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##### 1: 17.14 Constant Term Identities
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17.14.2 $\sum_{n=0}^{\infty}\frac{q^{n(n+1)}}{\left(q^{2};q^{2}\right)_{n}\left(-q;q^{2% }\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)% _{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}% }{\left(z^{-1}q^{2};q^{2}\right)_{\infty}\left(-q;q^{2}\right)_{\infty}\left(z% ^{-1}q;q^{2}\right)_{\infty}}=\frac{1}{\left(-q;q^{2}\right)_{\infty}}\mbox{ % coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q% ;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-1}q;q\right% )_{\infty}}=\frac{H(q)}{\left(-q;q^{2}\right)_{\infty}},$
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17.14.3 $\sum_{n=0}^{\infty}\frac{q^{n(n+1)}}{\left(q^{2};q^{2}\right)_{n}\left(-q;q^{2% }\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)% _{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}% }{\left(z^{-1};q^{2}\right)_{\infty}\left(-q;q^{2}\right)_{\infty}\left(z^{-1}% q;q^{2}\right)_{\infty}}=\frac{1}{\left(-q;q^{2}\right)_{\infty}}\mbox{ coeff.% of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q;q^{2}% \right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-1};q\right)_{% \infty}}=\frac{G(q)}{\left(-q;q^{2}\right)_{\infty}},$
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17.14.4 $\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{\left(q^{2};q^{2}\right)_{n}\left(q;q^{2}% \right)_{n}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{% \infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{% \left(-z^{-1};q^{2}\right)_{\infty}\left(q;q^{2}\right)_{\infty}\left(z^{-1};q% ^{2}\right)_{\infty}}=\frac{1}{\left(q;q^{2}\right)_{\infty}}\mbox{ coeff. of % }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-z^{-1}q;q^{2}% \right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(z^{-2};q^{4}\right)_{% \infty}}=\frac{G(q^{4})}{\left(q;q^{2}\right)_{\infty}},$
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17.14.5 $\sum_{n=0}^{\infty}\frac{q^{n^{2}+2n}}{\left(q^{2};q^{2}\right)_{n}\left(q;q^{% 2}\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right% )_{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty% }}{\left(-q^{2}z^{-1};q^{2}\right)_{\infty}\left(q;q^{2}\right)_{\infty}\left(% z^{-1}q^{2};q^{2}\right)_{\infty}}=\frac{1}{\left(q;q^{2}\right)_{\infty}}% \mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-% z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(q^{4}z^{% -2};q^{4}\right)_{\infty}}=\frac{H(q^{4})}{\left(q;q^{2}\right)_{\infty}}.$
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##### 3: 17.13 Integrals
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17.13.1 $\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-ax/c;q\right)_{\infty}\left(bx/d;q\right)_{\infty}}\,{\mathrm{d}}_{q}x=% \frac{(1-q)\left(q;q\right)_{\infty}\left(ab;q\right)_{\infty}cd\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(a;q\right)_{\infty}\left(b% ;q\right)_{\infty}(c+d)\left(-bc/d;q\right)_{\infty}\left(-ad/c;q\right)_{% \infty}},$
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17.13.2 $\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-xq^{\alpha}/c;q\right)_{\infty}\left(xq^{\beta}/d;q\right)_{\infty}}\,{% \mathrm{d}}_{q}x=\frac{\Gamma_{q}\left(\alpha\right)\Gamma_{q}\left(\beta% \right)}{\Gamma_{q}\left(\alpha+\beta\right)}\frac{cd}{c+d}\frac{\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(-q^{\beta}c/d;q\right)_{% \infty}\left(-q^{\alpha}d/c;q\right)_{\infty}}.$
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17.13.3 $\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}% {\left(-t;q\right)_{\infty}}\,\mathrm{d}t=\frac{\Gamma\left(\alpha\right)% \Gamma\left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-% \alpha\right)\Gamma_{q}\left(\alpha+\beta\right)},$
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17.13.4 $\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-ctq^{\alpha+\beta};q\right)_{\infty}% }{\left(-ct;q\right)_{\infty}}\,{\mathrm{d}}_{q}t=\frac{\Gamma_{q}\left(\alpha% \right)\Gamma_{q}\left(\beta\right)\left(-cq^{\alpha};q\right)_{\infty}\left(-% q^{1-\alpha}/c;q\right)_{\infty}}{\Gamma_{q}\left(\alpha+\beta\right)\left(-c;% q\right)_{\infty}\left(-q/c;q\right)_{\infty}}.$
##### 4: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
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22.12.2 $2Kk\operatorname{sn}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{% \sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(% \sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-(n+\frac{1}{2})\tau}\right),$
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22.12.3 $2iKk\operatorname{cn}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n% }\pi}{\sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}% \left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t-m-(n+\frac{1}{2})\tau}% \right),$
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22.12.8 $2K\operatorname{dc}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin% \left(\pi(t+\frac{1}{2}-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=% -\infty}^{\infty}\frac{(-1)^{m}}{t+\frac{1}{2}-m-n\tau}\right),$
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22.12.11 $2K\operatorname{ns}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{\pi}{\sin% \left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{% \infty}\frac{(-1)^{m}}{t-m-n\tau}\right),$
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22.12.12 $2K\operatorname{ds}\left(2Kt,k\right)=\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}% \pi}{\sin\left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-% \infty}^{\infty}\frac{(-1)^{m+n}}{t-m-n\tau}\right),$
##### 5: 24.20 Tables
βΊAbramowitz and Stegun (1964, Chapter 23) includes exact values of $\sum_{k=1}^{m}k^{n}$, $m=1(1)100$, $n=1(1)10$; $\sum_{k=1}^{\infty}k^{-n}$, $\sum_{k=1}^{\infty}(-1)^{k-1}k^{-n}$, $\sum_{k=0}^{\infty}(2k+1)^{-n}$, $n=1,2,\dotsc$, 20D; $\sum_{k=0}^{\infty}(-1)^{k}(2k+1)^{-n}$, $n=1,2,\dotsc$, 18D. …
##### 6: 4.22 Infinite Products and Partial Fractions
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4.22.3 $\cot z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}-n^{2}\pi^{2}},$
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4.22.5 $\csc z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}-n^{2}\pi^{2}}.$
##### 7: 4.36 Infinite Products and Partial Fractions
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4.36.4 ${\operatorname{csch}}^{2}z=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}},$
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##### 8: 27.7 Lambert Series as Generating Functions
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27.7.1 $\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}.$
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27.7.2 $\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sum_{d% \mathbin{|}n}f(d)x^{n}.$
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27.7.3 $\sum_{n=1}^{\infty}\mu\left(n\right)\frac{x^{n}}{1-x^{n}}=x,$
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27.7.5 $\sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sigma_{% \alpha}\left(n\right)x^{n},$
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27.7.6 $\sum_{n=1}^{\infty}\lambda\left(n\right)\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{% \infty}x^{n^{2}}.$
##### 9: 32.6 Hamiltonian Structure
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32.6.16 $z\mathrm{H}_{\mbox{\scriptsize III}}(q,p,z)=q^{2}p^{2}-{\left(\kappa_{\infty}% zq^{2}+(2\theta_{0}+1)q-\kappa_{0}z\right)p}+\kappa_{\infty}(\theta_{0}+\theta% _{\infty})zq,$
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32.6.18 $zp^{\prime}=-2qp^{2}+2\kappa_{\infty}zqp+(2\theta_{0}+1)p-\kappa_{\infty}(% \theta_{0}+\theta_{\infty})z.$
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32.6.19 $(\alpha,\beta,\gamma,\delta)=\left(-2\kappa_{\infty}\theta_{\infty},2\kappa_{0% }(\theta_{0}+1),\kappa_{\infty}^{2},-\kappa_{0}^{2}\right).$
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32.6.21 $(z\sigma^{\prime\prime}-\sigma^{\prime})^{2}+2\left((\sigma^{\prime})^{2}-% \kappa_{0}^{2}\kappa_{\infty}^{2}z^{2}\right)(z\sigma^{\prime}-2\sigma)+8% \kappa_{0}\kappa_{\infty}\theta_{0}\theta_{\infty}z\sigma^{\prime}=4\kappa_{0}% ^{2}\kappa_{\infty}^{2}(\theta_{0}^{2}+\theta_{\infty}^{2})z^{2}.$
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32.6.29 $\zeta^{2}(\sigma^{\prime\prime})^{2}+\left(4(\sigma^{\prime})^{2}-\eta_{0}^{2}% \eta_{\infty}^{2}\right)(\zeta\sigma^{\prime}-\sigma)+\eta_{0}\eta_{\infty}% \theta_{0}\theta_{\infty}\sigma^{\prime}=\tfrac{1}{4}\eta_{0}^{2}\eta_{\infty}% ^{2}(\theta_{0}^{2}+\theta_{\infty}^{2}).$
##### 10: 20.6 Power Series
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20.6.2 $\theta_{1}\left(\pi z\middle|\tau\right)=\pi z\theta_{1}'\left(0\middle|\tau% \right)\exp\left(-\sum_{j=1}^{\infty}\frac{1}{2j}\delta_{2j}(\tau)z^{2j}\right),$
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20.6.6 $\delta_{2j}(\tau)=\left.\sum_{n=-\infty}^{\infty}\sum_{\begin{subarray}{c}m=-% \infty\\ \left|m\right|+\left|n\right|\neq 0\end{subarray}}^{\infty}\right.(m+n\tau)^{-% 2j},$
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20.6.7 $\alpha_{2j}(\tau)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m-\tfrac{% 1}{2}+n\tau)^{-2j},$
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20.6.8 $\beta_{2j}(\tau)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m-\tfrac{1% }{2}+(n-\tfrac{1}{2})\tau)^{-2j},$
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20.6.9 $\gamma_{2j}(\tau)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m+(n-% \tfrac{1}{2})\tau)^{-2j},$