π q β’ f β’ ( z ) := { f β’ ( z ) - f β’ ( q β’ z ) ( 1 - q ) β’ z , z β 0 f β² β’ ( 0 ) , z = 0 . assign q-derivative π π π§ cases π π§ π π π§ 1 π π§ π§ 0 superscript π β² 0 π§ 0 {\displaystyle{\displaystyle{\displaystyle{}\mathcal{D}_{q}f(z):=\left\{\begin% {array}[]{ll}\displaystyle\frac{f(z)-f(qz)}{(1-q)z},&z\neq 0\\ f^{\prime}(0),&z=0.\end{array}\right.}}} {\displaystyle \index{q-Derivative operator@$q$-Derivative operator} \qderiv{q}f(z):=\left\{\begin{array}{ll} \displaystyle \frac{f(z)-f(qz)}{(1-q)z}, & z\neq 0\[5mm] f'(0), & z=0.\end{array}\right. } π q 0 β’ f := f β and β π q n β’ f := π q β‘ ( π q n - 1 β’ f ) β n = 1 , 2 , 3 , β¦ formulae-sequence assign q-derivative 0 π π π and formulae-sequence assign q-derivative π π π q-derivative π q-derivative π 1 π π π 1 2 3 β¦ {\displaystyle{\displaystyle{\displaystyle\mathcal{D}^{0}_{q}f:=f\quad\textrm{% and}\quad\mathcal{D}^{n}_{q}f:=\mathcal{D}_{q}\left(\mathcal{D}^{n-1}_{q}f% \right)\quad n=1,2,3,\ldots}}} {\displaystyle \qderiv[0]{q}f:=f\quad\textrm{and}\quad \qderiv[n]{q}f:=\qderiv{q}@{\qderiv[n-1]{q}f} \quad n=1,2,3,\ldots } lim q β 1 β‘ π q β’ f β’ ( z ) = f β² β’ ( z ) subscript β π 1 q-derivative π π π§ superscript π β² π§ {\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\mathcal% {D}_{q}f(z)=f^{\prime}(z)}}} {\displaystyle \lim\limits_{q\rightarrow 1}\qderiv{q}f(z)=f'(z) } π q β‘ ( f β’ ( Ξ³ β’ x ) ) = Ξ³ β’ ( π q β’ f ) β’ ( Ξ³ β’ x ) q-derivative π π πΎ π₯ πΎ q-derivative π π πΎ π₯ {\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}\left(f(\gamma x)% \right)=\gamma\left(\mathcal{D}_{q}f\right)(\gamma x)}}} {\displaystyle \qderiv{q}@{f(\gamma x)}=\gamma\left(\qderiv{q}f\right)(\gamma x) } π q n β‘ ( f β’ ( Ξ³ β’ x ) ) = Ξ³ n β’ ( π q n β’ f ) β’ ( Ξ³ β’ x ) q-derivative π π π πΎ π₯ superscript πΎ π q-derivative π π π πΎ π₯ {\displaystyle{\displaystyle{\displaystyle\mathcal{D}^{n}_{q}\left(f(\gamma x)% \right)=\gamma^{n}\left(\mathcal{D}^{n}_{q}f\right)(\gamma x)}}} {\displaystyle \qderiv[n]{q}@{f(\gamma x)}=\gamma^n\left(\qderiv[n]{q}f\right)(\gamma x) }
π q β‘ ( f β’ ( x ) β’ g β’ ( x ) ) = f β’ ( q β’ x ) β’ π q β’ g β’ ( x ) + g β’ ( x ) β’ π q β’ f β’ ( x ) q-derivative π π π₯ π π₯ π π π₯ q-derivative π π π₯ π π₯ q-derivative π π π₯ {\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}\left(f(x)g(x)\right)% =f(qx)\mathcal{D}_{q}g(x)+g(x)\mathcal{D}_{q}f(x)}}} {\displaystyle \qderiv{q}@{f(x)g(x)}= f(qx)\qderiv{q}g(x)+g(x)\qderiv{q}f(x) } π q n β‘ ( f β’ ( x ) β’ g β’ ( x ) ) = β k = 0 n [ n k ] q β’ ( π q n - k β’ f ) β’ ( q k β’ x ) β’ ( π q k β’ g ) β’ ( x ) q-derivative π π π π₯ π π₯ superscript subscript π 0 π q-binomial π π π q-derivative π π π π superscript π π π₯ q-derivative π π π π₯ {\displaystyle{\displaystyle{\displaystyle\mathcal{D}^{n}_{q}\left(f(x)g(x)% \right)=\sum_{k=0}^{n}\genfrac{[}{]}{0.0pt}{}{n}{k}_{q}\left(\mathcal{D}^{n-k}% _{q}f\right)(q^{k}x)\left(\mathcal{D}^{k}_{q}g\right)(x)}}} {\displaystyle \qderiv[n]{q}@{f(x)g(x)}=\sum_{k=0}^n\qBinomial{n}{k}{q} \left(\qderiv[n-k]{q}f\right)(q^kx)\left(\qderiv[k]{q}g\right)(x) }
β« 0 z f β’ ( t ) β’ d q β’ t := z β’ ( 1 - q ) β’ β n = 0 β f β’ ( q n β’ z ) β’ q n assign superscript subscript 0 π§ π π‘ subscript π π π‘ π§ 1 π superscript subscript π 0 π superscript π π π§ superscript π π {\displaystyle{\displaystyle{\displaystyle{}{}\int_{0}^{z}f(t)\,d_{q}t:=z(1-q)% \sum_{n=0}^{\infty}f(q^{n}z)q^{n}}}} {\displaystyle \index{q-Integral@$q$-Integral}\index{Jackson-Thomae q-integral@Jackson-Thomae $q$-integral} \int_0^zf(t)\,d_qt:=z(1-q)\sum_{n=0}^{\infty}f(q^nz)q^n }
β« 0 β f β’ ( t ) β’ d q β’ t := ( 1 - q ) β’ β n = - β β f β’ ( q n ) β’ q n assign superscript subscript 0 π π‘ subscript π π π‘ 1 π superscript subscript π π superscript π π superscript π π {\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}f(t)\,d_{q}t:=(1-q)% \sum_{n=-\infty}^{\infty}f(q^{n})q^{n}}}} {\displaystyle \int_0^{\infty}f(t)\,d_qt:=(1-q)\sum_{n=-\infty}^{\infty}f(q^n)q^n }
lim q β 1 β‘ β« 0 z f β’ ( t ) β’ d q β’ t = β« 0 z f β’ ( t ) β’ π t subscript β π 1 superscript subscript 0 π§ π π‘ subscript π π π‘ superscript subscript 0 π§ π π‘ differential-d π‘ {\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\int_{0}% ^{z}f(t)\,d_{q}t=\int_{0}^{z}f(t)\,dt}}} {\displaystyle \lim\limits_{q\rightarrow 1}\int_0^zf(t)\,d_qt=\int_0^zf(t)\,dt } β« a b f β’ ( t ) β’ d q β’ t = b β’ ( 1 - q ) β’ β n = 0 β f β’ ( b β’ q n ) β’ q n - a β’ ( 1 - q ) β’ β n = 0 β f β’ ( a β’ q n ) β’ q n superscript subscript π π π π‘ subscript π π π‘ π 1 π superscript subscript π 0 π π superscript π π superscript π π π 1 π superscript subscript π 0 π π superscript π π superscript π π {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}f(t)\,d_{q}t=b(1-q)\sum_% {n=0}^{\infty}f(bq^{n})q^{n}-a(1-q)\sum_{n=0}^{\infty}f(aq^{n})q^{n}}}} {\displaystyle \int_a^bf(t)\,d_qt=b(1-q)\sum_{n=0}^{\infty}f(bq^n)q^n-a(1-q)\sum_{n=0}^{\infty}f(aq^n)q^n }
β« - β β f β’ ( t ) β’ d q β’ t = ( 1 - q ) β’ β n = - β β { f β’ ( q n ) + f β’ ( - q n ) } β’ q n superscript subscript π π‘ subscript π π π‘ 1 π superscript subscript π π superscript π π π superscript π π superscript π π {\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}f(t)\,d_{q}t=% (1-q)\sum_{n=-\infty}^{\infty}\left\{f(q^{n})+f(-q^{n})\right\}q^{n}}}} {\displaystyle \int_{-\infty}^{\infty}f(t)\,d_qt=(1-q)\sum_{n=-\infty}^{\infty}\left\{f(q^n)+f(-q^n)\right\}q^n }
F β’ ( z ) = β« 0 z f β’ ( t ) β’ d q β’ t β βΉ β π q β’ F β’ ( z ) = f β’ ( z ) formulae-sequence πΉ π§ superscript subscript 0 π§ π π‘ subscript π π π‘ βΉ q-derivative π πΉ π§ π π§ {\displaystyle{\displaystyle{\displaystyle F(z)=\int_{0}^{z}f(t)\,d_{q}t\quad% \Longrightarrow\quad\mathcal{D}_{q}F(z)=f(z)}}} {\displaystyle F(z)=\int_0^zf(t)\,d_qt\quad\Longrightarrow\quad\qderiv{q}F(z)=f(z) } β« a b π q β’ f β’ ( t ) β’ d q β’ t = f β’ ( b ) - f β’ ( a ) superscript subscript π π q-derivative π π π‘ subscript π π π‘ π π π π {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}\mathcal{D}_{q}f(t)\,d_{% q}t=f(b)-f(a)}}} {\displaystyle \int_a^b\qderiv{q}f(t)\,d_qt=f(b)-f(a) } β« a b g β’ ( x ) β’ π q β’ f β’ ( x ) β’ d q β’ x = [ f β’ ( x ) β’ g β’ ( x ) ] a b - β« a b f β’ ( q β’ x ) β’ π q β’ g β’ ( x ) β’ d q β’ x superscript subscript π π π π₯ q-derivative π π π₯ subscript π π π₯ superscript subscript delimited-[] π π₯ π π₯ π π superscript subscript π π π π π₯ q-derivative π π π₯ subscript π π π₯ {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}g(x)\mathcal{D}_{q}f(x)% \,d_{q}x=\Big{[}f(x)g(x)\Big{]}_{a}^{b}-\int_{a}^{b}f(qx)\mathcal{D}_{q}g(x)\,% d_{q}x}}} {\displaystyle \int_a^bg(x)\qderiv{q}f(x)\,d_qx=\Big[f(x)g(x)\Big]_a^b-\int_a^bf(qx)\qderiv{q}g(x)\,d_qx } β« a b ( a - 1 β’ q β’ t , b - 1 β’ q β’ t , c β’ t ; q ) β ( d β’ t , e β’ t , f β’ t ; q ) β β’ d q β’ t = ( b - a ) β’ ( 1 - q ) β’ ( q , a - 1 β’ b β’ q , a β’ b - 1 β’ q , c β’ d - 1 , c β’ e - 1 , c β’ f - 1 ; q ) β ( a β’ d , a β’ e , a β’ f , b β’ d , b β’ e , b β’ f ; q ) β superscript subscript π π q-Pochhammer-symbol superscript π 1 π π‘ superscript π 1 π π‘ π π‘ π q-Pochhammer-symbol π π‘ π π‘ π π‘ π subscript π π π‘ π π 1 π q-Pochhammer-symbol π superscript π 1 π π π superscript π 1 π π superscript π 1 π 1 π superscript π 1 π q-Pochhammer-symbol π π π π π π π π π π π π π {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}\frac{\left(a^{-1}qt,b^{% -1}qt,ct;q\right)_{\infty}}{\left(dt,et,ft;q\right)_{\infty}}\,d_{q}t{}=(b-a)(% 1-q)\,\frac{\left(q,a^{-1}bq,ab^{-1}q,cd^{-1},c{\mathrm{e}^{-1}},cf^{-1};q% \right)_{\infty}}{\left(ad,ae,af,bd,be,bf;q\right)_{\infty}}}}} {\displaystyle \int_a^b\frac{\qPochhammer{a^{-1}qt,b^{-1}qt,ct}{q}{\infty}}{\qPochhammer{dt,et,ft}{q}{\infty}}\,d_qt {}=(b-a)(1-q)\,\frac{\qPochhammer{q,a^{-1}bq,ab^{-1}q,cd^{-1},c\expe^{-1},cf^{-1}}{q}{\infty}} {\qPochhammer{ad,ae,af,bd,be,bf}{q}{\infty}} } β« a b ( a - 1 β’ q β’ t , b - 1 β’ q β’ t ; q ) β ( d β’ t , e β’ t ; q ) β β’ d q β’ t = ( b - a ) β’ ( 1 - q ) β’ ( q , a - 1 β’ b β’ q , a β’ b - 1 β’ q , a β’ b β’ d β’ e ; q ) β ( a β’ d , a β’ e , b β’ d , b β’ e ; q ) β superscript subscript π π q-Pochhammer-symbol superscript π 1 π π‘ superscript π 1 π π‘ π q-Pochhammer-symbol π π‘ π π‘ π subscript π π π‘ π π 1 π q-Pochhammer-symbol π superscript π 1 π π π superscript π 1 π π π π π π q-Pochhammer-symbol π π π π π π π π π {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}\frac{\left(a^{-1}qt,b^{% -1}qt;q\right)_{\infty}}{\left(dt,et;q\right)_{\infty}}\,d_{q}t=(b-a)(1-q)\,% \frac{\left(q,a^{-1}bq,ab^{-1}q,abde;q\right)_{\infty}}{\left(ad,ae,bd,be;q% \right)_{\infty}}}}} {\displaystyle \int_a^b\frac{\qPochhammer{a^{-1}qt,b^{-1}qt}{q}{\infty}}{\qPochhammer{dt,et}{q}{\infty}}\,d_qt =(b-a)(1-q)\,\frac{\qPochhammer{q,a^{-1}bq,ab^{-1}q,abde}{q}{\infty}}{\qPochhammer{ad,ae,bd,be}{q}{\infty}} } β« a β ( a - 1 β’ q β’ t ; q ) β ( d β’ t , e β’ t ; q ) β β’ d q β’ t = ( 1 - q ) β’ ( q , a , a - 1 β’ q , a β’ d β’ e , a - 1 β’ d - 1 β’ e - 1 β’ q ; q ) β ( a β’ d , a β’ e , d , d - 1 β’ q , e , e - 1 β’ q ; q ) β superscript subscript π q-Pochhammer-symbol superscript π 1 π π‘ π q-Pochhammer-symbol π π‘ π π‘ π subscript π π π‘ 1 π q-Pochhammer-symbol π π superscript π 1 π π π π superscript π 1 superscript π 1 1 π π q-Pochhammer-symbol π π π π π superscript π 1 π π 1 π π {\displaystyle{\displaystyle{\displaystyle\int_{a}^{\infty}\frac{\left(a^{-1}% qt;q\right)_{\infty}}{\left(dt,et;q\right)_{\infty}}\,d_{q}t=(1-q)\,\frac{% \left(q,a,a^{-1}q,ade,a^{-1}d^{-1}{\mathrm{e}^{-1}}q;q\right)_{\infty}}{\left(% ad,ae,d,d^{-1}q,e,{\mathrm{e}^{-1}}q;q\right)_{\infty}}}}} {\displaystyle \int_a^{\infty}\frac{\qPochhammer{a^{-1}qt}{q}{\infty}}{\qPochhammer{dt,et}{q}{\infty}}\,d_qt =(1-q)\,\frac{\qPochhammer{q,a,a^{-1}q,ade,a^{-1}d^{-1}\expe^{-1}q}{q}{\infty}} {\qPochhammer{ad,ae,d,d^{-1}q,e,\expe^{-1}q}{q}{\infty}} } β« - β β 1 ( d β’ t , e β’ t ; q ) β β’ d q β’ t = ( 1 - q ) β’ ( q , - q , - 1 , - d β’ e , - d - 1 β’ e - 1 β’ q ; q ) β ( d , - d , d - 1 β’ q , - d - 1 β’ q , e , - e , e - 1 β’ q , - e - 1 β’ q ; q ) β superscript subscript 1 q-Pochhammer-symbol π π‘ π π‘ π subscript π π π‘ 1 π q-Pochhammer-symbol π π 1 π π superscript π 1 1 π π q-Pochhammer-symbol π π superscript π 1 π superscript π 1 π π π 1 π 1 π π {\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}\frac{1}{% \left(dt,et;q\right)_{\infty}}\,d_{q}t{}=(1-q)\,\frac{\left(q,-q,-1,-de,-d^{-1% }{\mathrm{e}^{-1}}q;q\right)_{\infty}}{\left(d,-d,d^{-1}q,-d^{-1}q,e,-e,{% \mathrm{e}^{-1}}q,-{\mathrm{e}^{-1}}q;q\right)_{\infty}}}}} {\displaystyle \int_{-\infty}^{\infty}\frac{1}{\qPochhammer{dt,et}{q}{\infty}}\,d_qt {}=(1-q)\,\frac{\qPochhammer{q,-q,-1,-de,-d^{-1}\expe^{-1}q}{q}{\infty}} {\qPochhammer{d,-d,d^{-1}q,-d^{-1}q,e,-e,\expe^{-1}q,-\expe^{-1}q}{q}{\infty}} }