# §17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

## §17.7(i) ${{}_{2}\phi_{2}}$ Functions

### $q$-Analog of Bailey’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

 17.7.1 ${{}_{2}\phi_{2}}\left({a,q/a\atop-q,b};q,-b\right)=\frac{\left(ab,bq/a;q^{2}% \right)_{\infty}}{\left(b;q\right)_{\infty}},$ $|b|<1$.

### $q$-Analog of Gauss’s ${{}_{2}F_{1}}\left(-1\right)$ Sum

 17.7.2 ${{}_{2}\phi_{2}}\left({a^{2},b^{2}\atop abq^{\frac{1}{2}},-abq^{\frac{1}{2}}};% q,-q\right)=\frac{\left(a^{2}q,b^{2}q;q^{2}\right)_{\infty}}{\left(q,a^{2}b^{2% }q;q^{2}\right)_{\infty}}.$

### Sum Related to (17.6.4)

 17.7.3 ${{}_{2}\phi_{2}}\left({\ifrac{c^{2}}{b^{2}},b^{2}\atop c,cq};q^{2},q\right)=% \frac{1}{2}\frac{\left(b^{2},q;q^{2}\right)_{\infty}}{\left(c,cq;q^{2}\right)_% {\infty}}{\left(\frac{\left(c/b;q\right)_{\infty}}{\left(b;q\right)_{\infty}}+% \frac{\left(-c/b;q\right)_{\infty}}{\left(-b;q\right)_{\infty}}\right)}.$

## §17.7(ii) ${{}_{3}\phi_{2}}$ Functions

### $q$-Pfaff–Saalschütz Sum

 17.7.4 ${{}_{3}\phi_{2}}\left({a,b,q^{-n}\atop c,abq^{1-n}/c};q,q\right)=\frac{\left(c% /a,c/b;q\right)_{n}}{\left(c,c/(ab);q\right)_{n}}.$

### Nonterminating Form of the $q$-Saalschütz Sum

 17.7.5 ${{}_{3}\phi_{2}}\left({a,b,c\atop e,f};q,q\right)+\frac{\left(q/e,a,b,c,qf/e;q% \right)_{\infty}}{\left(e/q,aq/e,bq/e,cq/e,f;q\right)_{\infty}}\*{{}_{3}\phi_{% 2}}\left({aq/e,bq/e,cq/e\atop q^{2}/e,qf/e};q,q\right)=\frac{\left(q/e,f/a,f/b% ,f/c;q\right)_{\infty}}{\left(aq/e,bq/e,cq/e,f;q\right)_{\infty}},$

where $ef=abcq$.

### F. H. Jackson’s Terminating $q$-Analog of Dixon’s Sum

 17.7.6 ${{}_{3}\phi_{2}}\left({q^{-2n},b,c\atop q^{1-2n}/b,q^{1-2n}/c};q,\frac{q^{2-n}% }{bc}\right)=\frac{\left(b,c;q\right)_{n}\left(q,bc;q\right)_{2n}}{\left(q,bc;% q\right)_{n}\left(b,c;q\right)_{2n}}.$

### Continued Fractions

For continued-fraction representations of a ratio of ${{}_{3}\phi_{2}}$ functions, see Cuyt et al. (2008, pp. 399–400).

## §17.7(iii) Other ${{}_{r}\phi_{s}}$ Functions

### $q$-Analog of Dixon’s ${{}_{3}F_{2}}\left(1\right)$ Sum

 17.7.7 ${{}_{4}\phi_{3}}\left({a,-qa^{\frac{1}{2}},b,c\atop-a^{\frac{1}{2}},aq/b,aq/c}% ;q,\frac{qa^{\frac{1}{2}}}{bc}\right)=\frac{\left(aq,qa^{\frac{1}{2}}/b,qa^{% \frac{1}{2}}/c,aq/(bc);q\right)_{\infty}}{\left(aq/b,aq/c,qa^{\frac{1}{2}},qa^% {\frac{1}{2}}/(bc);q\right)_{\infty}}.$

### Gasper–Rahman $q$-Analog of Watson’s ${{}_{3}F_{2}}$ Sum

 17.7.8 ${{}_{8}\phi_{7}}\left({\lambda,q\lambda^{\frac{1}{2}},-q\lambda^{\frac{1}{2}},% a,b,c,-c,\lambda q/c^{2}\atop\lambda^{\frac{1}{2}},-\lambda^{\frac{1}{2}},% \lambda q/a,\lambda q/b,\lambda q/c,-\lambda q/c,c^{2}};q,-\frac{\lambda q}{ab% }\right)=\frac{\left(\lambda q,c^{2}/\lambda;q\right)_{\infty}\left(aq,bq,c^{2% }q/a,c^{2}q/b;q^{2}\right)_{\infty}}{\left(\lambda q/a,\lambda q/b;q\right)_{% \infty}\left(q,abq,c^{2}q,c^{2}q/(ab);q^{2}\right)_{\infty}},$

where $\lambda=-c(ab/q)^{\frac{1}{2}}$.

### Andrews’ Terminating $q$-Analog of (17.7.8)

 17.7.9 ${{}_{4}\phi_{3}}\left({q^{-n},aq^{n},c,-c\atop(aq)^{\frac{1}{2}},-(aq)^{\frac{% 1}{2}},c^{2}};q,q\right)=\begin{cases}0,&\mbox{n odd,}\\ \dfrac{c^{n}\left(q,aq/c^{2};q^{2}\right)_{n/2}}{\left(aq,c^{2}q;q^{2}\right)_% {n/2}},&\mbox{n even.}\end{cases}$

### Gasper–Rahman $q$-Analog of Whipple’s ${{}_{3}F_{2}}$ Sum

 17.7.10 ${{}_{8}\phi_{7}}\left({-c,q(-c)^{\frac{1}{2}},-q(-c)^{\frac{1}{2}},a,q/a,c,-d,% -q/d\atop(-c)^{\frac{1}{2}},-(-c)^{\frac{1}{2}},-cq/a,-ac,-q,cq/d,cd};q,c% \right)=\frac{\left(-c,-cq;q\right)_{\infty}\left(acd,acq/d,cdq/a,cq^{2}/(ad);% q^{2}\right)_{\infty}}{\left(cd,cq/d,-ac,-cq/a;q\right)_{\infty}}.$

### Andrews’ Terminating $q$-Analog

 17.7.11 ${{}_{4}\phi_{3}}\left({q^{-n},q^{n+1},c,-c\atop e,c^{2}q/e,-q};q,q\right)=% \frac{\left(eq^{-n},eq^{n+1},c^{2}q^{1-n}/e,c^{2}q^{n+2}/e;q^{2}\right)_{% \infty}}{\left(e,c^{2}q/e;q\right)_{\infty}}.$

### First $q$-Analog of Bailey’s ${{}_{4}F_{3}}\left(1\right)$ Sum

 17.7.12 ${{}_{4}\phi_{3}}\left({a,aq,b^{2}q^{2n},q^{-2n}\atop b,bq,a^{2}q^{2}};q^{2},q^% {2}\right)=\frac{a^{n}\left(-q,b/a;q\right)_{n}}{\left(-aq,b;q\right)_{n}}.$

### Second $q$-Analog of Bailey’s ${{}_{4}F_{3}}\left(1\right)$ Sum

 17.7.13 ${{}_{4}\phi_{3}}\left({a,aq,b^{2}q^{2n-2},q^{-2n}\atop b,bq,a^{2}};q^{2},q^{2}% \right)=\frac{a^{n}\left(-q,b/a;q\right)_{n}(1-bq^{n-1})}{\left(-a,b;q\right)_% {n}(1-bq^{2n-1})}.$

### F. H. Jackson’s $q$-Analog of Dougall’s ${{}_{7}F_{6}}\left(1\right)$ Sum

 17.7.14 ${{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,q^{-n}% \atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq^{n+1}};q,q\right% )=\frac{\left(aq,aq/(bc),aq/(bd),aq/(cd);q\right)_{n}}{\left(aq/b,aq/c,aq/d,aq% /(bcd);q\right)_{n}},$

where $a^{2}q=bcdeq^{-n}$.

### Limiting Cases of (17.7.14)

 17.7.15 ${{}_{6}\phi_{5}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d};q,\frac{aq}{bcd}\right)=\frac{% \left(aq,aq/(bc),aq/(bd),aq/(cd);q\right)_{\infty}}{\left(aq/b,aq/c,aq/d,aq/(% bcd);q\right)_{\infty}},$

and when $d=q^{-n}$,

 17.7.16 ${{}_{6}\phi_{5}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,q^{-n}\atop a^% {\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq^{n+1}};q,\frac{aq^{n+1}}{bc}\right% )=\frac{\left(aq,aq/(bc);q\right)_{n}}{\left(aq/b,aq/c;q\right)_{n}}.$

### Bailey’s Nonterminating Extension of Jackson’s ${{}_{8}\phi_{7}}$ Sum

 17.7.17 ${{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,f\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};q,q\right)-\frac{b}{a}% \frac{\left(aq,c,d,e,f,bq/a,bq/c,bq/d,bq/e,bq/f;q\right)_{\infty}}{\left(aq/b,% aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a,b^{2}q/a;q\right)_{\infty}}\*{{}_{8}% \phi_{7}}\left({b^{2}/a,qba^{-\frac{1}{2}},-qba^{-\frac{1}{2}},b,bc/a,bd/a,be/% a,bf/a\atop ba^{-\frac{1}{2}},-ba^{-\frac{1}{2}},bq/a,bq/c,bq/d,bq/e,bq/f};q,q% \right)=\frac{\left(aq,b/a,aq/(cd),aq/(ce),aq/(cf),aq/(de),aq/(df),aq/(ef);q% \right)_{\infty}}{\left(aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a;q\right)_{% \infty}},$

where $qa^{2}=bcdef$.

### Gasper–Rahman $q$-Analogs of the Karlsson–Minton Sums

 17.7.18 ${{}_{r+2}\phi_{r+1}}\left({a,b,b_{1}q^{m_{1}},\dots,b_{r}q^{m_{r}}\atop bq,b_{% 1},\dots,b_{r}};q,a^{-1}q^{1-(m_{1}+\cdots+m_{r})}\right)=\frac{\left(q,bq/a;q% \right)_{\infty}\left(b_{1}/b;q\right)_{m_{1}}\cdots\left(b_{r}/b;q\right)_{m_% {r}}}{\left(bq,q/a;q\right)_{\infty}\left(b_{1};q\right)_{m_{1}}\cdots\left(b_% {r};q\right)_{m_{r}}}b^{m_{1}+\cdots+m_{r}},$

and

 17.7.19 ${{}_{r+1}\phi_{r}}\left({a,b_{1}q^{m_{1}},\dots,b_{r}q^{m_{r}}\atop b_{1},% \dots,b_{r}};q,a^{-1}q^{1-(m_{1}+\cdots+m_{r})}\right)=0,$

where $m_{1},m_{2},\dots,m_{r}$ are arbitrary nonnegative integers.

### Gosper’s Bibasic Sum

 17.7.20 $\sum_{k=0}^{n}\frac{1-ap^{k}q^{k}}{1-a}\frac{\left(a;p\right)_{k}\left(c;q% \right)_{k}}{\left(q;q\right)_{k}\left(ap/c;p\right)_{k}}c^{-k}=\frac{\left(ap% ;p\right)_{n}\left(cq;q\right)_{n}}{\left(q;q\right)_{n}\left(ap/c;p\right)_{n% }}c^{-n}.$

### Gasper’s Extensions of Gosper’s Bibasic Sum

 17.7.21 $\sum_{k=0}^{n}\frac{(1-ap^{k}q^{k})(1-bp^{k}q^{-k})}{(1-a)(1-b)}\frac{\left(a,% b;p\right)_{k}\left(c,a/(bc);q\right)_{k}}{\left(q,aq/b;q\right)_{k}\left(ap/c% ,bcp;p\right)_{k}}q^{k}=\frac{\left(ap,bp;p\right)_{n}\left(cq,aq/(bc);q\right% )_{n}}{\left(q,aq/b;q\right)_{n}\left(ap/c,bcp;p\right)_{n}},$
 17.7.22 $\sum_{k=-m}^{n}\frac{(1-adp^{k}q^{k})(1-bp^{k}/(dq^{k}))}{(1-ad)(1-(b/d))}% \frac{\left(a,b;p\right)_{k}\left(c,ad^{2}/(bc);q\right)_{k}}{\left(dq,adq/b;q% \right)_{k}\left(adp/c,bcp/d;p\right)_{k}}q^{k}=\frac{(1-a)(1-b)(1-c)(1-(ad^{2% }/(bc)))}{d(1-ad)(1-(b/d))(1-(c/d))(1-(ad/(bc)))}\left(\frac{\left(ap,bp;p% \right)_{n}\left(cq,ad^{2}q/(bc);q\right)_{n}}{\left(dq,adq/b;q\right)_{n}% \left(adp/c,bcp/d;p\right)_{n}}-\frac{\left(c/(ad),d/(bc);p\right)_{m+1}\left(% 1/d,b/(ad);q\right)_{m+1}}{\left(1/c,bc/(ad^{2});q\right)_{m+1}\left(1/a,1/b;p% \right)_{m+1}}\right),$

and $n$-th difference generalization:

 17.7.23 $\left(1-\frac{a}{q}\right)\left(1-\frac{b}{q}\right)\sum_{k=0}^{n}\frac{\left(% ap^{k},bp^{-k};q\right)_{n-1}(1-(ap^{2k}/b))}{\left(p;p\right)_{n}\left(p;p% \right)_{n-k}\left(ap^{k}/b;q\right)_{n+1}}(-1)^{k}p^{\genfrac{(}{)}{0.0pt}{}{% k}{2}}=\delta_{n,0}.$