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

Permalink:: http://dlmf.nist.gov/17.6

§17.6(i) Special Values
§17.6(ii) $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits$ Transformations
§17.6(iii) Contiguous Relations
§17.6(iv) Differential Equations
§17.6(v) Integral Representations
§17.6(vi) Continued Fractions

§17.6(i) Special Values

Notes:: See Gasper and Rahman (2004, pp. 13, 14, 18, 26, 27).
Permalink:: http://dlmf.nist.gov/17.6.i

¶ $q$ -Gauss Sum

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.6.E1
Encodings:: TeX, pMML, png

¶ First $q$ -Chu–Vandermonde Sum

Keywords:: $q$ -hypergeometric function

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $n$ : nonnegative integer
Referenced by:: ¶ ‣ §17.6(i)
Permalink:: http://dlmf.nist.gov/17.6.E2
Encodings:: TeX, pMML, png

¶ Second $q$ -Chu–Vandermonde Sum

This reverses the order of summation in (17.6.2):

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.6.E3
Encodings:: TeX, pMML, png

¶ Andrews–Askey Sum

Keywords:: $q$ -hypergeometric function

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function and $q$ : complex base
Referenced by:: ¶ ‣ §17.7(i), §17.7(i)
Permalink:: http://dlmf.nist.gov/17.6.E4
Encodings:: TeX, pMML, png

¶ Bailey–Daum $q$ -Kummer Sum

Keywords:: $q$ -hypergeometric function

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.6.E5
Encodings:: TeX, pMML, png

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

Notes:: See Gasper and Rahman (2004, pp. 13, 356, 363). For (17.6.9)–(17.6.12) see Fine (1988, pp. 12–15), and for (17.6.14) see Andrews (1966a).
Permalink:: http://dlmf.nist.gov/17.6.ii

¶ Heine’s First Transformation

Keywords:: $q$ -hypergeometric function

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E6
Encodings:: TeX, pMML, png

¶ Heine’s Second Tranformation

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E7
Encodings:: TeX, pMML, png

¶ Heine’s Third Transformation

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E8
Encodings:: TeX, pMML, png

¶ Fine’s First Transformation

Keywords:: $q$ -hypergeometric function

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.6(ii)
Permalink:: http://dlmf.nist.gov/17.6.E9
Encodings:: TeX, pMML, png

¶ Fine’s Second Transformation

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E10
Encodings:: TeX, pMML, png

¶ Fine’s Third Transformation

Keywords:: $q$ -hypergeometric function

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E11
Encodings:: TeX, pMML, png

¶ Rogers–Fine Identity

Keywords:: $q$ -hypergeometric function

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.6(ii)
Permalink:: http://dlmf.nist.gov/17.6.E12
Encodings:: TeX, pMML, png

¶ Nonterminating Form of the $q$ -Vandermonde Sum

Keywords:: $q$ -hypergeometric function

17.6.13 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left(a,b;c;q,q\right)+\frac{\left(q/c,a,b;q\right)_{{\infty}}}{\left(c/q,aq/c,bq/c;q\right)_{{\infty}}}\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left(aq/c,bq/c;q^{2}/c;q,q\right)=\frac{\left(q/c,abq/c;q\right)_{{\infty}}}{\left(aq/c,bq/c;q\right)_{{\infty}}},$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.6.E13
Encodings:: TeX, pMML, png

17.6.14 $\sum _{{n=0}}^{\infty}\frac{\left(a;q\right)_{{n}}\left(b;q^{2}\right)_{{n}}z^{n}}{\left(q;q\right)_{{n}}\left(azb;q^{2}\right)_{{n}}}=\frac{\left(az,bz;q^{2}\right)_{{\infty}}}{\left(z,azb;q^{2}\right)_{{\infty}}}\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop bz};q^{2},zq\right).$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.6(ii)
Permalink:: http://dlmf.nist.gov/17.6.E14
Encodings:: TeX, pMML, png

¶ Three-Term $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits$ Transformations

Keywords:: $q$ -hypergeometric function

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E15
Encodings:: TeX, pMML, png

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

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E16
Encodings:: TeX, pMML, png

§17.6(iii) Contiguous Relations

Notes:: See Gasper and Rahman (2004, pp. 26–27).
Permalink:: http://dlmf.nist.gov/17.6.iii

¶ Heine’s Contiguous Relations

Keywords:: $q$ -hypergeometric function

17.6.17 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c/q};q,z\right)-\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=cz\frac{(1-a)(1-b)}{(q-c)(1-c)}\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({aq,bq\atop cq};q,z\right),$

Symbols:: $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E17
Encodings:: TeX, pMML, png

17.6.18 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({aq,b\atop c};q,z\right)-\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=az\frac{1-b}{1-c}\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({aq,bq\atop cq};q,z\right),$

Symbols:: $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E18
Encodings:: TeX, pMML, png

17.6.19 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({aq,b\atop cq};q,z\right)-\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=az\frac{(1-b)(1-(c/a))}{(1-c)(1-cq)}\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({aq,bq\atop cq^{2}};q,z\right),$

Symbols:: $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E19
Encodings:: TeX, pMML, png

17.6.20 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({aq,b/q\atop c};q,z\right)-\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=az\frac{(1-b/(aq))}{1-c}\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({aq,b\atop cq};q,z\right),$

Symbols:: $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E20
Encodings:: TeX, pMML, png

17.6.21 $b(1-a)\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({aq,b\atop c};q,z\right)-a(1-b)\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,bq\atop c};q,z\right)=(b-a)\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right),$

Symbols:: $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E21
Encodings:: TeX, pMML, png

17.6.22 $a\left(1-\frac{b}{c}\right)\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b/q\atop c};q,z\right)-b\left(1-\frac{a}{c}\right)\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a/q,b\atop c};q,z\right)=(a-b)\left(1-\frac{abz}{cq}\right)\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right),$

Symbols:: $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E22
Encodings:: TeX, pMML, png

17.6.23 $q\left(1-\frac{a}{c}\right)\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a/q,b\atop c};q,z\right)+(1-a)\left(1-\frac{abz}{c}\right)\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({aq,b\atop c};q,z\right)=\left(1+q-a-\frac{aq}{c}+\frac{a^{2}z}{c}-\frac{abz}{c}\right)\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right),$

Symbols:: $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E23
Encodings:: TeX, pMML, png

17.6.24 $(1-c)(q-c)(abz-c)\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c/q};q,z\right)+z(c-a)(c-b)\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop cq};q,z\right)=(c-1)(c(q-c)+z(ca+cb-ab-abq))\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right).$

Symbols:: $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E24
Encodings:: TeX, pMML, png

§17.6(iv) Differential Equations

Notes:: See Gasper and Rahman (2004, pp. 26–28).
Keywords:: differential equations, $q$ -differential equations, $q$ -hypergeometric function
Referenced by:: ¶ ‣ §17.2(iv)
Permalink:: http://dlmf.nist.gov/17.6.iv

¶ Iterations of $\mathcal{D}$

17.6.25 $\mathcal{D}_{q}^{n}\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,zd\right)=\frac{\left(a,b;q\right)_{{n}}d^{n}}{\left(c;q\right)_{{n}}(1-q)^{n}}\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({aq^{n},bq^{n}\atop cq^{n}};q,dz\right),$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E25
Encodings:: TeX, pMML, png

17.6.26 $\mathcal{D}_{q}^{n}\left(\frac{\left(z;q\right)_{{\infty}}}{\left(abz/c;q\right)_{{\infty}}}\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right)\right)=\frac{\left(c/a,c/b;q\right)_{{n}}}{\left(c;q\right)_{{n}}(1-q)^{n}}\left(\frac{ab}{c}\right)^{n}\frac{\left(zq^{n};q\right)_{{\infty}}}{\left(abz/c;q\right)_{{\infty}}}\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop cq^{n}};q,zq^{n}\right).$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E26
Encodings:: TeX, pMML, png

¶ $q$ -Differential Equation

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

Symbols:: $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Referenced by:: ¶ ‣ §17.6(iv)
Permalink:: http://dlmf.nist.gov/17.6.E27
Encodings:: TeX, pMML, png

(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

Notes:: See Gasper and Rahman (2004, pp. 23, 115).
Keywords:: $q$ -hypergeometric function
Permalink:: http://dlmf.nist.gov/17.6.v

17.6.28 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({q^{{\alpha}},q^{{\beta}}\atop q^{{\gamma}}};q,z\right)=\frac{\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\gamma\right)}{\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\beta\right)\mathop{\Gamma _{{q}}\/}\nolimits\!\left(\gamma-\beta\right)}\int _{0}^{1}\frac{t^{{\beta-1}}\left(tq;q\right)_{{\gamma-\beta-1}}}{\left(xt;q\right)_{{\alpha}}}{d}_{q}t.$

Symbols:: $\int$ : integral, $\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)$ : $q$ -gamma function, ${d}_{q}x$ : $q$ -differential, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $z$ : complex variable and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.6.E28
Encodings:: TeX, pMML, png

17.6.29 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=\left(\frac{-1}{2\pi i}\right)\frac{\left(a,b;q\right)_{{\infty}}}{\left(q,c;q\right)_{{\infty}}}\int _{{-i\infty}}^{{i\infty}}\frac{\left(q^{{1+\zeta}},cq^{\zeta};q\right)_{{\infty}}}{\left(aq^{\zeta},bq^{\zeta};q\right)_{{\infty}}}\frac{\pi(-z)^{\zeta}}{\mathop{\sin\/}\nolimits\!\left(\pi\zeta\right)}d\zeta,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\mathop{\sin\/}\nolimits z$ : sine function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.6.E29
Encodings:: TeX, pMML, png

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

Keywords:: continued fractions, $q$ -hypergeometric function
Permalink:: http://dlmf.nist.gov/17.6.vi

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

§17.6 Function

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