# Pochhammer symbol

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##### 1: 17.2 Calculus
For $n=0,1,2,\dots$, … …
17.2.6 $\left(a_{1},a_{2},\dots,a_{r};q\right)_{\infty}=\prod_{j=1}^{r}\left(a_{j};q% \right)_{\infty}.$
17.2.19 $\left(a;q\right)_{2n}=\left(a,aq;q^{2}\right)_{n},$
17.2.20 $\left(a;q\right)_{kn}=\left(a,aq,\dots,aq^{k-1};q^{k}\right)_{n}.$
##### 2: 5.2 Definitions
###### §5.2(iii) Pochhammer’s Symbol
5.2.5 ${\left(a\right)_{n}}=\Gamma\left(a+n\right)/\Gamma\left(a\right),$ $a\neq 0,-1,-2,\dots$.
5.2.6 ${\left(-a\right)_{n}}=(-1)^{n}{\left(a-n+1\right)_{n}},$
5.2.7 ${\left(-m\right)_{n}}=\begin{cases}\frac{(-1)^{n}m!}{(m-n)!},&0\leq n\leq m,\\ 0,&n>m,\end{cases}$
##### 3: 15.15 Sums
15.15.1 $\mathbf{F}\left({a,b\atop c};\frac{1}{z}\right)=\left(1-\frac{z_{0}}{z}\right)% ^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{s!}\*\mathbf{F}\left({-s,b% \atop c};\frac{1}{z_{0}}\right)\left(1-\frac{z}{z_{0}}\right)^{-s}.$
##### 4: 17.11 Transformations of $q$-Appell Functions
17.11.1 $\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx,b^{\prime}y;q% \right)_{\infty}}{\left(c,x,y;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c/a,x,y% \atop bx,b^{\prime}y};q,a\right),$
17.11.2 $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)=\frac{\left(b,ax;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a,b% ^{\prime};q\right)_{n}\left(c/b,x;q\right)_{r}b^{r}y^{n}}{\left(q,c^{\prime};q% \right)_{n}\left(q;q\right)_{r}\left(ax;q\right)_{n+r}},$
17.11.3 $\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a^{% \prime},b^{\prime};q\right)_{n}\left(x;q\right)_{r}\left(c/a;q\right)_{n+r}a^{% r}y^{n}}{\left(q,c/a;q\right)_{n}\left(q,bx;q\right)_{r}}.$
17.11.4 $\sum_{m_{1},\dots,m_{n}\geqq 0}\frac{\left(a;q\right)_{m_{1}+m_{2}+\cdots+m_{n% }}\left(b_{1};q\right)_{m_{1}}\left(b_{2};q\right)_{m_{2}}\cdots\left(b_{n};q% \right)_{m_{n}}x_{1}^{m_{1}}x_{2}^{m_{2}}\cdots x_{n}^{m_{n}}}{\left(q;q\right% )_{m_{1}}\left(q;q\right)_{m_{2}}\cdots\left(q;q\right)_{m_{n}}\left(c;q\right% )_{m_{1}+m_{2}+\cdots+m_{n}}}=\frac{\left(a,b_{1}x_{1},b_{2}x_{2},\dots,b_{n}x% _{n};q\right)_{\infty}}{\left(c,x_{1},x_{2},\dots,x_{n};q\right)_{\infty}}{{}_% {n+1}\phi_{n}}\left({c/a,x_{1},x_{2},\dots,x_{n}\atop b_{1}x_{1},b_{2}x_{2},% \dots,b_{n}x_{n}};q,a\right).$
##### 5: 17.12 Bailey Pairs
17.12.3 $\beta_{n}=\sum_{j=0}^{n}\frac{\alpha_{j}}{\left(q;q\right)_{n-j}\left(aq;q% \right)_{n+j}}.$
17.12.4 $\sum_{n=0}^{\infty}q^{n^{2}}a^{n}\beta_{n}=\frac{1}{\left(aq;q\right)_{\infty}% }\sum_{n=0}^{\infty}q^{n^{2}}a^{n}\alpha_{n}.$
##### 6: 17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions
17.5.1 ${{}_{0}\phi_{0}}\left(-;-;q,z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{% \genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(q;q\right)_{n}}=\left(z;q\right)_{% \infty},$ $|z|<1$;
17.5.2 ${{}_{1}\phi_{0}}\left(a;-;q,z\right)=\frac{\left(az;q\right)_{\infty}}{\left(z% ;q\right)_{\infty}},$ $|z|<1$;
17.5.3 ${{}_{1}\phi_{0}}\left(q^{-n};-;q,z\right)=\left(zq^{-n};q\right)_{n}.$
17.5.4 ${{}_{1}\phi_{0}}\left(0;-;q,z\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{\left(q;q% \right)_{n}}=\frac{1}{\left(z;q\right)_{\infty}},$ $|z|<1$;
17.5.5 ${{}_{1}\phi_{1}}\left({a\atop c};q,c/a\right)=\frac{\left(c/a;q\right)_{\infty% }}{\left(c;q\right)_{\infty}},$ $|c|<|a|$.
##### 7: 17.3 $q$-Elementary and $q$-Special Functions
17.3.1 $e_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}x^{n}}{\left(q;q\right)_% {n}}=\frac{1}{\left((1-q)x;q\right)_{\infty}},$
17.3.2 $E_{q}\left(x\right)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}q^{\genfrac{(}{)}{0.0pt}% {}{n}{2}}x^{n}}{\left(q;q\right)_{n}}=\left(-(1-q)x;q\right)_{\infty}.$
17.3.3 $\mathrm{sin}_{q}\left(x\right)=\frac{1}{2i}(e_{q}\left(ix\right)-e_{q}\left(-% ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}(-1)^{n}x^{2n+1}}{\left(q;q% \right)_{2n+1}},$
17.3.4 $\mathrm{Sin}_{q}\left(x\right)=\frac{1}{2i}(E_{q}\left(ix\right)-E_{q}\left(-% ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}q^{n(2n+1)}(-1)^{n}x^{2n+1}}{% \left(q;q\right)_{2n+1}}.$
17.3.5 $\mathrm{cos}_{q}\left(x\right)=\frac{1}{2}(e_{q}\left(ix\right)+e_{q}\left(-ix% \right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}(-1)^{n}x^{2n}}{\left(q;q\right)_{% 2n}},$
##### 8: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions
17.8.1 $\sum_{n=-\infty}^{\infty}(-z)^{n}q^{n(n-1)/2}=\left(q,z,q/z;q\right)_{\infty};$
17.8.2 ${{}_{1}\psi_{1}}\left({a\atop b};q,z\right)=\frac{\left(q,b/a,az,q/(az);q% \right)_{\infty}}{\left(b,q/a,z,b/(az);q\right)_{\infty}}.$
17.8.3 $\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n(3n-1)/2}z^{3n}(1+zq^{n})=\left(q,-z,-q/z% ;q\right)_{\infty}\left(qz^{2},q/{z^{2}};q^{2}\right)_{\infty}.$
17.8.4 ${{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)=\frac{\left(aq/(bc);q% \right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c^{2},q^{2},aq,q/a;q^{2}\right)_{% \infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q\right)_{\infty}},$
17.8.8 ${{}_{2}\psi_{2}}\left({b^{2},\ifrac{b^{2}}{c}\atop q,cq};q^{2},\ifrac{cq^{2}}{% b^{2}}\right)=\frac{1}{2}\frac{\left(q^{2},qb^{2},\ifrac{q}{b^{2}},\ifrac{cq}{% b^{2}};q^{2}\right)_{\infty}}{\left(cq,\ifrac{cq^{2}}{b^{2}},\ifrac{q^{2}}{b^{% 2}},\ifrac{c}{b^{2}};q^{2}\right)_{\infty}}\left(\frac{\left(\ifrac{c\sqrt{q}}% {b};q\right)_{\infty}}{\left(b\sqrt{q};q\right)_{\infty}}+\frac{\left(\ifrac{-% c\sqrt{q}}{b};q\right)_{\infty}}{\left(-b\sqrt{q};q\right)_{\infty}}\right),$ $\left|cq^{2}\right|<\left|b^{2}\right|$.
##### 9: 17.13 Integrals
17.13.1 $\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-ax/c;q\right)_{\infty}\left(bx/d;q\right)_{\infty}}{\mathrm{d}}_{q}x=% \frac{(1-q)\left(q;q\right)_{\infty}\left(ab;q\right)_{\infty}cd\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(a;q\right)_{\infty}\left(b% ;q\right)_{\infty}(c+d)\left(-bc/d;q\right)_{\infty}\left(-ad/c;q\right)_{% \infty}},$
17.13.2 $\int_{-c}^{d}\frac{\left(-qx/c;q\right)_{\infty}\left(qx/d;q\right)_{\infty}}{% \left(-xq^{\alpha}/c;q\right)_{\infty}\left(xq^{\beta}/d;q\right)_{\infty}}{% \mathrm{d}}_{q}x=\frac{\Gamma_{q}\left(\alpha\right)\Gamma_{q}\left(\beta% \right)}{\Gamma_{q}\left(\alpha+\beta\right)}\frac{cd}{c+d}\frac{\left(-c/d;q% \right)_{\infty}\left(-d/c;q\right)_{\infty}}{\left(-q^{\beta}c/d;q\right)_{% \infty}\left(-q^{\alpha}d/c;q\right)_{\infty}}.$
17.13.3 $\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}% {\left(-t;q\right)_{\infty}}\mathrm{d}t=\frac{\Gamma\left(\alpha\right)\Gamma% \left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-\alpha% \right)\Gamma_{q}\left(\alpha+\beta\right)},$
17.13.4 $\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-ctq^{\alpha+\beta};q\right)_{\infty}% }{\left(-ct;q\right)_{\infty}}{\mathrm{d}}_{q}t=\frac{\Gamma_{q}\left(\alpha% \right)\Gamma_{q}\left(\beta\right)\left(-cq^{\alpha};q\right)_{\infty}\left(-% q^{1-\alpha}/c;q\right)_{\infty}}{\Gamma_{q}\left(\alpha+\beta\right)\left(-c;% q\right)_{\infty}\left(-q/c;q\right)_{\infty}}.$
##### 10: 18.28 Askey–Wilson Class
18.28.3 $2\pi\sin\theta\,w(\cos\theta)={\left|\frac{\left(e^{2i\theta};q\right)_{\infty% }}{\left(ae^{i\theta},be^{i\theta},ce^{i\theta},de^{i\theta};q\right)_{\infty}% }\right|}^{2},$
18.28.4 $h_{0}=\frac{\left(abcd;q\right)_{\infty}}{\left(q,ab,ac,ad,bc,bd,cd;q\right)_{% \infty}},$
18.28.5 $h_{n}=h_{0}\frac{(1-abcdq^{n-1})\left(q,ab,ac,ad,bc,bd,cd;q\right)_{n}}{(1-% abcdq^{2n-1})\left(abcd;q\right)_{n}},$ $n=1,2,\dotsc$.
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 $\frac{1}{2\pi}\int_{0}^{\pi}C_{n}\left(\cos\theta;\beta\,|\,q\right)C_{m}\left% (\cos\theta;\beta\,|\,q\right)\*{\left|\frac{\left(e^{2\mathrm{i}\theta};q% \right)_{\infty}}{\left(\beta e^{2\mathrm{i}\theta};q\right)_{\infty}}\right|}% ^{2}\mathrm{d}\theta=\frac{\left(\beta,\beta q;q\right)_{\infty}}{\left(\beta^% {2},q;q\right)_{\infty}}\frac{(1-\beta)\left(\beta^{2};q\right)_{n}}{(1-\beta q% ^{n})\left(q;q\right)_{n}}\delta_{n,m},$ $-1<\beta<1$.