# Shift operators and Rodrigues-type formulas

## Shift operators and Rodrigues-type formulas

$\displaystyle {\displaystyle f'(x)=\frac{d}{dx}f(x)=\frac{df}{dx}(x)=\frac{df(x)}{dx} }$
$\displaystyle {\displaystyle \Delta f(x)=f(x+1)-f(x) }$
$\displaystyle {\displaystyle \nabla f(x)=f(x)-f(x-1) }$
$\displaystyle {\displaystyle \delta f(x)=f(x+\textstyle\frac{1}{2}\iunit)-f(x-\textstyle\frac{1}{2}\iunit) }$
$\displaystyle {\displaystyle \delta^2 f(x)=f(x+\iunit)-f(x)-f(x)+f(x-\iunit)=f(x+\iunit)-2f(x)+f(x-\iunit) }$
$\displaystyle {\displaystyle \delta x=x+\textstyle\frac{1}{2}\iunit-(x-\textstyle\frac{1}{2}\iunit)=\iunit \quad\textrm{and}\quad \delta x^2=(x+\textstyle\frac{1}{2}\iunit)^2-(x-\textstyle\frac{1}{2}\iunit)^2=2\iunit x }$
$\displaystyle {\displaystyle \Delta\lambda(x)=2x+\gamma+\delta+2\quad\textrm{and}\quad\nabla\lambda(x)=2x+\gamma+\delta }$
$\displaystyle {\displaystyle \Delta\mu(x)=q^{-x-1}(1-q)(1-\gamma\delta q^{2x+2})\quad\textrm{and}\quad \nabla\mu(x)=q^{-x}(1-q)(1-\gamma\delta q^{2x}) }$
$\displaystyle {\displaystyle \Delta\lambda(x)=q^{-x-1}(1-q)(1-cq^{2x-N+1})\quad\textrm{and}\quad \nabla\lambda(x)=q^{-x}(1-q)(1-cq^{2x-N-1}) }$
$\displaystyle {\displaystyle \Delta q^{-x}=q^{-x-1}(1-q)\quad\textrm{and}\quad\nabla q^{-x}=q^{-x}(1-q) }$
$\displaystyle {\displaystyle D_qf(x):=\frac{\delta_qf(x)}{\delta_qx} }$

Substitution(s): $\displaystyle {\displaystyle x=\cos@@{\theta}}$

$\displaystyle {\displaystyle \delta_qf(\expe^{\iunit\theta}) =f(q^{\frac{1}{2}}\expe^{\iunit\theta})-f(q^{-\frac{1}{2}}\expe^{\iunit\theta}) }$
$\displaystyle {\displaystyle \delta_qx=-\textstyle\frac{1}{2}q^{-\frac{1}{2}}(1-q)(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) }$

Substitution(s): $\displaystyle {\displaystyle x=\cos@@{\theta}}$