Formula:KLS:14.04:18

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formula in Continuous q-Hahn

Formula:KLS:14.04:19 >>

${\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;a,b,c,d;q)\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}\right]{}=q^{-\frac{1}{2}(n+1)}({\mathrm{e}^{\mathrm{i}(\theta+\phi)}}-{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}}){\tilde{w}}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{1}{2}};q){}\ctsqHahn{n+1}@{x}{aq^{-\frac{1}{2}}}{bq^{-\frac{1}{2}}}{cq^{-\frac{1}{2}}}{dq^{-\frac{1}{2}}}{q}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c,d;q):=\frac{w(x;a,b,c,d;q)}{\sqrt{1-x^{2}}}}}}$ &

${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c,d;q)=\left|\frac{\qPochhammer{\expe^{2\iunit(\theta+\phi)}}{q}{\infty}}{\qPochhammer{a\expe^{\iunit(\theta+2\phi)},b\expe^{\iunit(\theta+2\phi)}c\expe^{\iunit\theta},d\expe^{\iunit\theta}}{q}{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a\expe^{\iunit\phi})h(x,b\expe^{\iunit\phi})h(x,c\expe^{-\iunit\phi})h(x,d\expe^{-\iunit\phi})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\qPochhammer{\alpha\expe^{\iunit(\theta+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{q}{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqHahn
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\Pi}}}$ : product : http://drmf.wmflabs.org/wiki/Definition:prod
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.4 of KLS.

URL links

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formula in Continuous q-Hahn

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