f Ⲡ⢠( x ) = d d ⢠x ⢠f ⢠( x ) = d ⢠f d ⢠x ⢠( x ) = d ⢠f ⢠( x ) d ⢠x superscript ð â² ð¥ ð ð ð¥ ð ð¥ ð ð ð ð¥ ð¥ ð ð ð¥ ð ð¥ {\displaystyle{\displaystyle{\displaystyle f^{\prime}(x)=\frac{d}{dx}f(x)=% \frac{df}{dx}(x)=\frac{df(x)}{dx}}}} {\displaystyle f'(x)=\frac{d}{dx}f(x)=\frac{df}{dx}(x)=\frac{df(x)}{dx} } Π⢠f ⢠( x ) = f ⢠( x + 1 ) - f ⢠( x ) Î ð ð¥ ð ð¥ 1 ð ð¥ {\displaystyle{\displaystyle{\displaystyle\Delta f(x)=f(x+1)-f(x)}}} {\displaystyle \Delta f(x)=f(x+1)-f(x) } â â¡ f ⢠( x ) = f ⢠( x ) - f ⢠( x - 1 ) â ð ð¥ ð ð¥ ð ð¥ 1 {\displaystyle{\displaystyle{\displaystyle\nabla f(x)=f(x)-f(x-1)}}} {\displaystyle \nabla f(x)=f(x)-f(x-1) } δ ⢠f ⢠( x ) = f ⢠( x + 1 2 ⢠i ) - f ⢠( x - 1 2 ⢠i ) ð¿ ð ð¥ ð ð¥ 1 2 imaginary-unit ð ð¥ 1 2 imaginary-unit {\displaystyle{\displaystyle{\displaystyle\delta f(x)=f(x+\textstyle\frac{1}{2% }\mathrm{i})-f(x-\textstyle\frac{1}{2}\mathrm{i})}}} {\displaystyle \delta f(x)=f(x+\textstyle\frac{1}{2}\iunit)-f(x-\textstyle\frac{1}{2}\iunit) } δ 2 ⢠f ⢠( x ) = f ⢠( x + i ) - f ⢠( x ) - f ⢠( x ) + f ⢠( x - i ) = f ⢠( x + i ) - 2 ⢠f ⢠( x ) + f ⢠( x - i ) superscript ð¿ 2 ð ð¥ ð ð¥ imaginary-unit ð ð¥ ð ð¥ ð ð¥ imaginary-unit ð ð¥ imaginary-unit 2 ð ð¥ ð ð¥ imaginary-unit {\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 \delta^2 f(x)=f(x+\iunit)-f(x)-f(x)+f(x-\iunit)=f(x+\iunit)-2f(x)+f(x-\iunit) } δ ⢠x = x + 1 2 ⢠i - ( x - 1 2 ⢠i ) = i â and â δ ⢠x 2 = ( x + 1 2 ⢠i ) 2 - ( x - 1 2 ⢠i ) 2 = 2 ⢠i ⢠x formulae-sequence ð¿ ð¥ ð¥ 1 2 imaginary-unit ð¥ 1 2 imaginary-unit imaginary-unit and ð¿ superscript ð¥ 2 superscript ð¥ 1 2 imaginary-unit 2 superscript ð¥ 1 2 imaginary-unit 2 2 imaginary-unit ð¥ {\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 \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 } Π⢠λ ⢠( x ) = 2 ⢠x + γ + δ + 2 â and â â ⡠λ ⢠( x ) = 2 ⢠x + γ + δ formulae-sequence Î ð ð¥ 2 ð¥ ð¾ ð¿ 2 and â ð ð¥ 2 ð¥ ð¾ ð¿ {\displaystyle{\displaystyle{\displaystyle\Delta\lambda(x)=2x+\gamma+\delta+2% \quad\textrm{and}\quad\nabla\lambda(x)=2x+\gamma+\delta}}} {\displaystyle \Delta\lambda(x)=2x+\gamma+\delta+2\quad\textrm{and}\quad\nabla\lambda(x)=2x+\gamma+\delta } Π⢠μ ⢠( x ) = q - x - 1 ⢠( 1 - q ) ⢠( 1 - γ ⢠δ ⢠q 2 ⢠x + 2 ) â and â â ⡠μ ⢠( x ) = q - x ⢠( 1 - q ) ⢠( 1 - γ ⢠δ ⢠q 2 ⢠x ) formulae-sequence Î ð ð¥ superscript ð ð¥ 1 1 ð 1 ð¾ ð¿ superscript ð 2 ð¥ 2 and â ð ð¥ superscript ð ð¥ 1 ð 1 ð¾ ð¿ superscript ð 2 ð¥ {\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 \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}) } Π⢠λ ⢠( x ) = q - x - 1 ⢠( 1 - q ) ⢠( 1 - c ⢠q 2 ⢠x - N + 1 ) â and â â ⡠λ ⢠( x ) = q - x ⢠( 1 - q ) ⢠( 1 - c ⢠q 2 ⢠x - N - 1 ) formulae-sequence Î ð ð¥ superscript ð ð¥ 1 1 ð 1 ð superscript ð 2 ð¥ ð 1 and â ð ð¥ superscript ð ð¥ 1 ð 1 ð superscript ð 2 ð¥ ð 1 {\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 \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}) } Π⢠q - x = q - x - 1 ⢠( 1 - q ) â and â â â¡ q - x = q - x ⢠( 1 - q ) formulae-sequence Î superscript ð ð¥ superscript ð ð¥ 1 1 ð and â superscript ð ð¥ superscript ð ð¥ 1 ð {\displaystyle{\displaystyle{\displaystyle\Delta q^{-x}=q^{-x-1}(1-q)\quad% \textrm{and}\quad\nabla q^{-x}=q^{-x}(1-q)}}} {\displaystyle \Delta q^{-x}=q^{-x-1}(1-q)\quad\textrm{and}\quad\nabla q^{-x}=q^{-x}(1-q) } D q ⢠f ⢠( x ) := δ q ⢠f ⢠( x ) δ q ⢠x assign subscript ð· ð ð ð¥ subscript ð¿ ð ð ð¥ subscript ð¿ ð ð¥ {\displaystyle{\displaystyle{\displaystyle D_{q}f(x):=\frac{\delta_{q}f(x)}{% \delta_{q}x}}}} {\displaystyle D_qf(x):=\frac{\delta_qf(x)}{\delta_qx} }
δ q ⢠f ⢠( \expe \iunit ⢠θ ) = f ⢠( q 1 2 ⢠\expe \iunit ⢠θ ) - f ⢠( q - 1 2 ⢠\expe \iunit ⢠θ ) subscript ð¿ ð ð superscript \expe \iunit ð ð superscript ð 1 2 superscript \expe \iunit ð ð superscript ð 1 2 superscript \expe \iunit ð {\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 \delta_qf(\expe^{\iunit\theta}) =f(q^{\frac{1}{2}}\expe^{\iunit\theta})-f(q^{-\frac{1}{2}}\expe^{\iunit\theta}) } δ q ⢠x = - 1 2 ⢠q - 1 2 ⢠( 1 - q ) ⢠( e i ⢠θ - e - i ⢠θ ) subscript ð¿ ð ð¥ 1 2 superscript ð 1 2 1 ð imaginary-unit ð imaginary-unit ð {\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}})}}} {\displaystyle \delta_qx=-\textstyle\frac{1}{2}q^{-\frac{1}{2}}(1-q)(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) }