Big q-Jacobi: Special case

From DRMF
Jump to navigation Jump to search

Big q-Jacobi: Special case

Big q-Legendre

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqLegendre{n}@{x}{c}{q}=\qHyperrphis{3}{2}@@{q^{-n},q^{n+1},x}{q,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}^{q}\bigqLegendre{m}@{x}{c}{q}\bigqLegendre{n}@{x}{c}{q}\,d_qx {}=q(1-c)\frac{(1-q)}{(1-q^{2n+1})} \frac{\qPochhammer{c^{-1}q}{q}{n}}{\qPochhammer{cq}{q}{n}}(-cq^2)^nq^{\binomial{n}{2}}\,\Kronecker{m}{n} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle c<0}}


Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (x-1)\bigqLegendre{n}@{x}{c}{q}=A_n\bigqLegendre{n+1}@{x}{c}{q}-\left(A_n+C_n\right)\bigqLegendre{n}@{x}{c}{q} {}+C_n\bigqLegendre{n-1}@{x}{c}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=-cq^{n+1}\frac{(1-q^n)(1-c^{-1}q^n)}{(1+q^n)(1-q^{2n+1})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-q^{n+1})(1-cq^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}}}


Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicbigqLegendre{n}@@{x}{c}{q}=\monicbigqLegendre{n+1}@@{x}{c}{q}+\left[1-(A_n+C_n)\right]\monicbigqLegendre{n}@@{x}{c}{q}+A_{n-1}C_n\monicbigqLegendre{n-1}@@{x}{c}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=-cq^{n+1}\frac{(1-q^n)(1-c^{-1}q^n)}{(1+q^n)(1-q^{2n+1})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-q^{n+1})(1-cq^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqLegendre{n}@{x}{c}{q}=\frac{\qPochhammer{q^{n+1}}{q}{n}}{\qPochhammer{q,cq}{q}{n}}\monicbigqLegendre{n}@@{x}{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-q^{n+1})x^2y(x) {}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D(x)=(x-q)(x-cq)}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=q(x-1)(x-c)}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\bigqLegendre{n}@{x}{c}{q}}}


Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqLegendre{n}@{x}{c}{q}=\frac{c^nq^{n(n+1)}(1-q)^n}{\qPochhammer{q,cq}{q}{n}} \left(\qderiv{q}\right)^n\left[\qPochhammer{q^{-n}x,c^{-1}q^{-n}x}{q}{n}\right] }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqLegendre{n}@{x}{c}{q}=\frac{(1-q)^n}{\qPochhammer{q,cq}{q}{n}}\left(\qderiv{q}\right)^n \left[\qPochhammer{qx^{-1},cqx^{-1}}{q}{n}x^{2n}\right] }}

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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{\qPochhammer{cq}{q}{n}}{\qPochhammer{q,q}{q}{n}}\bigqLegendre{n}@{x}{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}@@{c^{-1}x}{c^{-1}q}{q}{qt} =\sum_{n=0}^{\infty}\frac{\bigqLegendre{n}@{x}{c}{q}}{\qPochhammer{c^{-1}q}{q}{n}}t^n }}

Limit relations

Big q-Legendre polynomial to Legendre / Spherical polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\bigqLegendre{n}@{x}{0}{q}=\LegendrePoly{n}@{2x-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\bigqLegendre{n}@{x}{-q^{\gamma}}{q}=\LegendrePoly{n}@{x} }}

Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Big_q-Jacobi:_Special_case&oldid=505"