# Formula:KLS:14.10:12

$\displaystyle {\displaystyle x\monicctsqJacobi{\alpha}{\beta}{n}@@{x}{q}=\monicctsqJacobi{\alpha}{\beta}{n+1}@@{x}{q}+\frac{1}{2}\left[q^{\frac{1}{2}\alpha+\frac{1}{4}}+ q^{-\frac{1}{2}\alpha-\frac{1}{4}}-(A_n+C_n)\right]\monicctsqJacobi{\alpha}{\beta}{n}@@{x}{q} {}+\frac{1}{4}A_{n-1}C_n\monicctsqJacobi{\alpha}{\beta}{n-1}@@{x}{q} }$

## Substitution(s)

$\displaystyle {\displaystyle C_n=\frac{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^n)(1-q^{n+\beta})(1+q^{n+\frac{1}{2}(\alpha+\beta)})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})} {(1-q^{2n+\alpha+\beta})(1-q^{2n+\alpha+\beta+1})}}$ &
$\displaystyle {\displaystyle A_n=\frac{(1-q^{n+\alpha+1})(1-q^{n+\alpha+\beta+1})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})(1+q^{n+\frac{1}{2}(\alpha+\beta+2)})} {q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{2n+\alpha+\beta+1})(1-q^{2n+\alpha+\beta+2})}}$

## Proof

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## Symbols List

& : logical and
$\displaystyle {\displaystyle {\widehat P}^{(\alpha,\beta)}_{n}}$  : monic continuous $\displaystyle {\displaystyle q}$ -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi