# §17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

## §17.9(i) ${{}_{2}\phi_{1}}\to{{}_{2}\phi_{2}}$, ${{}_{3}\phi_{1}}$, or ${{}_{3}\phi_{2}}$

### F. H. Jackson’s Transformations

 17.9.1 $\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)$ $\displaystyle=\frac{\left(za;q\right)_{\infty}}{\left(z;q\right)_{\infty}}{{}_% {2}\phi_{2}}\left({a,c/b\atop c,az};q,bz\right),$ 17.9.2 $\displaystyle{{}_{2}\phi_{1}}\left({q^{-n},b\atop c};q,z\right)$ $\displaystyle=\frac{\left(c/b;q\right)_{n}}{\left(c;q\right)_{n}}b^{n}{{}_{3}% \phi_{1}}\left({q^{-n},b,q/z\atop bq^{1-n}/c};q,z/c\right),$ ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $n$: nonnegative integer and $z$: complex variable Referenced by: Equation (17.9.2) Permalink: http://dlmf.nist.gov/17.9.E2 Encodings: TeX, pMML, png Correction (effective with 1.0.14): The entry $q/c$ in the first row of ${{}_{3}\phi_{1}}\left({q^{-n},b,q/c\atop bq^{1-n}/c};q,z/c\right)$ was replaced by $q/z$. Reported 2016-08-30 by Xinrong Ma See also: Annotations for 17.9(i), 17.9(i), 17.9 and 17 17.9.3 $\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)$ $\displaystyle=\frac{\left(abz/c;q\right)_{\infty}}{\left(bz/c;q\right)_{\infty% }}{{}_{3}\phi_{2}}\left({a,c/b,0\atop c,cq/bz};q,q\right),$ 17.9.4 $\displaystyle{{}_{2}\phi_{1}}\left({q^{-n},b\atop c};q,z\right)$ $\displaystyle=\frac{\left(c/b;q\right)_{n}}{\left(c;q\right)_{n}}\left(\frac{% bz}{q}\right)^{n}{{}_{3}\phi_{2}}\left({q^{-n},q/z,q^{1-n}/c\atop bq^{1-n}/c,0% };q,q\right),$ 17.9.5 $\displaystyle{{}_{2}\phi_{1}}\left({q^{-n},b\atop c};q,z\right)$ $\displaystyle=\frac{\left(c/b;q\right)_{n}}{\left(c;q\right)_{n}}{{}_{3}\phi_{% 2}}\left({q^{-n},b,bzq^{-n}/c\atop bq^{1-n}/c,0};q,q\right).$

## §17.9(ii) ${{}_{3}\phi_{2}}\to{{}_{3}\phi_{2}}$

### Transformations of ${{}_{3}\phi_{2}}$-Series

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

### $q$-Sheppard Identity

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

## §17.9(iii) Further ${{}_{r}\phi_{s}}$ Functions

### Sears’ Balanced ${{}_{4}\phi_{3}}$ Transformations

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

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

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

With $n$ a nonnegative integer

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

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

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

### Sears–Carlitz Transformation

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

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

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

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

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

## §17.9(iv) Bibasic Series

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