# §17.6 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits$ Function

## §17.6(i) Special Values

### $q$-Gauss Sum

 17.6.1 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,\ifrac{c}{(ab)}% \right)=\frac{\left(c/a,c/b;q\right)_{\infty}}{\left(c,c/(ab);q\right)_{\infty% }}.$

### First $q$-Chu–Vandermonde Sum

 17.6.2 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,q^{-n}\atop c};q,\ifrac{cq^{n}}% {a}\right)=\frac{\left(c/a;q\right)_{n}}{\left(c;q\right)_{n}}.$

### Second $q$-Chu–Vandermonde Sum

This reverses the order of summation in (17.6.2):

 17.6.3 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,q^{-n}\atop c};q,q\right)=\frac% {a^{n}\left(c/a;q\right)_{n}}{\left(c;q\right)_{n}}.$

 17.6.4 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({b^{2},\ifrac{b^{2}}{c}\atop c};q^% {2},\ifrac{cq}{b^{2}}\right)=\frac{1}{2}\frac{\left(b^{2},q;q^{2}\right)_{% \infty}}{\left(c,cq/b^{2};q^{2}\right)_{\infty}}\left(\frac{\left(c/b;q\right)% _{\infty}}{\left(b;q\right)_{\infty}}+\frac{\left(-c/b;q\right)_{\infty}}{% \left(-b;q\right)_{\infty}}\right),$ $|cq|<|b^{2}|$.

### Bailey–Daum $q$-Kummer Sum

 17.6.5 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop aq/b};q,-q/b\right)=% \frac{\left(-q;q\right)_{\infty}\left(aq,\ifrac{aq^{2}}{b^{2}};q^{2}\right)_{% \infty}}{\left(-q/b,aq/b;q\right)_{\infty}},$ $|b|>|q|$.

## §17.6(ii) $\mathop{{{}_{2}\phi_{1}}\/}\nolimits$ Transformations

### Heine’s First Transformation

 17.6.6 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=\frac{% \left(b,az;q\right)_{\infty}}{\left(c,z;q\right)_{\infty}}\mathop{{{}_{2}\phi_% {1}}\/}\nolimits\!\left({c/b,z\atop az};q,b\right),$ $|z|<1,|b|<1$.

### Heine’s Second Tranformation

 17.6.7 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=\frac{% \left(c/b,bz;q\right)_{\infty}}{\left(c,z;q\right)_{\infty}}\mathop{{{}_{2}% \phi_{1}}\/}\nolimits\!\left({\ifrac{abz}{c},b\atop bz};q,c/b\right),$ $|z|<1,|c|<|b|$.

### Heine’s Third Transformation

 17.6.8 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=\frac{% \left(\ifrac{abz}{c};q\right)_{\infty}}{\left(z;q\right)_{\infty}}\mathop{{{}_% {2}\phi_{1}}\/}\nolimits\!\left({c/a,c/b\atop c};q,\ifrac{abz}{c}\right),$ $|z|<1,|abz|<|c|$.

### Fine’s First Transformation

 17.6.9 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({q,aq\atop bq};q,z\right)=-\frac{(% 1-b)(aq/b)}{(1-(\ifrac{aq}{b}))}\sum_{n=0}^{\infty}\frac{\left(aq,azq/b;q% \right)_{n}q^{n}}{\left(azq^{2}/b;q\right)_{n}}+\frac{\left(aq,azq/b;q\right)_% {\infty}}{\left(aq/b;q\right)_{\infty}}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!% \left({q,0\atop bq};q,z\right),$ $|z|<1$.

### Fine’s Second Transformation

 17.6.10 $(1-z)\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({q,aq\atop bq};q,z\right)=% \sum_{n=0}^{\infty}\frac{\left(b/a;q\right)_{n}(-az)^{n}q^{(n^{2}+n)/2}}{\left% (bq,zq;q\right)_{n}},$ $|z|<1$.

### Fine’s Third Transformation

 17.6.11 $\frac{1-z}{1-b}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({q,aq\atop bq};q,z% \right)=\sum_{n=0}^{\infty}\frac{\left(aq;q\right)_{n}\left(azq/b;q\right)_{2n% }b^{n}}{\left(zq,aq/b;q\right)_{n}}-aq\sum_{n=0}^{\infty}\frac{\left(aq;q% \right)_{n}\left(azq/b;q\right)_{2n+1}(bq)^{n}}{\left(zq;q\right)_{n}\left(aq/% b;q\right)_{n+1}},$ $|z|<1,|b|<1$.

### Rogers–Fine Identity

 17.6.12 $(1-z)\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({q,aq\atop bq};q,z\right)=% \sum_{n=0}^{\infty}\frac{\left(aq,azq/b;q\right)_{n}}{\left(bq,zq;q\right)_{n}% }(1-azq^{2n+1})(bz)^{n}q^{n^{2}},$ $|z|<1$.

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

 17.6.13 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left(a,b;c;q,q\right)+\frac{\left(q/c,a% ,b;q\right)_{\infty}}{\left(c/q,aq/c,bq/c;q\right)_{\infty}}\mathop{{{}_{2}% \phi_{1}}\/}\nolimits\!\left(aq/c,bq/c;q^{2}/c;q,q\right)=\frac{\left(q/c,abq/% c;q\right)_{\infty}}{\left(aq/c,bq/c;q\right)_{\infty}},$
 17.6.14 $\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}\left(b;q^{2}\right)_{n}z^{n}}{% \left(q;q\right)_{n}\left(azb;q^{2}\right)_{n}}=\frac{\left(az,bz;q^{2}\right)% _{\infty}}{\left(z,azb;q^{2}\right)_{\infty}}\mathop{{{}_{2}\phi_{1}}\/}% \nolimits\!\left({a,b\atop bz};q^{2},zq\right).$

### Three-Term $\mathop{{{}_{2}\phi_{1}}\/}\nolimits$ Transformations

 17.6.15 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=\frac{% \left(abz/c,q/c;q\right)_{\infty}}{\left(az/c,q/a;q\right)_{\infty}}\mathop{{{% }_{2}\phi_{1}}\/}\nolimits\!\left({c/a,cq/(abz)\atop cq/(az)};q,bq/c\right)-% \frac{\left(b,q/c,c/a,az/q,q^{2}/(az);q\right)_{\infty}}{\left(c/q,bq/c,q/a,az% /c,cq/(az);q\right)_{\infty}}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({aq/c% ,bq/c\atop q^{2}/c};q,z\right),$ $|z|<1,|bq|<|c|$.
 17.6.16 $\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=\frac{% \left(b,c/a,az,q/(az);q\right)_{\infty}}{\left(c,b/a,z,q/z;q\right)_{\infty}}% \mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,aq/c\atop aq/b};q,cq/(abz)% \right)+\frac{\left(a,c/b,bz,q/(bz);q\right)_{\infty}}{\left(c,a/b,z,q/z;q% \right)_{\infty}}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({b,bq/c\atop bq/a% };q,cq/(abz)\right),$ $|z|<1$, $|abz|<|cq|$.

For a similar result for $q$-confluent hypergeometric functions see Morita (2013).

## §17.6(iii) Contiguous Relations

### Heine’s Contiguous Relations

 17.6.17 $\displaystyle\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c/q};q,z% \right)-\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)$ $\displaystyle=cz\frac{(1-a)(1-b)}{(q-c)(1-c)}\mathop{{{}_{2}\phi_{1}}\/}% \nolimits\!\left({aq,bq\atop cq};q,z\right),$ 17.6.18 $\displaystyle\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({aq,b\atop c};q,z% \right)-\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)$ $\displaystyle=az\frac{1-b}{1-c}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({aq% ,bq\atop cq};q,z\right),$ 17.6.19 $\displaystyle\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({aq,b\atop cq};q,z% \right)-\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)$ $\displaystyle=az\frac{(1-b)(1-(c/a))}{(1-c)(1-cq)}\mathop{{{}_{2}\phi_{1}}\/}% \nolimits\!\left({aq,bq\atop cq^{2}};q,z\right),$ 17.6.20 $\displaystyle\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({aq,b/q\atop c};q,z% \right)-\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)$ $\displaystyle=az\frac{(1-b/(aq))}{1-c}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!% \left({aq,b\atop cq};q,z\right),$
 17.6.21 $\displaystyle b(1-a)\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({aq,b\atop c};% q,z\right)-a(1-b)\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,bq\atop c};q,z\right)$ $\displaystyle=(b-a)\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,% z\right),$ 17.6.22 $\displaystyle a\left(1-\frac{b}{c}\right)\mathop{{{}_{2}\phi_{1}}\/}\nolimits% \!\left({a,b/q\atop c};q,z\right)-b\left(1-\frac{a}{c}\right)\mathop{{{}_{2}% \phi_{1}}\/}\nolimits\!\left({a/q,b\atop c};q,z\right)$ $\displaystyle=(a-b)\left(1-\frac{abz}{cq}\right)\mathop{{{}_{2}\phi_{1}}\/}% \nolimits\!\left({a,b\atop c};q,z\right),$
 17.6.23 $q\left(1-\frac{a}{c}\right)\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a/q,b% \atop c};q,z\right)+(1-a)\left(1-\frac{abz}{c}\right)\mathop{{{}_{2}\phi_{1}}% \/}\nolimits\!\left({aq,b\atop c};q,z\right)=\left(1+q-a-\frac{aq}{c}+\frac{a^% {2}z}{c}-\frac{abz}{c}\right)\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b% \atop c};q,z\right),$
 17.6.24 $(1-c)(q-c)(abz-c)\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c/q};q,% z\right)+z(c-a)(c-b)\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop cq};% q,z\right)=(c-1)(c(q-c)+z(ca+cb-ab-abq))\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!% \left({a,b\atop c};q,z\right).$

## §17.6(iv) Differential Equations

### Iterations of $\mathcal{D}$

 17.6.25 $\displaystyle\mathcal{D}_{q}^{n}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a% ,b\atop c};q,zd\right)$ $\displaystyle=\frac{\left(a,b;q\right)_{n}d^{n}}{\left(c;q\right)_{n}(1-q)^{n}% }\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({aq^{n},bq^{n}\atop cq^{n}};q,dz% \right),$ 17.6.26 $\displaystyle\mathcal{D}_{q}^{n}\left(\frac{\left(z;q\right)_{\infty}}{\left(% abz/c;q\right)_{\infty}}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c% };q,z\right)\right)$ $\displaystyle=\frac{\left(c/a,c/b;q\right)_{n}}{\left(c;q\right)_{n}(1-q)^{n}}% \left(\frac{ab}{c}\right)^{n}\frac{\left(zq^{n};q\right)_{\infty}}{\left(abz/c% ;q\right)_{\infty}}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop cq^{n% }};q,zq^{n}\right).$

### $q$-Differential Equation

 17.6.27 $z(c-abqz)\mathcal{D}_{q}^{2}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b% \atop c};q,z\right)+\left(\frac{1-c}{1-q}+\frac{(1-a)(1-b)-(1-abq)}{1-q}z% \right)\mathcal{D}_{q}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c}% ;q,z\right)-\frac{(1-a)(1-b)}{(1-q)^{2}}\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!% \left({a,b\atop c};q,z\right)=0.$

(17.6.27) reduces to the hypergeometric equation (15.10.1) with the substitutions $a\to q^{a}$, $b\to q^{b}$, $c\to q^{c}$, followed by $\lim_{q\to 1-}$.

## §17.6(v) Integral Representations

 17.6.28 $\displaystyle\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({q^{\alpha},q^{\beta}% \atop q^{\gamma}};q,z\right)$ $\displaystyle=\frac{\mathop{\Gamma_{q}\/}\nolimits\!\left(\gamma\right)}{% \mathop{\Gamma_{q}\/}\nolimits\!\left(\beta\right)\mathop{\Gamma_{q}\/}% \nolimits\!\left(\gamma-\beta\right)}\int_{0}^{1}\frac{t^{\beta-1}\left(tq;q% \right)_{\gamma-\beta-1}}{\left(xt;q\right)_{\alpha}}{\mathrm{d}}_{q}t.$ 17.6.29 $\displaystyle\mathop{{{}_{2}\phi_{1}}\/}\nolimits\!\left({a,b\atop c};q,z\right)$ $\displaystyle=\left(\frac{-1}{2\pi i}\right)\frac{\left(a,b;q\right)_{\infty}}% {\left(q,c;q\right)_{\infty}}\int_{-i\infty}^{i\infty}\frac{\left(q^{1+\zeta},% cq^{\zeta};q\right)_{\infty}}{\left(aq^{\zeta},bq^{\zeta};q\right)_{\infty}}% \frac{\pi(-z)^{\zeta}}{\mathop{\sin\/}\nolimits\!\left(\pi\zeta\right)}\mathrm% {d}\zeta,$

where $|z|<1$, $|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)|<\pi$, and the contour of integration separates the poles of $\left(q^{1+\zeta},cq^{\zeta};q\right)_{\infty}/\mathop{\sin\/}\nolimits\!\left% (\pi\zeta\right)$ from those of $1/\left(aq^{\zeta},bq^{\zeta};q\right)_{\infty}$, and the infimum of the distances of the poles from the contour is positive.

## §17.6(vi) Continued Fractions

For continued-fraction representations of the $\mathop{{{}_{2}\phi_{1}}\/}\nolimits$ function, see Cuyt et al. (2008, pp. 395–399).