{q}=\frac{\qPochhammer{qa^2}{q^2}{n}}{\qPochhammer{qa^2}{q}{n}} (qax)^n \qHyperrphis{2}{1}@@{q^{-n},q^{-n+1}}{q^{-2n+1}a^{-2}}{q^2}{(ax)^{-2}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobiIVparam{n}@{x}{a}{a}{1}{1}{q} =\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{qa^2}{q}{n}} (qa)^n \sum_{k=0}^{[\frac12 n]}(-1)^k q^{k(k-1)} \frac{\qPochhammer{qa^2}{q^2}{n-k}}{\qPochhammer{q^2}{q^2}{k} \qPochhammer{q}{q}{n-2k}} x^{n-2k} }}
q-Chebyshev polynomials
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \CiglerqChebyU{n}@{x}{q}{b}=(q^{-3} b)^{\frac12 n} \frac{1-q^{n+1}}{1-q} \bigqJacobiIVparam{n}@{b^{-\frac12}x}{q^\frac12}{q^\frac12}{1}{1}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \CiglerqChebyT{n}@{x}{s}{q}=(-s)^{\half n} \bigqJacobiIVparam{n}@{(-qs)^{-\half} x}{q^{-\half}}{q^{-\half}}{1}{1}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \CiglerqChebyU{n}@{x}{s}{q}=(-q^{-2}s)^{\half n} \frac{1-q^{n+1}}{1-q} \bigqJacobiIVparam{n}@{(-qs)^{-\half} x}{q^{\half}}{q^{\half}}{1}{1}{q} }}
Limit to Discrete q-Hermite I
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{a\to0} a^{-n} \bigqJacobiIVparam{n}@{x}{a}{a}{1}{1}{q}=q^n \discrqHermiteI{n}@{x}{q} }}
Pseudo big q-Jacobi polynomials
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{z_- q^\mathbb{Z}\cup z_+ q^\mathbb{Z}}\bigqJacobi{m}@{cx}{c/b}{d/a}{c/a}{q} \bigqJacobi{n}@{cx}{c/b}{d/a}{c/a}{q} \frac{\qPochhammer{ax,bx}{q}{\infty}}{\qPochhammer{cx,dx}{q}{\infty}} d_qx=h_n\Kronecker{m}{n} (m,n=0,1,\ldots,N) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h_0 =(1-q)z_+ \frac{\qPochhammer{q,a/c,a/d,b/c,b/d}{q}{\infty}}{\qPochhammer{ab/(qcd)}{q}{\infty}} \frac{\theta(z_-/z_+,cdz_-z_+;q)}{\theta(cz_-,dz_-,cz_+,dz_+;q)}}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h_0=\int_{z_- q^\mathbb{Z}\cup z_+ q^\mathbb{Z}}\frac{\qPochhammer{ax,bx}{q}{\infty}}{\qPochhammer{cx,dx}{q}{\infty}} d_qx }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h_0 =(1-q)z_+ \frac{\qPochhammer{q,a/c,a/d,b/c,b/d}{q}{\infty}}{\qPochhammer{ab/(qcd)}{q}{\infty}} \frac{\theta(z_-/z_+,cdz_-z_+;q)}{\theta(cz_-,dz_-,cz_+,dz_+;q)}}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobi{n}@{cx}{c/b}{d/a}{c/a}{q}=\bigqJacobiIVparam{n}@{-q^{-1}ax}{c/b}{d/a}{-a/b}{1}{q} }}
Pseudo big q-Jacobi \longrightarrow Discrete Hermite II
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{a\to\infty}\iunit^n q^{\frac12 n(n-1)} \bigqJacobiIVparam{n}@{q^{-1}a^{-1}\iunit x}{a}{a}{1}{1}{q}= \discrqHermiteII{n}@{x}{q} }}
Pseudo big q-Jacobi \longrightarrow Pseudo Jacobi
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\uparrow1}\bigqJacobi{n}@{\iunit q^{\frac12(-N-1+\iunit\nu)}x}{-q^{-N-1}}{-q^{-N-1}}{q^{-N+\iunit\nu-1}}{q} =\frac{\pseudoJacobi{n}@{x}{\nu}{N}}{\pseudoJacobi{n}@{-\iunit}{\nu}{N}} }}