e q ( z ) := \qHyperrphis 10 @ @ 0 - q z = ∑ n = 0 ∞ z n ( q ; q ) n = 1 ( z ; q ) ∞ , 0 < | q | < 1 formulae-sequence assign KLS-q-exp 𝑞 𝑧 \qHyperrphis 10 @ @ 0 𝑞 𝑧 superscript subscript 𝑛 0 superscript 𝑧 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 1 q-Pochhammer-symbol 𝑧 𝑞 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle{}\mathrm{e}_{q}\!\left(z\right):=% \qHyperrphis{1}{0}@@{0}{-}{q}{z}=\sum_{n=0}^{\infty}\frac{z^{n}}{\left(q;q% \right)_{n}}=\frac{1}{\left(z;q\right)_{\infty}},\quad 0<|q|<1}}} {\displaystyle \index{q-Exponential function@$q$-Exponential function} \qexpKLS{q}@{z}:=\qHyperrphis{1}{0}@@{0}{-}{q}{z}=\sum_{n=0}^{\infty}\frac{z^n}{\qPochhammer{q}{q}{n}} =\frac{1}{\qPochhammer{z}{q}{\infty}},\quad 0<|q|<1 }
E q ( z ) := \qHyperrphis 00 @ @ - - q - z = ∑ n = 0 ∞ q \binomial n 2 ( q ; q ) n z n = ( - z ; q ) ∞ fragments KLS-q-Exp 𝑞 𝑧 assign \qHyperrphis 00 @ @ q z superscript subscript 𝑛 0 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑧 𝑛 q-Pochhammer-symbol 𝑧 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathrm{E}_{q}\!\left(z\right):=% \qHyperrphis{0}{0}@@{-}{-}{q}{-z}=\sum_{n=0}^{\infty}\frac{q^{\binomial{n}{2}}% }{\left(q;q\right)_{n}}z^{n}=\left(-z;q\right)_{\infty}}}} {\displaystyle \qExpKLS{q}@{z}:=\qHyperrphis{0}{0}@@{-}{-}{q}{-z}= \sum_{n=0}^{\infty}\frac{q^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}z^n=\qPochhammer{-z}{q}{\infty} }
e q ( z ) \E q ( - z ) = 1 KLS-q-exp 𝑞 fragments z ) \E 𝑞 ( z 1 {\displaystyle{\displaystyle{\displaystyle\mathrm{e}_{q}\!\left(z)\E_{q}(-z% \right)=1}}} {\displaystyle \qexpKLS{q}@{z)\E_q(-z}=1 } lim q → 1 e q ( ( 1 - q ) z ) = lim q → 1 E q ( ( 1 - q ) z ) = e z subscript → 𝑞 1 KLS-q-exp 𝑞 1 𝑞 𝑧 subscript → 𝑞 1 KLS-q-Exp 𝑞 1 𝑞 𝑧 𝑧 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\mathrm{% e}_{q}\!\left((1-q)z\right)=\lim\limits_{q\rightarrow 1}\mathrm{E}_{q}\!\left(% (1-q)z\right)={\mathrm{e}^{z}}}}} {\displaystyle \lim\limits_{q\rightarrow 1}\qexpKLS{q}@{(1-q)z}=\lim\limits_{q\rightarrow 1}\qExpKLS{q}@{(1-q)z}=\expe^z } cos q ( z ) := e q ( i z ) + e q ( - i z ) 2 = ∑ n = 0 ∞ ( - 1 ) n z 2 n ( q ; q ) 2 n assign KLS-q-cos 𝑞 𝑧 KLS-q-exp 𝑞 fragments imaginary-unit z ) e 𝑞 ( imaginary-unit z 2 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑧 2 𝑛 q-Pochhammer-symbol 𝑞 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\mathrm{cos}_{q}\!\left(z\right):=% \frac{\mathrm{e}_{q}\!\left(\mathrm{i}z)+{\mathrm{e}}_{q}(-\mathrm{i}z\right)}% {2}=\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n}}{\left(q;q\right)_{2n}}}}} {\displaystyle \qcosKLS{q}@{z}:=\frac{\qexpKLS{q}@{\iunit z)+{\mathrm e}_q(-\iunit z}}{2}= \sum_{n=0}^{\infty}\frac{(-1)^nz^{2n}}{\qPochhammer{q}{q}{2n}} } sin q ( z ) := e q ( i z ) - e q ( - i z ) 2 i = ∑ n = 0 ∞ ( - 1 ) n z 2 n + 1 ( q ; q ) 2 n + 1 assign KLS-q-sin 𝑞 𝑧 KLS-q-exp 𝑞 fragments imaginary-unit z ) e 𝑞 ( imaginary-unit z 2 imaginary-unit superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑧 2 𝑛 1 q-Pochhammer-symbol 𝑞 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\mathrm{sin}_{q}\!\left(z\right):=% \frac{\mathrm{e}_{q}\!\left(\mathrm{i}z)-{\mathrm{e}}_{q}(-\mathrm{i}z\right)}% {2\mathrm{i}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{\left(q;q\right)_{2n+% 1}}}}} {\displaystyle \qsinKLS{q}@{z}:=\frac{\qexpKLS{q}@{\iunit z)-{\mathrm e}_q(-\iunit z}}{2\iunit}= \sum_{n=0}^{\infty}\frac{(-1)^nz^{2n+1}}{\qPochhammer{q}{q}{2n+1}} } Cos q ( z ) := E q ( i z ) + E q ( - i z ) 2 = ∑ n = 0 ∞ ( - 1 ) n q \binomial 2 n 2 z 2 n ( q ; q ) 2 n assign KLS-q-Cos 𝑞 𝑧 KLS-q-Exp 𝑞 imaginary-unit 𝑧 KLS-q-Exp 𝑞 imaginary-unit 𝑧 2 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 \binomial 2 𝑛 2 superscript 𝑧 2 𝑛 q-Pochhammer-symbol 𝑞 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\mathrm{Cos}_{q}\!\left(z\right):=% \frac{\mathrm{E}_{q}\!\left(\mathrm{i}z\right)+\mathrm{E}_{q}\!\left(-\mathrm{% i}z\right)}{2}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{2n}{2}}z^{2n}}{% \left(q;q\right)_{2n}}}}} {\displaystyle \qCosKLS{q}@{z}:=\frac{\qExpKLS{q}@{\iunit z}+\qExpKLS{q}@{-\iunit z}}{2} =\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{2n}{2}}z^{2n}}{\qPochhammer{q}{q}{2n}} } Sin q ( z ) := E q ( i z ) - E q ( - i z ) 2 i = ∑ n = 0 ∞ ( - 1 ) n q \binomial 2 n + 12 z 2 n + 1 ( q ; q ) 2 n + 1 assign KLS-q-Sin 𝑞 𝑧 KLS-q-Exp 𝑞 imaginary-unit 𝑧 KLS-q-Exp 𝑞 imaginary-unit 𝑧 2 imaginary-unit superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 \binomial 2 𝑛 12 superscript 𝑧 2 𝑛 1 q-Pochhammer-symbol 𝑞 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\mathrm{Sin}_{q}\!\left(z\right):=% \frac{\mathrm{E}_{q}\!\left(\mathrm{i}z\right)-\mathrm{E}_{q}\!\left(-\mathrm{% i}z\right)}{2\mathrm{i}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{2n+1}{2% }}z^{2n+1}}{\left(q;q\right)_{2n+1}}}}} {\displaystyle \qSinKLS{q}@{z}:=\frac{\qExpKLS{q}@{\iunit z}-\qExpKLS{q}@{-\iunit z}}{2\iunit} =\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{2n+1}{2}}z^{2n+1}}{\qPochhammer{q}{q}{2n+1}} } e q ( i z ) = cos q ( z ) + i sin q ( z ) and E q ( i z ) = Cos q ( z ) + i Sin q ( z ) formulae-sequence KLS-q-exp 𝑞 imaginary-unit 𝑧 KLS-q-cos 𝑞 𝑧 imaginary-unit KLS-q-sin 𝑞 𝑧 and KLS-q-Exp 𝑞 imaginary-unit 𝑧 KLS-q-Cos 𝑞 𝑧 imaginary-unit KLS-q-Sin 𝑞 𝑧 {\displaystyle{\displaystyle{\displaystyle\mathrm{e}_{q}\!\left(\mathrm{i}z% \right)=\mathrm{cos}_{q}\!\left(z\right)+\mathrm{i}\mathrm{sin}_{q}\!\left(z% \right)\quad\textrm{and}\quad\mathrm{E}_{q}\!\left(\mathrm{i}z\right)=\mathrm{% Cos}_{q}\!\left(z\right)+\mathrm{i}\>\mathrm{Sin}_{q}\!\left(z\right)}}} {\displaystyle \qexpKLS{q}@{\iunit z}=\qcosKLS{q}@{z}+\iunit\qsinKLS{q}@{z}\quad\textrm{and}\quad \qExpKLS{q}@{\iunit z}=\qCosKLS{q}@{z}+\iunit\:\qSinKLS{q}@{z} } cos q ( z ) Cos q ( z ) + sin q ( z ) Sin q ( z ) = 1 KLS-q-cos 𝑞 𝑧 KLS-q-Cos 𝑞 𝑧 KLS-q-sin 𝑞 𝑧 KLS-q-Sin 𝑞 𝑧 1 {\displaystyle{\displaystyle{\displaystyle\mathrm{cos}_{q}\!\left(z\right)% \mathrm{Cos}_{q}\!\left(z\right)+\mathrm{sin}_{q}\!\left(z\right)\mathrm{Sin}_% {q}\!\left(z\right)=1}}} {\displaystyle \qcosKLS{q}@{z}\qCosKLS{q}@{z}+\qsinKLS{q}@{z}\qSinKLS{q}@{z}=1 } sin q ( z ) Cos q ( z ) - cos q ( z ) Sin q ( z ) = 0 KLS-q-sin 𝑞 𝑧 KLS-q-Cos 𝑞 𝑧 KLS-q-cos 𝑞 𝑧 KLS-q-Sin 𝑞 𝑧 0 {\displaystyle{\displaystyle{\displaystyle\mathrm{sin}_{q}\!\left(z\right)% \mathrm{Cos}_{q}\!\left(z\right)-\mathrm{cos}_{q}\!\left(z\right)\mathrm{Sin}_% {q}\!\left(z\right)=0}}} {\displaystyle \qsinKLS{q}@{z}\qCosKLS{q}@{z}-\qcosKLS{q}@{z}\qSinKLS{q}@{z}=0 } J ν ( 1 ) ( z ; q ) := ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ ( z 2 ) ν \qHyperrphis 21 @ @ 0 , 0 q ν + 1 q - z 2 4 assign Jackson-q-Bessel-1-J 𝜈 𝑧 𝑞 q-Pochhammer-symbol superscript 𝑞 𝜈 1 𝑞 q-Pochhammer-symbol 𝑞 𝑞 superscript 𝑧 2 𝜈 \qHyperrphis 21 @ @ 0 0 superscript 𝑞 𝜈 1 𝑞 superscript 𝑧 2 4 {\displaystyle{\displaystyle{\displaystyle{}{}J^{(1)}_{\nu}\!\left(z;q\right):% =\frac{\left(q^{\nu+1};q\right)_{\infty}}{\left(q;q\right)_{\infty}}\left(% \frac{z}{2}\right)^{\nu}\,\qHyperrphis{2}{1}@@{0,0}{q^{\nu+1}}{q}{-\frac{z^{2}% }{4}}}}} {\displaystyle \index{q-Bessel function@$q$-Bessel function}\index{Jackson's q-Bessel function@Jackson's $q$-Bessel function} \JacksonqBesselI{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}} \left(\frac{z}{2}\right)^{\nu}\,\qHyperrphis{2}{1}@@{0,0}{q^{\nu+1}}{q}{-\frac{z^2}{4}} }
J ν ( 2 ) ( z ; q ) := ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ ( z 2 ) ν \qHyperrphis 01 @ @ - q ν + 1 q - q ν + 1 z 2 4 assign Jackson-q-Bessel-2-J 𝜈 𝑧 𝑞 q-Pochhammer-symbol superscript 𝑞 𝜈 1 𝑞 q-Pochhammer-symbol 𝑞 𝑞 superscript 𝑧 2 𝜈 \qHyperrphis 01 @ @ superscript 𝑞 𝜈 1 𝑞 superscript 𝑞 𝜈 1 superscript 𝑧 2 4 {\displaystyle{\displaystyle{\displaystyle J^{(2)}_{\nu}\!\left(z;q\right):=% \frac{\left(q^{\nu+1};q\right)_{\infty}}{\left(q;q\right)_{\infty}}\left(\frac% {z}{2}\right)^{\nu}\,\qHyperrphis{0}{1}@@{-}{q^{\nu+1}}{q}{-\frac{q^{\nu+1}z^{% 2}}{4}}}}} {\displaystyle \JacksonqBesselII{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}} \left(\frac{z}{2}\right)^{\nu}\,\qHyperrphis{0}{1}@@{-}{q^{\nu+1}}{q}{-\frac{q^{\nu+1}z^2}{4}} } J ν ( 2 ) ( z ; q ) = ( - z 2 4 ; q ) ∞ ⋅ J ν ( 1 ) ( z ; q ) Jackson-q-Bessel-2-J 𝜈 𝑧 𝑞 ⋅ q-Pochhammer-symbol superscript 𝑧 2 4 𝑞 Jackson-q-Bessel-1-J 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle{\displaystyle J^{(2)}_{\nu}\!\left(z;q\right)=% \left(-\frac{z^{2}}{4};q\right)_{\infty}\cdot J^{(1)}_{\nu}\!\left(z;q\right)}}} {\displaystyle \JacksonqBesselII{\nu}@{z}{q}=\qPochhammer{-\frac{z^2}{4}}{q}{\infty}\cdot \JacksonqBesselI{\nu}@{z}{q} }
lim q → 1 J ν ( k ) ( ( 1 - q ) z ; q ) = J ν ( z ) subscript → 𝑞 1 superscript subscript 𝐽 𝜈 𝑘 1 𝑞 𝑧 𝑞 Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}J_{\nu}^% {(k)}((1-q)z;q)=J_{\nu}\left(z\right)}}} {\displaystyle \lim\limits_{q\rightarrow 1}J_{\nu}^{(k)}((1-q)z;q)=\BesselJ{\nu}@{z} }
J ν ( 3 ) ( z ; q ) := ( q ν + 1 ; q ) ∞ ( q ; q ) ∞ z ν \qHyperrphis 11 @ @ 0 q ν + 1 q q z 2 assign Jackson-q-Bessel-3-J 𝜈 𝑧 𝑞 q-Pochhammer-symbol superscript 𝑞 𝜈 1 𝑞 q-Pochhammer-symbol 𝑞 𝑞 superscript 𝑧 𝜈 \qHyperrphis 11 @ @ 0 superscript 𝑞 𝜈 1 𝑞 𝑞 superscript 𝑧 2 {\displaystyle{\displaystyle{\displaystyle J^{(3)}_{\nu}\!\left(z;q\right):=% \frac{\left(q^{\nu+1};q\right)_{\infty}}{\left(q;q\right)_{\infty}}z^{\nu}\,% \qHyperrphis{1}{1}@@{0}{q^{\nu+1}}{q}{qz^{2}}}}} {\displaystyle \JacksonqBesselIII{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}} z^{\nu}\,\qHyperrphis{1}{1}@@{0}{q^{\nu+1}}{q}{qz^2} } lim q → 1 J ν ( 3 ) ( ( 1 - q ) z ; q ) = J ν ( 2 z ) subscript → 𝑞 1 Jackson-q-Bessel-3-J 𝜈 1 𝑞 𝑧 𝑞 Bessel-J 𝜈 2 𝑧 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}J^{(3)}_% {\nu}\!\left((1-q)z;q\right)=J_{\nu}\left(2z\right)}}} {\displaystyle \lim\limits_{q\rightarrow 1}\JacksonqBesselIII{\nu}@{(1-q)z}{q}=\BesselJ{\nu}@{2z} }