# §17.10 Transformations of $\mathop{{{}_{r}\psi_{r}}\/}\nolimits$ Functions

## Bailey’s $\mathop{{{}_{2}\psi_{2}}\/}\nolimits$ Transformations

 17.10.1 $\mathop{{{}_{2}\psi_{2}}\/}\nolimits\!\left({a,b\atop c,d};q,z\right)=\frac{% \left(az,d/a,c/b,dq/(abz);q\right)_{\infty}}{\left(z,d,q/b,cd/(abz);q\right)_{% \infty}}\mathop{{{}_{2}\psi_{2}}\/}\nolimits\!\left({a,abz/d\atop az,c};q,% \frac{d}{a}\right),$
 17.10.2 $\mathop{{{}_{2}\psi_{2}}\/}\nolimits\!\left({a,b\atop c,d};q,z\right)=\frac{% \left(az,bz,cq/(abz),dq/(abz);q\right)_{\infty}}{\left(q/a,q/b,c,d;q\right)_{% \infty}}\mathop{{{}_{2}\psi_{2}}\/}\nolimits\!\left({abz/c,abz/d\atop az,bz};q% ,\frac{cd}{abz}\right).$

## Other Transformations

 17.10.3 $\mathop{{{}_{8}\psi_{8}}\/}\nolimits\!\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}% },c,d,e,f,aq^{-n},q^{-n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,% aq/f,q^{n+1},aq^{n+1}};q,\frac{a^{2}q^{2n+2}}{cdef}\right)=\frac{\left(aq,q/a,% aq/(cd),aq/(ef);q\right)_{n}}{\left(q/c,q/d,aq/e,aq/f;q\right)_{n}}\*\mathop{{% {}_{4}\psi_{4}}\/}\nolimits\!\left({e,f,aq^{n+1}/(cd),q^{-n}\atop aq/c,aq/d,q^% {n+1},ef/(aq^{n})};q,q\right),$
 17.10.4 $\mathop{{{}_{2}\psi_{2}}\/}\nolimits\!\left({e,f\atop aq/c,aq/d};q,\frac{aq}{% ef}\right)=\frac{\left(q/c,q/d,aq/e,aq/f;q\right)_{\infty}}{\left(aq,q/a,aq/(% cd),aq/(ef);q\right)_{\infty}}\*\sum_{n=-\infty}^{\infty}\frac{(1-aq^{2n})% \left(c,d,e,f;q\right)_{n}}{(1-a)\left(aq/c,aq/d,aq/e,aq/f;q\right)_{n}}\left(% \frac{qa^{3}}{cdef}\right)^{n}q^{n^{2}}.$
 17.10.5 $\frac{\left(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q\right)_{\infty}}% {\left(fa,ga,f/a,g/a,qa^{2},q/a^{2};q\right)_{\infty}}\*\mathop{{{}_{8}\psi_{8% }}\/}\nolimits\!\left({qa,-qa,ba,ca,da,ea,fa,ga\atop a,-a,aq/b,aq/c,aq/d,aq/e,% aq/f,aq/g};q,\frac{q^{2}}{bcdefg}\right)=\frac{\left(q,q/(bf),q/(cf),q/(df),q/% (ef),qf/b,qf/c,qf/d,qf/e;q\right)_{\infty}}{\left(fa,q/(fa),aq/f,f/a,g/f,fg,qf% ^{2};q\right)_{\infty}}\*\mathop{{{}_{8}\phi_{7}}\/}\nolimits\!\left({f^{2},qf% ,-qf,fb,fc,fd,fe,fg\atop f,-f,fq/b,fq/c,fq/d,fq/e,fq/g};q,\frac{q^{2}}{bcdefg}% \right)+\mathop{\mathrm{idem}\/}\nolimits\!\left(f;g\right).$
 17.10.6 $\frac{\left(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q% \right)_{\infty}}{\left(ag,ah,ak,g/a,h/a,k/a,qa^{2},q/a^{2};q\right)_{\infty}}% \*\mathop{{{}_{10}\psi_{10}}\/}\nolimits\!\left({qa,-qa,ba,ca,da,ea,fa,ga,ha,% ka\atop a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^{2}}{bcdefghk}% \right)=\frac{\left(q,q/(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,% qg/f;q\right)_{\infty}}{\left(gh,gk,h/g,ag,q/(ag),g/a,aq/g,qg^{2};q\right)_{% \infty}}\*\mathop{{{}_{10}\phi_{9}}\/}\nolimits\!\left({g^{2},qg,-qg,gb,gc,gd,% ge,gf,gh,gk\atop g,-g,qg/b,qg/c,qg/d,qg/e,qg/f,qg/h,qg/k};q,\frac{q^{2}}{% bcdefghk}\right)+\mathop{\mathrm{idem}\/}\nolimits\!\left(g;h,k\right).$