# Formula:KLS:14.04:19

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

## Substitution(s)

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

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

$\displaystyle {\displaystyle x=\cos@{\theta+\phi}}$

## Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

## Symbols List

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