Formula:KLS:14.08:14

$\displaystyle {\displaystyle (1-q)^2D_q\left[{\tilde w}(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}}|q)D_qy(x)\right] {}+4q^{-n+1}(1-q^n){\tilde w}(x;a,b|q)y(x)=0 }$

Substitution(s)

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

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

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

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
$\displaystyle {\displaystyle Q_{n}}$  : Al-Salam-Chihara polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -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