F. H. Jackson’s Transformations

 17.9.1 $\displaystyle\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)$ $\displaystyle=\frac{\left(za;q\right)_{\infty}}{\left(z;q\right)_{\infty}}% \mathop{{{}_{2}\phi_{2}}\/}\nolimits\!\left({a,c/b\atop c,az};q,bz\right),$ 17.9.2 $\displaystyle\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({q^{-n},b\atop c};q,z\right)$ $\displaystyle=\frac{\left(c/b;q\right)_{n}}{\left(c;q\right)_{n}}b^{n}\mathop{% {{}_{3}\phi_{1}}\/}\nolimits\!\left({q^{-n},b,q/c\atop bq^{1-n}/c};q,z/c\right),$ 17.9.3 $\displaystyle\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)$ $\displaystyle=\frac{\left(abz/c;q\right)_{\infty}}{\left(bz/c;q\right)_{\infty% }}\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({a,c/b,0\atop c,cq/bz};q,q\right),$ 17.9.4 $\displaystyle\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({q^{-n},b\atop c};q,z\right)$ $\displaystyle=\frac{\left(c/b;q\right)_{n}}{\left(c;q\right)_{n}}\left(\frac{% bz}{q}\right)^{n}\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({q^{-n},q/z,q^{1-% n}/c\atop bq^{1-n}/c,0};q,q\right),$ 17.9.5 $\displaystyle\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({q^{-n},b\atop c};q,z\right)$ $\displaystyle=\frac{\left(c/b;q\right)_{n}}{\left(c;q\right)_{n}}\mathop{{{}_{% 3}\phi_{2}}\/}\nolimits\!\left({q^{-n},b,bzq^{-n}/c\atop bq^{1-n}/c,0};q,q% \right).$

Transformations of $\mathop{{{}_{3}\phi_{2}}\/}\nolimits$-Series

 17.9.6 $\displaystyle\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({a,b,c\atop d,e};q,de% /(abc)\right)$ $\displaystyle=\frac{\left(e/a,de/(bc);q\right)_{\infty}}{\left(e,de/(abc);q% \right)_{\infty}}\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({a,d/b,d/c\atop d% ,de/(bc)};q,e/a\right),$ 17.9.7 $\displaystyle\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({a,b,c\atop d,e};q,de% /(abc)\right)$ $\displaystyle=\frac{\left(b,de/(ab),de/(bc);q\right)_{\infty}}{\left(d,e,de/(% abc);q\right)_{\infty}}\*\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({d/b,e/b,% de/(abc)\atop de/(ab),de/(bc)};q,b\right),$ 17.9.8 $\displaystyle\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({q^{-n},b,c\atop d,e}% ;q,q\right)$ $\displaystyle=\frac{\left(de/(bc);q\right)_{n}}{\left(e;q\right)_{n}}\left(% \frac{bc}{d}\right)^{n}\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({q^{-n},d/b% ,d/c\atop d,de/(bc)};q,q\right),$ 17.9.9 $\displaystyle\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({q^{-n},b,c\atop d,e}% ;q,q\right)$ $\displaystyle=\frac{\left(e/c;q\right)_{n}}{\left(e;q\right)_{n}}c^{n}\mathop{% {{}_{3}\phi_{2}}\/}\nolimits\!\left({q^{-n},c,d/b\atop d,cq^{1-n}/e};q,\frac{% bq}{e}\right),$ 17.9.10 $\displaystyle\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({q^{-n},b,c\atop d,e}% ;q,\frac{deq^{n}}{bc}\right)$ $\displaystyle=\frac{\left(e/c;q\right)_{n}}{\left(e;q\right)_{n}}\mathop{{{}_{% 3}\phi_{2}}\/}\nolimits\!\left({q^{-n},c,d/b\atop d,cq^{1-n}/e};q,q\right).$

$q$-Sheppard Identity

 17.9.11 $\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({q^{-n},b,c\atop d,e};q,q\right)=% \frac{\left(e/c,d/c;q\right)_{n}}{\left(e,d;q\right)_{n}}c^{n}\mathop{{{}_{3}% \phi_{2}}\/}\nolimits\!\left({q^{-n},c,\ifrac{cbq^{1-n}}{(de)}\atop\ifrac{cq^{% 1-n}}{e},\ifrac{cq^{1-n}}{d}};q,q\right),$
 17.9.12 $\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({a,b,c\atop d,e};q,\frac{de}{abc}% \right)=\frac{\left(e/b,e/c,cq/a,q/d;q\right)_{\infty}}{\left(e,cq/d,q/a,e/(bc% );q\right)_{\infty}}\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({c,d/a,cq/e% \atop cq/a,bcq/e};q,\frac{bq}{d}\right)-\frac{\left(q/d,eq/d,b,c,d/a,de/(bcq),% bcq^{2}/(de);q\right)_{\infty}}{\left(d/q,e,bq/d,cq/d,q/a,e/(bc),bcq/e;q\right% )_{\infty}}\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({aq/d,bq/d,cq/d\atop q^% {2}/d,eq/d};q,\frac{de}{abc}\right),$
 17.9.13 $\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({a,b,c\atop d,e};q,\frac{de}{abc}% \right)=\frac{\left(e/b,e/c;q\right)_{\infty}}{\left(e,e/(bc);q\right)_{\infty% }}\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({d/a,b,c\atop d,bcq/e};q,q\right% )+\frac{\left(d/a,b,c,de/(bc);q\right)_{\infty}}{\left(d,e,bc/e,de/(abc);q% \right)_{\infty}}\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({e/b,e/c,de/(abc)% \atop de/(bc),eq/(bc)};q,q\right).$

Sears’ Balanced $\mathop{{{}_{4}\phi_{3}}\/}\nolimits$ Transformations

With $def=abcq^{1-n}$

 17.9.14 $\mathop{{{}_{4}\phi_{3}}\/}\nolimits\!\left({q^{-n},a,b,c\atop d,e,f};q,q% \right)=\frac{\left(e/a,f/a;q\right)_{n}}{\left(e,f;q\right)_{n}}a^{n}\mathop{% {{}_{4}\phi_{3}}\/}\nolimits\!\left({q^{-n},a,d/b,d/c\atop d,aq^{1-n}/e,aq^{1-% n}/f};q,q\right)=\frac{\left(a,ef/(ab),ef/(ac);q\right)_{n}}{\left(e,f,ef/(abc% );q\right)_{n}}\mathop{{{}_{4}\phi_{3}}\/}\nolimits\!\left({q^{-n},e/a,f/a,ef/% (abc)\atop ef/(ab),ef/(ac),q^{1-n}/a};q,q\right).$

Watson’s $q$-Analog of Whipple’s Theorem

With $n$ a nonnegative integer

 17.9.15 $\frac{\left(aq,aq/(de);q\right)_{n}}{\left(aq/d,aq/e;q\right)_{n}}\mathop{{{}_% {4}\phi_{3}}\/}\nolimits\!\left({aq/(bc),d,e,q^{-n}\atop aq/b,aq/c,deq^{-n}/a}% ;q,q\right)=\mathop{{{}_{8}\phi_{7}}\/}\nolimits\!\left({a,qa^{\frac{1}{2}},-% qa^{\frac{1}{2}},b,c,d,e,q^{-n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/% c,aq/d,aq/e,aq^{n+1}};q,\frac{a^{2}q^{2+n}}{bcde}\right).$

Bailey’s Transformation of Very-Well-Poised $\mathop{{{}_{8}\phi_{7}}\/}\nolimits$

 17.9.16 $\mathop{{{}_{8}\phi_{7}}\/}\nolimits\!\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{% 2}},b,c,d,e,f\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};% q,\frac{a^{2}q^{2}}{bcdef}\right)=\frac{\left(aq,aq/(de),aq/(df),aq/(ef);q% \right)_{\infty}}{\left(aq/d,aq/e,aq/f,aq/(def);q\right)_{\infty}}\mathop{{{}_% {4}\phi_{3}}\/}\nolimits\!\left({aq/(bc),d,e,f\atop aq/b,aq/c,def/a};q,q\right% )+\frac{\left(aq,aq/(bc),d,e,f,a^{2}q^{2}/(bdef),a^{2}q^{2}/(cdef);q\right)_{% \infty}}{\left(aq/b,aq/c,aq/d,aq/e,aq/f,a^{2}q^{2}/(bcdef),def/(aq);q\right)_{% \infty}}\*\mathop{{{}_{4}\phi_{3}}\/}\nolimits\!\left({aq/(de),aq/(df),aq/(ef)% ,a^{2}q^{2}/(bcdef)\atop a^{2}q^{2}/(bdef),a^{2}q^{2}/(cdef),aq^{2}/(def)};q,q% \right).$

Sears–Carlitz Transformation

With $a=q^{-n}$ and $n$ a nonnegative integer,

 17.9.17 $\mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({a,b,c\atop aq/b,aq/c};q,\frac{aqz% }{bc}\right)=\frac{\left(az;q\right)_{\infty}}{\left(z;q\right)_{\infty}}\*% \mathop{{{}_{5}\phi_{4}}\/}\nolimits\!\left({a^{\frac{1}{2}},-a^{\frac{1}{2}},% (aq)^{\frac{1}{2}},-(aq)^{\frac{1}{2}},aq/(bc)\atop aq/b,aq/c,az,q/z};q,q% \right).$

Gasper’s $q$-Analog of Clausen’s Formula

 17.9.18 $\left(\mathop{{{}_{4}\phi_{3}}\/}\nolimits\!\left({a,b,abz,ab/z\atop abq^{% \frac{1}{2}},-abq^{\frac{1}{2}},-ab};q,q\right)\right)^{2}=\mathop{{{}_{5}\phi% _{4}}\/}\nolimits\!\left({a^{2},b^{2},ab,abz,ab/z\atop abq^{\frac{1}{2}},-abq^% {\frac{1}{2}},-ab,a^{2}b^{2}};q,q\right),$

provided that the series expansions of both $\phi$’s terminate.

Mixed-Base Heine-Type Transformations

 17.9.19 $\sum_{n=0}^{\infty}\frac{\left(a;q^{2}\right)_{n}\left(b;q\right)_{n}}{\left(q% ^{2};q^{2}\right)_{n}\left(c;q\right)_{n}}z^{n}=\frac{\left(b;q\right)_{\infty% }\left(az;q^{2}\right)_{\infty}}{\left(c;q\right)_{\infty}\left(z;q^{2}\right)% _{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{2n}\left(z;q^{2}\right)% _{n}b^{2n}}{\left(q;q\right)_{2n}\left(az;q^{2}\right)_{n}}+\frac{\left(b;q% \right)_{\infty}\left(azq;q^{2}\right)_{\infty}}{\left(c;q\right)_{\infty}% \left(zq;q^{2}\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{2n% +1}\left(zq;q^{2}\right)_{n}b^{2n+1}}{\left(q;q\right)_{2n+1}\left(azq;q^{2}% \right)_{n}}.$
 17.9.20 $\sum_{n=0}^{\infty}\frac{\left(a;q^{k}\right)_{n}\left(b;q\right)_{kn}z^{n}}{% \left(q^{k};q^{k}\right)_{n}\left(c;q\right)_{kn}}=\frac{\left(b;q\right)_{% \infty}\left(az;q^{k}\right)_{\infty}}{\left(c;q\right)_{\infty}\left(z;q^{k}% \right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{n}\left(z;q^{k}% \right)_{n}b^{n}}{\left(q;q\right)_{n}\left(az;q^{k}\right)_{n}},$ $k=1,2,3,\dots$.