Askey-Wilson: Difference between revisions
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Latest revision as of 23:33, 5 March 2017
Askey-Wilson
Basic hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{a^n\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab,ac,ad}{q}{n}} {}=\qHyperrphis{4}{3}@@{q^{-n},abcdq^{n-1},a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{ab,ac,ad}{q}{q} }}
Orthogonality relation(s)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\AskeyWilson{m}@{x}{a}{b}{c}{d}{q}\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}\,dx=h_n\,\Kronecker{m}{n} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c,d|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta} c\expe^{\iunit\theta},d\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a)h(x,b)h(x,c)h(x,d)}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\AskeyWilson{m}@{x}{a}{b}{c}{d}{q}\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}\,dx {}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1<aq^k\leq a\end{array}} w_k\AskeyWilson{m}@{x_k}{a}{b}{c}{d}{q}\AskeyWilson{n}@{x_k}{a}{b}{c}{d}{q}=h_n\,\Kronecker{m}{n} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x_k=\frac{aq^k+\left(aq^k\right)^{-1}}{2}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h_n=\frac{\qPochhammer{abcdq^{n-1}}{q}{n}\qPochhammer{abcdq^{2n}}{q}{\infty}}{\qPochhammer{q^{n+1},abq^n,acq^n,adq^n,bcq^n,bdq^n,cdq^n}{q}{\infty}}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c,d|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta} c\expe^{\iunit\theta},d\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a)h(x,b)h(x,c)h(x,d)}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 2x\normAskeyWilsonptilde{n}@@{x}{a}{b}{c}{d}{q}=A_n\normAskeyWilsonptilde{n+1}@@{x}{a}{b}{c}{d}{q}+\left[a+a^{-1}-\left(A_n+C_n\right)\right]\normAskeyWilsonptilde{n}@@{x}{a}{b}{c}{d}{q}+C_n\normAskeyWilsonptilde{n-1}@@{x}{a}{b}{c}{d}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-abq^n)(1-acq^n)(1-adq^n)(1-abcdq^{n-1})}{a(1-abcdq^{2n-1})(1-abcdq^{2n})}}}
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}:=\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}} }}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q}=\monicAskeyWilson{n+1}@@{x}{a}{b}{c}{d}{q}+\frac{1}{2}\left[a+a^{-1}-(A_n+C_n)\right]\monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q}+ \frac{1}{4}A_{n-1}C_n\monicAskeyWilson{n-1}@@{x}{a}{b}{c}{d}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-abq^n)(1-acq^n)(1-adq^n)(1-abcdq^{n-1})}{a(1-abcdq^{2n-1})(1-abcdq^{2n})}}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AskeyWilson{n}@{x}{a}{b}{c}{d}{q}=2^n\qPochhammer{abcdq^{n-1}}{q}{n}\monicAskeyWilson{n}@@{x}{a}{b}{c}{d}{q} }}
q-Difference equation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-q)^2D_q\left[{\tilde w}(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}} cq^{\frac{1}{2}},dq^{\frac{1}{2}}|q)D_qy(x)\right] {}+\lambda_n{\tilde w}(x;a,b,c,d|q)y(x)=0 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda_n=4q^{-n+1}(1-q^n)(1-abcdq^{n-1})}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;a,b,c,d|q):=\frac{w(x;a,b,c,d|q)}{\sqrt{1-x^2}}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c,d|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta} c\expe^{\iunit\theta},d\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a)h(x,b)h(x,c)h(x,d)}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q^{-n}(1-q^n)(1-abcdq^{n-1}){\mathcal P}_n(z) {}=A(z){\mathcal P}_n(qz)-\left[A(z)+A(z^{-1})\right]{\mathcal P}_n(z)+A(z^{-1}){\mathcal P}_n(q^{-1}z) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\mathcal P}_n(z):=\frac{\qPochhammer{ab,ac,ad}{q}{n}}{a^n}\,\qHyperrphis{4}{3}@@{q^{-n},abcdq^{n-1},az,az^{-1}}{ab,ac,ad}{q}{q}}}
Forward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \delta_q \AskeyWilson{n}@{x}{a}{b}{c}{d}{q}=-q^{-\frac{1}{2}n}(1-q^n)(1-abcdq^{n-1})(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {} \AskeyWilson{n-1}@{x}{aq^{\frac{1}{2}}}{bq^{\frac{1}{2}}}{cq^{\frac{1}{2}}}{dq^{\frac{1}{2}}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D_q \AskeyWilson{n}@{x}{a}{b}{c}{d}{q}=2q^{-\frac{1}{2}(n-1)}\frac{(1-q^n)(1-abcdq^{n-1})}{1-q} {} \AskeyWilson{n-1}@{x}{aq^{\frac{1}{2}}}{bq^{\frac{1}{2}}}{cq^{\frac{1}{2}}}{dq^{\frac{1}{2}}}{q} }}
Backward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \delta_q\left[{\tilde w}(x;a,b,c,d|q)\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}\right] {}=q^{-\frac{1}{2}(n+1)}(\expe^{\iunit\theta}-\expe^{-\iunit\theta}){\tilde w}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{1}{2}}|q) {} \AskeyWilson{n+1}@{x}{aq^{-\frac{1}{2}}}{bq^{-\frac{1}{2}}}{cq^{-\frac{1}{2}}}{dq^{-\frac{1}{2}}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c,d|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta} c\expe^{\iunit\theta},d\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a)h(x,b)h(x,c)h(x,d)}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D_q\left[{\tilde w}(x;a,b,c,d|q)\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}\right] {}=-\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde w}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{1}{2}}|q) {} \AskeyWilson{n+1}@{x}{aq^{-\frac{1}{2}}}{bq^{-\frac{1}{2}}}{cq^{-\frac{1}{2}}}{dq^{-\frac{1}{2}}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c,d|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta} c\expe^{\iunit\theta},d\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a)h(x,b)h(x,c)h(x,d)}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Rodrigues-type formula
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;a,b,c,d|q)\AskeyWilson{n}@{x}{a}{b}{c}{d}{q} {}=\left(\frac{q-1}{2}\right)^nq^{\frac{1}{4}n(n-1)}\left(D_q\right)^n \left[{\tilde w}(x;aq^{\frac{1}{2}n},bq^{\frac{1}{2}n},cq^{\frac{1}{2}n},dq^{\frac{1}{2}n}|q)\right] }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c,d|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta} c\expe^{\iunit\theta},d\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a)h(x,b)h(x,c)h(x,d)}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}
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Generating functions
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{a\expe^{\iunit\theta},b\expe^{\iunit\theta}}{ab}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{c\expe^{-\iunit\theta},d\expe^{-\iunit\theta}}{cd}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab,cd,q}{q}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{a\expe^{\iunit\theta},c\expe^{\iunit\theta}}{ac}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{b\expe^{-\iunit\theta},d\expe^{-\iunit\theta}}{bd}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ac,bd,q}{q}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{a\expe^{\iunit\theta},d\expe^{\iunit\theta}}{ad}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{b\expe^{-\iunit\theta},c\expe^{-\iunit\theta}}{bc}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ad,bc,q}{q}{n}}t^n }}
Limit relations
Askey-Wilson polynomial to Continuous dual q-Hahn polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AskeyWilson{n}@{x}{a}{b}{c}{0}{q}=\ctsdualqHahn{n}@{x}{a}{b}{c}{q} }}
Askey-Wilson polynomial to Continuous q-Hahn polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AskeyWilson{n}@{\cos@{\theta+\phi}}{a\expe^{\iunit\phi}}{b\expe^{\iunit\phi}}{c\expe^{-\iunit\phi}}{d\expe^{-\iunit\phi}}{q}=\ctsqHahn{n}@{\cos@{\theta+\phi}}{a}{b}{c}{d}{q} }}
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} }}
Askey-Wilson polynomial to Continuous q-Jacobi polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}\AskeyWilson{n}@{x}{q^{\frac{1}{2}\alpha+\frac{1}{4}}}{q^{\frac{1}{2}\alpha+\frac{3}{4}}}{ -q^{\frac{1}{2}\beta+\frac{1}{4}}}{-q^{\frac{1}{2}\beta+\frac{3}{4}}}{q}} {\qPochhammer{q,-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{n}} =\ctsqJacobi{\alpha}{\beta}{n}@{x}{q} }}
Askey-Wilson polynomial to Continuous q-ultraspherical / Rogers polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{\beta^2}{q}{n}\AskeyWilson{n}@{x}{\beta^{\frac{1}{2}}}{\beta^{\frac{1}{2}}q^{\frac{1}{2}}}{ -\beta^{\frac{1}{2}}}{-\beta^{\frac{1}{2}}q^{\frac{1}{2}}}{q}} {\qPochhammer{\beta q^{\frac{1}{2}},-\beta,-\beta q^{\frac{1}{2}},q}{q}{n}}=\ctsqUltra{n}@{x}{\beta}{q} }}
Askey-Wilson polynomial to Wilson polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\frac{\AskeyWilson{n}@{\frac{1}{2}\left(q^{\iunit x}+q^{-\iunit x}\right)}{q^a}{q^b}{q^c}{q^d}{q}}{(1-q)^{3n}}=\Wilson{n}@{x^2}{a}{b}{c}{d} }}
Remarks
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q} {}=\frac{(\gamma\delta q)^{\frac{1}{2}n} \AskeyWilson{n}@{\nu(x)}{\gamma^{\frac{1}{2}}\delta^{\frac{1}{2}}q^{\frac{1}{2}}}{ \alpha\gamma^{-\frac{1}{2}}\delta^{-\frac{1}{2}}q^{\frac{1}{2}}}{ \beta\gamma^{-\frac{1}{2}}\delta^{\frac{1}{2}}q^{\frac{1}{2}}}{ \gamma^{\frac{1}{2}}\delta^{-\frac{1}{2}}q^{\frac{1}{2}}}{q}}{\qPochhammer{\alpha q,\beta\delta q,\gamma q}{q}{n}} }}
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^{-1}}=\normAskeyWilsonptilde{n}@{x}{a^{-1}}{b^{-1}}{c^{-1}}{d^{-1}}{q} }}