§17.7 Special Cases of Higher $\mathop{{{}_{r}\phi_{s}}\/}\nolimits$ Functions

§17.7(i) $\mathop{{{}_{2}\phi_{2}}\/}\nolimits$ Functions

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

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

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

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

Sum Related to (17.6.4)

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

§17.7(ii) $\mathop{{{}_{3}\phi_{2}}\/}\nolimits$ Functions

$q$-Pfaff–Saalschütz Sum

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

Nonterminating Form of the $q$-Saalschütz Sum

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

where $ef=abcq$.

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

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

Continued Fractions

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

§17.7(iii) Other $\mathop{{{}_{r}\phi_{s}}\/}\nolimits$ Functions

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

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

Gasper–Rahman $q$-Analog of Watson’s $\mathop{{{}_{3}F_{2}}\/}\nolimits$ Sum

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

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

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

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

Gasper–Rahman $q$-Analog of Whipple’s $\mathop{{{}_{3}F_{2}}\/}\nolimits$ Sum

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

Andrews’ Terminating $q$-Analog

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

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

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

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

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

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

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

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

Limiting Cases of (17.7.14)

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

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

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

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

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

where $qa^{2}=bcdef$.

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

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

and

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

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