# q-exponential

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## 9 matching pages

##### 1: 17.17 Physical Applications
βΊSee Kassel (1995). …
##### 2: 17.3 $q$-Elementary and $q$-Special Functions
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###### $q$-Exponential Functions
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17.3.1 $e_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}x^{n}}{\left(q;q\right)_% {n}}=\frac{1}{\left((1-q)x;q\right)_{\infty}},$
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17.3.2 $E_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}q^{\genfrac{(}{)}{0.0pt}% {}{n}{2}}x^{n}}{\left(q;q\right)_{n}}=\left(-(1-q)x;q\right)_{\infty}.$
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17.3.3 $\mathrm{sin}_{q}\left(x\right)=\frac{1}{2i}(e_{q}\left(ix\right)-e_{q}\left(-% ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}(-1)^{n}x^{2n+1}}{\left(q;q% \right)_{2n+1}},$
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17.3.4 $\mathrm{Sin}_{q}\left(x\right)=\frac{1}{2i}(E_{q}\left(ix\right)-E_{q}\left(-% ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}q^{n(2n+1)}(-1)^{n}x^{2n+1}}{% \left(q;q\right)_{2n+1}}.$
##### 3: 17.18 Methods of Computation
βΊFor computation of the $q$-exponential function see Gabutti and Allasia (2008). …
##### 4: 20.11 Generalizations and Analogs
βΊWith the substitutions $a=qe^{2iz}$, $b=qe^{-2iz}$, with $q=e^{i\pi\tau}$, we have …
##### 6: 22.11 Fourier and Hyperbolic Series
βΊIf $q\exp\left(2|\Im\zeta|\right)<1$, then … βΊNext, if $q\exp\left(|\Im\zeta|\right)<1$, then … βΊNext, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and $q\exp\left(2|\Im\zeta|\right)<1$, …
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18.28.3 $2\pi\sin\theta\,w(\cos\theta)={\left|\frac{\left(e^{2i\theta};q\right)_{\infty% }}{\left(ae^{i\theta},be^{i\theta},ce^{i\theta},de^{i\theta};q\right)_{\infty}% }\right|}^{2},$
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18.28.8 $\frac{1}{2\pi}\int_{0}^{\pi}Q_{n}\left(\cos\theta;a,b\,|\,q\right)Q_{m}\left(% \cos\theta;a,b\,|\,q\right)\*{\left|\frac{\left(e^{2i\theta};q\right)_{\infty}% }{\left(ae^{i\theta},be^{i\theta};q\right)_{\infty}}\right|}^{2}\mathrm{d}% \theta=\frac{\delta_{n,m}}{\left(q^{n+1},abq^{n};q\right)_{\infty}},$ $a,b\in\mathbb{R}$ or $a=\overline{b}$; $|ab|<1$; $|a|,|b|\leq 1$.
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18.28.13 $C_{n}\left(\cos\theta;\beta\,|\,q\right)=\sum_{\ell=0}^{n}\frac{\left(\beta;q% \right)_{\ell}\left(\beta;q\right)_{n-\ell}}{\left(q;q\right)_{\ell}\left(q;q% \right)_{n-\ell}}e^{\mathrm{i}(n-2\ell)\theta}=\frac{\left(\beta;q\right)_{n}}% {\left(q;q\right)_{n}}e^{\mathrm{i}n\theta}{{}_{2}\phi_{1}}\left({q^{-n},\beta% \atop\beta^{-1}q^{1-n}};q,\beta^{-1}qe^{-2\mathrm{i}\theta}\right).$
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18.28.15 $\frac{1}{2\pi}\int_{0}^{\pi}C_{n}\left(\cos\theta;\beta\,|\,q\right)C_{m}\left% (\cos\theta;\beta\,|\,q\right)\*{\left|\frac{\left(e^{2\mathrm{i}\theta};q% \right)_{\infty}}{\left(\beta e^{2\mathrm{i}\theta};q\right)_{\infty}}\right|}% ^{2}\mathrm{d}\theta=\frac{\left(\beta,\beta q;q\right)_{\infty}}{\left(\beta^% {2},q;q\right)_{\infty}}\frac{(1-\beta)\left(\beta^{2};q\right)_{n}}{(1-\beta q% ^{n})\left(q;q\right)_{n}}\delta_{n,m},$ $-1<\beta<1$.
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18.28.17 $\frac{1}{2\pi}\int_{0}^{\pi}H_{n}\left(\cos\theta\,|\,q\right)H_{m}\left(\cos% \theta\,|\,q\right){\left|\left(e^{2\mathrm{i}\theta};q\right)_{\infty}\right|% }^{2}\mathrm{d}\theta=\frac{\delta_{n,m}}{\left(q^{n+1};q\right)_{\infty}}.$
##### 8: Bibliography O
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• A. B. Olde Daalhuis (1994) Asymptotic expansions for $q$-gamma, $q$-exponential, and $q$-Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.
• ##### 9: 18.27 $q$-Hahn Class
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18.27.20 $\int_{0}^{\infty}S_{n}\left(q^{\frac{1}{2}}x;q\right)S_{m}\left(q^{\frac{1}{2}% }x;q\right)\exp\left(-\frac{(\ln x)^{2}}{2\ln\left(q^{-1}\right)}\right)% \mathrm{d}x=\frac{\sqrt{2\pi q^{-1}\ln\left(q^{-1}\right)}}{q^{n}\left(q;q% \right)_{n}}\delta_{n,m}.$