# q-exponential function

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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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###### $q$-ExponentialFunctions
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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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17.3.4 $\operatorname{Sin}_{q}\left(x\right)=\frac{1}{2i}(E_{q}\left(ix\right)-E_{q}% \left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}q^{n(2n+1)}(-1)^{n}x^{2% n+1}}{\left(q;q\right)_{2n+1}}.$
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17.3.6 $\operatorname{Cos}_{q}\left(x\right)=\frac{1}{2}(E_{q}\left(ix\right)+E_{q}% \left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}q^{n(2n-1)}(-1)^{n}x^{2n}% }{\left(q;q\right)_{2n}}.$
##### 3: 17.18 Methods of Computation
βΊFor computation of the $q$-exponential function see Gabutti and Allasia (2008). …
##### 4: 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.
• ##### 5: 22.11 Fourier and Hyperbolic Series
βΊThroughout this section $q$ and $\zeta$ are defined as in §22.2. 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 $q\exp\left(2|\Im\zeta|\right)<1$, … βΊ
##### 7: 20.11 Generalizations and Analogs
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###### §20.11(ii) Ramanujan’s Theta Function and $q$-Series
βΊWith the substitutions $a=qe^{2iz}$, $b=qe^{-2iz}$, with $q=e^{i\pi\tau}$, we have … βΊIn the case $z=0$ identities for theta functions become identities in the complex variable $q$, with $\left|q\right|<1$, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … βΊAs in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions22.2), and Weierstrass elliptic functions23.6(ii)) can be expanded in $q$-series via (20.9.1). … βΊFor $m=1,2,3,4$, $n=1,2,3,4$, and $m\neq n$, define twelve combined theta functions $\varphi_{m,n}\left(z,q\right)$ by …
18.28.3 $2\pi\sin\theta\,w(\cos\theta)={\left|\frac{\left(e^{2i\theta};q\right)_{\infty% }}{\left(ae^{i\theta},be^{i\theta},ce^{i\theta},de^{i\theta};q\right)_{\infty}% }\right|}^{2},$
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 $a=\overline{b}$; $|ab|<1$; $|a|,|b|\leq 1$.
18.28.15 $\frac{1}{2\pi}\int_{0}^{\pi}C_{n}\left(\cos\theta;\beta\,|\,q\right)C_{m}\left% (\cos\theta;\beta\,|\,q\right)\*{\left|\frac{\left(e^{2\mathrm{i}\theta};q% \right)_{\infty}}{\left(\beta e^{2\mathrm{i}\theta};q\right)_{\infty}}\right|}% ^{2}\,\mathrm{d}\theta=\frac{\left(\beta,\beta q;q\right)_{\infty}}{\left(% \beta^{2},q;q\right)_{\infty}}\frac{(1-\beta)\left(\beta^{2};q\right)_{n}}{(1-% \beta q^{n})\left(q;q\right)_{n}}\delta_{n,m},$ $-1<\beta<1$.
18.28.17 $\frac{1}{2\pi}\int_{0}^{\pi}H_{n}\left(\cos\theta\,|\,q\right)H_{m}\left(\cos% \theta\,|\,q\right){\left|\left(e^{2\mathrm{i}\theta};q\right)_{\infty}\right|% }^{2}\,\mathrm{d}\theta=\frac{\delta_{n,m}}{\left(q^{n+1};q\right)_{\infty}}.$
18.28.18 $h_{n}\left(\sinh t\,|\,q\right)=\sum_{\ell=0}^{n}q^{\frac{1}{2}\ell(\ell+1)}% \frac{\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}e^{(n-2\ell)t}=e^{% nt}{{}_{1}\phi_{1}}\left({q^{-n}\atop 0};q,-qe^{-2t}\right)={\mathrm{i}}^{-n}H% _{n}\left(\mathrm{i}\sinh t\,|\,q^{-1}\right).$
##### 9: 18.27 $q$-Hahn Class
18.27.20 $\int_{0}^{\infty}S_{n}\left(q^{\frac{1}{2}}x;q\right)S_{m}\left(q^{\frac{1}{2}% }x;q\right)\exp\left(-\frac{(\ln x)^{2}}{2\ln\left(q^{-1}\right)}\right)\,% \mathrm{d}x=\frac{\sqrt{2\pi q^{-1}\ln\left(q^{-1}\right)}}{q^{n}\left(q;q% \right)_{n}}\delta_{n,m}.$