# §17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions

## Jacobi’s Triple Product

 17.8.1 $\sum_{n=-\infty}^{\infty}(-z)^{n}q^{n(n-1)/2}=\left(q,z,q/z;q\right)_{\infty};$

compare (20.5.9).

## Ramanujan’s ${{}_{1}\psi_{1}}$ Summation

 17.8.2 ${{}_{1}\psi_{1}}\left({a\atop b};q,z\right)=\frac{\left(q,b/a,az,q/(az);q% \right)_{\infty}}{\left(b,q/a,z,b/(az);q\right)_{\infty}}.$

## Quintuple Product Identity

 17.8.3 $\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n(3n-1)/2}z^{3n}(1+zq^{n})=\left(q,-z,-q/z% ;q\right)_{\infty}\left(qz^{2},q/{z^{2}};q^{2}\right)_{\infty}.$

## Bailey’s Bilateral Summations

 17.8.4 $\displaystyle{{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)$ $\displaystyle=\frac{\left(aq/(bc);q\right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c% ^{2},q^{2},aq,q/a;q^{2}\right)_{\infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q% \right)_{\infty}},$ 17.8.5 $\displaystyle{{}_{3}\psi_{3}}\left({b,c,d\atop q/b,q/c,q/d};q,\frac{q}{bcd}\right)$ $\displaystyle=\frac{\left(q,q/(bc),q/(bd),q/(cd);q\right)_{\infty}}{\left(q/b,% q/c,q/d,q/(bcd);q\right)_{\infty}},$
 17.8.6 ${{}_{4}\psi_{4}}\left({-qa^{\frac{1}{2}},b,c,d\atop-a^{\frac{1}{2}},aq/b,aq/c,% aq/d};q,\frac{qa^{\frac{3}{2}}}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/% (cd),qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,qa^{\frac{1}{2}}/d,q,q/a;q\right)_{% \infty}}{\left(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa^{\frac{1}{2}},qa^{-\frac{1}{2}},% qa^{\frac{3}{2}}/(bcd);q\right)_{\infty}},$
 17.8.7 ${{}_{6}\psi_{6}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e};q,\frac{qa^{2}}{bcde}\right% )=\frac{\left(aq,aq/(bc),aq/(bd),aq/(be),aq/(cd),aq/(ce),aq/(de),q,q/a;q\right% )_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/(bcde);q\right)_{% \infty}}.$