# §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: Erratum (V1.0.14) for 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 $q/z$. Suggested 2016-08-30 by Xinrong Ma See also: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17 17.9.3 $\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)$ $\displaystyle=\frac{\left(abz/c;q\right)_{\infty}}{\left(bz/c;q\right)_{\infty% }}{{}_{3}\phi_{2}}\left({a,c/b,0\atop c,cq/(bz)};q,q\right)+\frac{\left(a,bz,c% /b;q\right)_{\infty}}{\left(c,z,c/(bz);q\right)_{\infty}}{{}_{3}\phi_{2}}\left% ({z,abz/c,0\atop bz,bzq/c};q,q\right),$ ⓘ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable Referenced by: Erratum (V1.0.23) for Equation (17.9.3) Permalink: http://dlmf.nist.gov/17.9.E3 Encodings: TeX, pMML, png Correction (effective with 1.0.23): The second term on the right-hand side was missing. The form of the equation where the second term is missing is correct if the ${{}_{2}\phi_{1}}$ is terminating. It is this form which appeared in the first edition of Gasper and Rahman (1990). The more general version which appears now is what is reproduced in Gasper and Rahman (2004, (III.5)). Suggested 2019-04-26 by Roberto S. Costas-Santos See also: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17 17.9.3_5 $\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)$ $\displaystyle=\frac{\left(c/a,c/b;q\right)_{\infty}}{\left(c,c/(ab);q\right)_{% \infty}}{{}_{3}\phi_{2}}\left({a,b,abz/c\atop qab/c,0};q,q\right)+\frac{\left(% a,b,abz/c;q\right)_{\infty}}{\left(c,ab/c,z;q\right)_{\infty}}{{}_{3}\phi_{2}}% \left({c/a,c/b,z\atop qc/(ab),0};q,q\right),$ ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $q$: complex base and $z$: complex variable Source: Derivable from (17.9.13) after replacing $a$ with $d/a$ and then taking $d\rightarrow 0$. Then replace $a$, $b$, $c$, $e$ with $abz/c$, $a$, $b$, $c$ respectively. Referenced by: §17.9(i), Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/17.9.E3_5 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17 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 $\sum_{n=0}^{\infty}\frac{\left(a;q^{k}\right)_{n}\left(b;q\right)_{kn}z^{n}}{% \left(q^{k};q^{k}\right)_{n}\left(c;q\right)_{kn}}=\frac{\left(b;q\right)_{% \infty}\left(az;q^{k}\right)_{\infty}}{\left(c;q\right)_{\infty}\left(z;q^{k}% \right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(c/b;q\right)_{n}\left(z;q^{k}% \right)_{n}b^{n}}{\left(q;q\right)_{n}\left(az;q^{k}\right)_{n}},$ $k=1,2,3,\dots$.