# Formula:KLS:14.04:11

$\displaystyle {\displaystyle x\monicctsqHahn{n}@@{x}{a}{b}{c}{d}{q}=\monicctsqHahn{n+1}@@{x}{a}{b}{c}{d}{q}+\frac{1}{2}\left[a\expe^{\iunit\phi}+a^{-1}\expe^{-\iunit\phi}-(A_n+C_n)\right]\monicctsqHahn{n}@@{x}{a}{b}{c}{d}{q} {}+\frac{1}{4}A_{n-1}C_n\monicctsqHahn{n-1}@@{x}{a}{b}{c}{d}{q} }$

## Substitution(s)

$\displaystyle {\displaystyle C_n=\frac{a\expe^{\iunit\phi}(1-q^n)(1-bcq^{n-1})(1-bdq^{n-1})(1-cd\expe^{-2\iunit\phi}q^{n-1})}{(1-abcdq^{2n-2})(1-abcdq^{2n-1})}}$ &
$\displaystyle {\displaystyle A_n=\frac{(1-ab\expe^{2\iunit\phi}q^n)(1-acq^n)(1-adq^n)(1-abcdq^{n-1})}{a\expe^{\iunit\phi}(1-abcdq^{2n-1})(1-abcdq^{2n})}}$

## 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
$\displaystyle {\displaystyle {\widehat p}_{n}}$  : monic continuous $\displaystyle {\displaystyle q}$ -Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:monicctsqHahn
$\displaystyle {\displaystyle \mathrm{e}}$  : 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