\qderiv q f ( z ) := { f ( z ) - f ( q z ) ( 1 - q ) z , z ≠ 0 f ′ ( 0 ) , z = 0 . assign \qderiv 𝑞 𝑓 𝑧 cases 𝑓 𝑧 𝑓 𝑞 𝑧 1 𝑞 𝑧 𝑧 0 superscript 𝑓 ′ 0 𝑧 0 {\displaystyle{\displaystyle{\displaystyle{}\qderiv{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. } \qderiv [ 0 ] q f := f and \qderiv [ n ] q f := \qderiv q @ \qderiv [ n - 1 ] q f n = 1 , 2 , 3 , … formulae-sequence assign \qderiv delimited-[] 0 𝑞 𝑓 𝑓 and formulae-sequence assign \qderiv delimited-[] 𝑛 𝑞 𝑓 \qderiv 𝑞 @ \qderiv delimited-[] 𝑛 1 𝑞 𝑓 𝑛 1 2 3 … {\displaystyle{\displaystyle{\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}}} {\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 → 1 \qderiv q f ( z ) = f ′ ( z ) subscript → 𝑞 1 \qderiv 𝑞 𝑓 𝑧 superscript 𝑓 ′ 𝑧 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\qderiv{% q}f(z)=f^{\prime}(z)}}} {\displaystyle \lim\limits_{q\rightarrow 1}\qderiv{q}f(z)=f'(z) } \qderiv q @ f ( γ x ) = γ ( \qderiv q f ) ( γ x ) \qderiv 𝑞 @ 𝑓 𝛾 𝑥 𝛾 \qderiv 𝑞 𝑓 𝛾 𝑥 {\displaystyle{\displaystyle{\displaystyle\qderiv{q}@{f(\gamma x)}=\gamma\left% (\qderiv{q}f\right)(\gamma x)}}} {\displaystyle \qderiv{q}@{f(\gamma x)}=\gamma\left(\qderiv{q}f\right)(\gamma x) } \qderiv [ n ] q @ f ( γ x ) = γ n ( \qderiv [ n ] q f ) ( γ x ) \qderiv delimited-[] 𝑛 𝑞 @ 𝑓 𝛾 𝑥 superscript 𝛾 𝑛 \qderiv delimited-[] 𝑛 𝑞 𝑓 𝛾 𝑥 {\displaystyle{\displaystyle{\displaystyle\qderiv[n]{q}@{f(\gamma x)}=\gamma^{% n}\left(\qderiv[n]{q}f\right)(\gamma x)}}} {\displaystyle \qderiv[n]{q}@{f(\gamma x)}=\gamma^n\left(\qderiv[n]{q}f\right)(\gamma x) }
\qderiv q @ f ( x ) g ( x ) = f ( q x ) \qderiv q g ( x ) + g ( x ) \qderiv q f ( x ) \qderiv 𝑞 @ 𝑓 𝑥 𝑔 𝑥 𝑓 𝑞 𝑥 \qderiv 𝑞 𝑔 𝑥 𝑔 𝑥 \qderiv 𝑞 𝑓 𝑥 {\displaystyle{\displaystyle{\displaystyle\qderiv{q}@{f(x)g(x)}=f(qx)\qderiv{q% }g(x)+g(x)\qderiv{q}f(x)}}} {\displaystyle \qderiv{q}@{f(x)g(x)}= f(qx)\qderiv{q}g(x)+g(x)\qderiv{q}f(x) } \qderiv [ n ] q @ f ( x ) g ( x ) = ∑ k = 0 n \qBinomial n k q ( \qderiv [ n - k ] q f ) ( q k x ) ( \qderiv [ k ] q g ) ( x ) \qderiv delimited-[] 𝑛 𝑞 @ 𝑓 𝑥 𝑔 𝑥 superscript subscript 𝑘 0 𝑛 \qBinomial 𝑛 𝑘 𝑞 \qderiv delimited-[] 𝑛 𝑘 𝑞 𝑓 superscript 𝑞 𝑘 𝑥 \qderiv delimited-[] 𝑘 𝑞 𝑔 𝑥 {\displaystyle{\displaystyle{\displaystyle\qderiv[n]{q}@{f(x)g(x)}=\sum_{k=0}^% {n}\qBinomial{n}{k}{q}\left(\qderiv[n-k]{q}f\right)(q^{k}x)\left(\qderiv[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) }
∫ 0 z f ( t ) d q t := z ( 1 - q ) ∑ n = 0 ∞ f ( q n z ) q n assign superscript subscript 0 𝑧 𝑓 𝑡 subscript 𝑑 𝑞 𝑡 𝑧 1 𝑞 superscript subscript 𝑛 0 𝑓 superscript 𝑞 𝑛 𝑧 superscript 𝑞 𝑛 {\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 }
∫ 0 ∞ f ( t ) d q t := ( 1 - q ) ∑ n = - ∞ ∞ f ( q n ) q n assign superscript subscript 0 𝑓 𝑡 subscript 𝑑 𝑞 𝑡 1 𝑞 superscript subscript 𝑛 𝑓 superscript 𝑞 𝑛 superscript 𝑞 𝑛 {\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 → 1 ∫ 0 z f ( t ) d q t = ∫ 0 z f ( t ) 𝑑 t subscript → 𝑞 1 superscript subscript 0 𝑧 𝑓 𝑡 subscript 𝑑 𝑞 𝑡 superscript subscript 0 𝑧 𝑓 𝑡 differential-d 𝑡 {\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 } ∫ a b f ( t ) d q t = b ( 1 - q ) ∑ n = 0 ∞ f ( b q n ) q n - a ( 1 - q ) ∑ n = 0 ∞ f ( a q n ) q n superscript subscript 𝑎 𝑏 𝑓 𝑡 subscript 𝑑 𝑞 𝑡 𝑏 1 𝑞 superscript subscript 𝑛 0 𝑓 𝑏 superscript 𝑞 𝑛 superscript 𝑞 𝑛 𝑎 1 𝑞 superscript subscript 𝑛 0 𝑓 𝑎 superscript 𝑞 𝑛 superscript 𝑞 𝑛 {\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 }
∫ - ∞ ∞ f ( t ) d q t = ( 1 - q ) ∑ n = - ∞ ∞ { f ( q n ) + f ( - q n ) } q n superscript subscript 𝑓 𝑡 subscript 𝑑 𝑞 𝑡 1 𝑞 superscript subscript 𝑛 𝑓 superscript 𝑞 𝑛 𝑓 superscript 𝑞 𝑛 superscript 𝑞 𝑛 {\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 ( z ) = ∫ 0 z f ( t ) d q t ⟹ \qderiv q F ( z ) = f ( z ) formulae-sequence 𝐹 𝑧 superscript subscript 0 𝑧 𝑓 𝑡 subscript 𝑑 𝑞 𝑡 ⟹ \qderiv 𝑞 𝐹 𝑧 𝑓 𝑧 {\displaystyle{\displaystyle{\displaystyle F(z)=\int_{0}^{z}f(t)\,d_{q}t\quad% \Longrightarrow\quad\qderiv{q}F(z)=f(z)}}} {\displaystyle F(z)=\int_0^zf(t)\,d_qt\quad\Longrightarrow\quad\qderiv{q}F(z)=f(z) } ∫ a b \qderiv q f ( t ) d q t = f ( b ) - f ( a ) superscript subscript 𝑎 𝑏 \qderiv 𝑞 𝑓 𝑡 subscript 𝑑 𝑞 𝑡 𝑓 𝑏 𝑓 𝑎 {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}\qderiv{q}f(t)\,d_{q}t=f% (b)-f(a)}}} {\displaystyle \int_a^b\qderiv{q}f(t)\,d_qt=f(b)-f(a) } ∫ a b g ( x ) \qderiv q f ( x ) d q x = [ f ( x ) g ( x ) ] a b - ∫ a b f ( q x ) \qderiv q g ( x ) d q x superscript subscript 𝑎 𝑏 𝑔 𝑥 \qderiv 𝑞 𝑓 𝑥 subscript 𝑑 𝑞 𝑥 superscript subscript delimited-[] 𝑓 𝑥 𝑔 𝑥 𝑎 𝑏 superscript subscript 𝑎 𝑏 𝑓 𝑞 𝑥 \qderiv 𝑞 𝑔 𝑥 subscript 𝑑 𝑞 𝑥 {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}g(x)\qderiv{q}f(x)\,d_{q% }x=\Big{[}f(x)g(x)\Big{]}_{a}^{b}-\int_{a}^{b}f(qx)\qderiv{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 } ∫ a b \qPochhammer a - 1 q t , b - 1 q t , c t q ∞ \qPochhammer d t , e t , f t q ∞ d q t = ( b - a ) ( 1 - q ) \qPochhammer q , a - 1 b q , a b - 1 q , c d - 1 , c \expe - 1 , c f - 1 q ∞ \qPochhammer a d , a e , a f , b d , b e , b f q ∞ superscript subscript 𝑎 𝑏 \qPochhammer superscript 𝑎 1 𝑞 𝑡 superscript 𝑏 1 𝑞 𝑡 𝑐 𝑡 𝑞 \qPochhammer 𝑑 𝑡 𝑒 𝑡 𝑓 𝑡 𝑞 subscript 𝑑 𝑞 𝑡 𝑏 𝑎 1 𝑞 \qPochhammer 𝑞 superscript 𝑎 1 𝑏 𝑞 𝑎 superscript 𝑏 1 𝑞 𝑐 superscript 𝑑 1 𝑐 superscript \expe 1 𝑐 superscript 𝑓 1 𝑞 \qPochhammer 𝑎 𝑑 𝑎 𝑒 𝑎 𝑓 𝑏 𝑑 𝑏 𝑒 𝑏 𝑓 𝑞 {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}\frac{\qPochhammer{a^{-1% }qt,b^{-1}qt,ct}{q}{\infty}}{\qPochhammer{dt,et,ft}{q}{\infty}}\,d_{q}t{}=(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}}}}} {\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}} } ∫ a b \qPochhammer a - 1 q t , b - 1 q t q ∞ \qPochhammer d t , e t q ∞ d q t = ( b - a ) ( 1 - q ) \qPochhammer q , a - 1 b q , a b - 1 q , a b d e q ∞ \qPochhammer a d , a e , b d , b e q ∞ superscript subscript 𝑎 𝑏 \qPochhammer superscript 𝑎 1 𝑞 𝑡 superscript 𝑏 1 𝑞 𝑡 𝑞 \qPochhammer 𝑑 𝑡 𝑒 𝑡 𝑞 subscript 𝑑 𝑞 𝑡 𝑏 𝑎 1 𝑞 \qPochhammer 𝑞 superscript 𝑎 1 𝑏 𝑞 𝑎 superscript 𝑏 1 𝑞 𝑎 𝑏 𝑑 𝑒 𝑞 \qPochhammer 𝑎 𝑑 𝑎 𝑒 𝑏 𝑑 𝑏 𝑒 𝑞 {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}\frac{\qPochhammer{a^{-1% }qt,b^{-1}qt}{q}{\infty}}{\qPochhammer{dt,et}{q}{\infty}}\,d_{q}t=(b-a)(1-q)\,% \frac{\qPochhammer{q,a^{-1}bq,ab^{-1}q,abde}{q}{\infty}}{\qPochhammer{ad,ae,bd% ,be}{q}{\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}} } ∫ a ∞ \qPochhammer a - 1 q t q ∞ \qPochhammer d t , e t q ∞ d q t = ( 1 - q ) \qPochhammer q , a , a - 1 q , a d e , a - 1 d - 1 \expe - 1 q q ∞ \qPochhammer a d , a e , d , d - 1 q , e , \expe - 1 q q ∞ superscript subscript 𝑎 \qPochhammer superscript 𝑎 1 𝑞 𝑡 𝑞 \qPochhammer 𝑑 𝑡 𝑒 𝑡 𝑞 subscript 𝑑 𝑞 𝑡 1 𝑞 \qPochhammer 𝑞 𝑎 superscript 𝑎 1 𝑞 𝑎 𝑑 𝑒 superscript 𝑎 1 superscript 𝑑 1 superscript \expe 1 𝑞 𝑞 \qPochhammer 𝑎 𝑑 𝑎 𝑒 𝑑 superscript 𝑑 1 𝑞 𝑒 superscript \expe 1 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\int_{a}^{\infty}\frac{\qPochhammer{% a^{-1}qt}{q}{\infty}}{\qPochhammer{dt,et}{q}{\infty}}\,d_{q}t=(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}}}}} {\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}} } ∫ - ∞ ∞ 1 \qPochhammer d t , e t q ∞ d q t = ( 1 - q ) \qPochhammer q , - q , - 1 , - d e , - d - 1 \expe - 1 q q ∞ \qPochhammer d , - d , d - 1 q , - d - 1 q , e , - e , \expe - 1 q , - \expe - 1 q q ∞ superscript subscript 1 \qPochhammer 𝑑 𝑡 𝑒 𝑡 𝑞 subscript 𝑑 𝑞 𝑡 1 𝑞 \qPochhammer 𝑞 𝑞 1 𝑑 𝑒 superscript 𝑑 1 superscript \expe 1 𝑞 𝑞 \qPochhammer 𝑑 𝑑 superscript 𝑑 1 𝑞 superscript 𝑑 1 𝑞 𝑒 𝑒 superscript \expe 1 𝑞 superscript \expe 1 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}\frac{1}{% \qPochhammer{dt,et}{q}{\infty}}\,d_{q}t{}=(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}}}}} {\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}} }