# Formula:KLS:14.02:28

$\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{q^{-x},\beta\gamma^{-1}q^{-x}}{\beta\delta q}{q}{\gamma\delta q^{x+1}t}\ \qHyperrphis{2}{1}@@{\alpha q^{x+1},\gamma q^{x+1}}{\alpha\delta^{-1}q}{q}{q^{-x}t} {}=\sum_{n=0}^N\frac{\qPochhammer{\alpha q,\gamma q}{q}{n}}{\qPochhammer{\alpha\delta^{-1}q,q}{q}{n}} \qRacah{n}@{\mu(x)}{\alpha}{\beta}{\gamma}{\delta}{q}t^n {} }$

## Constraint(s)

$\displaystyle {\displaystyle \textrm{if}\quad\alpha q=q^{-N}\quad\textrm{or}\quad\gamma q=q^{-N}}$

## Substitution(s)

$\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}$ &

$\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}$ &
$\displaystyle {\displaystyle \mu(x):=q^{-x}+\gamma\delta q^{x+1}}$ &
$\displaystyle {\displaystyle \mu(x)=q^{-x}+\gamma\delta q^{x+1} =\lambda(x)=q^{-x}+cq^{x-N} =q^{-x}+q^{x+\gamma+\delta+1} =2a\cos@@{\theta}}$ &

$\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}$

## 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
$\displaystyle {\displaystyle {{}_{r}\phi_{s}}}$  : basic hypergeometric (or $\displaystyle {\displaystyle q}$ -hypergeometric) function : http://dlmf.nist.gov/17.4#E1
$\displaystyle {\displaystyle \Sigma}$  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
$\displaystyle {\displaystyle R_{n}}$  : $\displaystyle {\displaystyle q}$ -Racah polynomial : http://dlmf.nist.gov/18.28#E19
$\displaystyle {\displaystyle \mathrm{cos}}$  : cosine function : http://dlmf.nist.gov/4.14#E2