Shift operators and Rodrigues-type formulas

<< The q-derivative and q-integral

Shift operators and Rodrigues-type formulas

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Shift operators and Rodrigues-type formulas

${\displaystyle{\displaystyle{\displaystyle f^{\prime}(x)=\frac{d}{dx}f(x)=\frac{df}{dx}(x)=\frac{df(x)}{dx}}}}$
${\displaystyle{\displaystyle{\displaystyle\Delta f(x)=f(x+1)-f(x)}}}$
${\displaystyle{\displaystyle{\displaystyle\nabla f(x)=f(x)-f(x-1)}}}$
${\displaystyle{\displaystyle{\displaystyle\delta f(x)=f(x+\textstyle\frac{1}{2}\mathrm{i})-f(x-\textstyle\frac{1}{2}\mathrm{i})}}}$
${\displaystyle{\displaystyle{\displaystyle\delta^{2}f(x)=f(x+\mathrm{i})-f(x)-f(x)+f(x-\mathrm{i})=f(x+\mathrm{i})-2f(x)+f(x-\mathrm{i})}}}$
${\displaystyle{\displaystyle{\displaystyle\delta x=x+\textstyle\frac{1}{2}\mathrm{i}-(x-\textstyle\frac{1}{2}\mathrm{i})=\mathrm{i}\quad\textrm{and}\quad\delta x^{2}=(x+\textstyle\frac{1}{2}\mathrm{i})^{2}-(x-\textstyle\frac{1}{2}\mathrm{i})^{2}=2\mathrm{i}x}}}$
${\displaystyle{\displaystyle{\displaystyle\Delta\lambda(x)=2x+\gamma+\delta+2\quad\textrm{and}\quad\nabla\lambda(x)=2x+\gamma+\delta}}}$
${\displaystyle{\displaystyle{\displaystyle\Delta\mu(x)=q^{-x-1}(1-q)(1-\gamma\delta q^{2x+2})\quad\textrm{and}\quad\nabla\mu(x)=q^{-x}(1-q)(1-\gamma\delta q^{2x})}}}$
${\displaystyle{\displaystyle{\displaystyle\Delta\lambda(x)=q^{-x-1}(1-q)(1-cq^{2x-N+1})\quad\textrm{and}\quad\nabla\lambda(x)=q^{-x}(1-q)(1-cq^{2x-N-1})}}}$
${\displaystyle{\displaystyle{\displaystyle\Delta q^{-x}=q^{-x-1}(1-q)\quad\textrm{and}\quad\nabla q^{-x}=q^{-x}(1-q)}}}$
${\displaystyle{\displaystyle{\displaystyle D_{q}f(x):=\frac{\delta_{q}f(x)}{\delta_{q}x}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\delta_{q}f(\expe^{\iunit\theta})=f(q^{\frac{1}{2}}\expe^{\iunit\theta})-f(q^{-\frac{1}{2}}\expe^{\iunit\theta})}}}$
${\displaystyle{\displaystyle{\displaystyle\delta_{q}x=-\textstyle\frac{1}{2}q^{-\frac{1}{2}}(1-q)({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}})}}}$

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

<< The q-derivative and q-integral

Shift operators and Rodrigues-type formulas

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