§17.9 Transformations of Higher $\mathop{{{}_{{r}}\phi _{{r}}}\/}\nolimits$ Functions

Keywords:: $q$ -hypergeometric function
Permalink:: http://dlmf.nist.gov/17.9

§17.9(i) $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\to\mathop{{{}_{{2}}\phi _{{2}}}\/}\nolimits$ , $\mathop{{{}_{{3}}\phi _{{1}}}\/}\nolimits$ , or $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits$
§17.9(ii) $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\to\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits$
§17.9(iii) Further $\mathop{{{}_{{r}}\phi _{{s}}}\/}\nolimits$ Functions
§17.9(iv) Bibasic Series

§17.9(i) $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\to\mathop{{{}_{{2}}\phi _{{2}}}\/}\nolimits$ , $\mathop{{{}_{{3}}\phi _{{1}}}\/}\nolimits$ , or $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits$

Notes:: See Gasper and Rahman (2004, p. 351).
Permalink:: http://dlmf.nist.gov/17.9.i

¶ F. H. Jackson’s Transformations

Keywords:: $q$ -hypergeometric function

17.9.1 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=\frac{\left(za;q\right)_{{\infty}}}{\left(z;q\right)_{{\infty}}}\mathop{{{}_{{2}}\phi _{{2}}}\/}\nolimits\!\left({a,c/b\atop c,az};q,bz\right),$

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 and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E1
Encodings:: TeX, pMML, png

17.9.2 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({q^{{-n}},b\atop c};q,z\right)=\frac{\left(c/b;q\right)_{{n}}}{\left(c;q\right)_{{n}}}b^{n}\mathop{{{}_{{3}}\phi _{{1}}}\/}\nolimits\!\left({q^{{-n}},b,q/c\atop bq^{{1-n}}/c};q,z/c\right),$

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/17.9.E2
Encodings:: TeX, pMML, png

17.9.3 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({a,b\atop c};q,z\right)=\frac{\left(abz/c;q\right)_{{\infty}}}{\left(bz/c;q\right)_{{\infty}}}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({a,c/b,0\atop c,cq/bz};q,q\right),$

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 and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E3
Encodings:: TeX, pMML, png

17.9.4 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({q^{{-n}},b\atop c};q,z\right)=\frac{\left(c/b;q\right)_{{n}}}{\left(c;q\right)_{{n}}}\left(\frac{bz}{q}\right)^{n}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({q^{{-n}},q/z,q^{{1-n}}/c\atop bq^{{1-n}}/c,0};q,q\right),$

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/17.9.E4
Encodings:: TeX, pMML, png

17.9.5 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits\!\left({q^{{-n}},b\atop c};q,z\right)=\frac{\left(c/b;q\right)_{{n}}}{\left(c;q\right)_{{n}}}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({q^{{-n}},b,bzq^{{-n}}/c\atop bq^{{1-n}}/c,0};q,q\right).$

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/17.9.E5
Encodings:: TeX, pMML, png

§17.9(ii) $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\to\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits$

Notes:: See Gasper and Rahman (2004, pp. 71–73). For (17.9.11) apply (17.9.10) to (17.9.9) and interchange the roles of $d$ and $e$ .
Permalink:: http://dlmf.nist.gov/17.9.ii

¶ Transformations of $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits$ -Series

17.9.6 $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({a,b,c\atop d,e};q,de/(abc)\right)=\frac{\left(e/a,de/(bc);q\right)_{{\infty}}}{\left(e,de/(abc);q\right)_{{\infty}}}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({a,d/b,d/c\atop d,de/(bc)};q,e/a\right),$

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 and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.9.E6
Encodings:: TeX, pMML, png

17.9.7 $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({a,b,c\atop d,e};q,de/(abc)\right)=\frac{\left(b,de/(ab),de/(bc);q\right)_{{\infty}}}{\left(d,e,de/(abc);q\right)_{{\infty}}}\*\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({d/b,e/b,de/(abc)\atop de/(ab),de/(bc)};q,b\right),$

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 and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.9.E7
Encodings:: TeX, pMML, png

17.9.8 $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({q^{{-n}},b,c\atop d,e};q,q\right)=\frac{\left(de/(bc);q\right)_{{n}}}{\left(e;q\right)_{{n}}}\left(\frac{bc}{d}\right)^{n}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({q^{{-n}},d/b,d/c\atop d,de/(bc)};q,q\right),$

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 and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.9.E8
Encodings:: TeX, pMML, png

17.9.9 $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({q^{{-n}},b,c\atop d,e};q,q\right)=\frac{\left(e/c;q\right)_{{n}}}{\left(e;q\right)_{{n}}}c^{n}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({q^{{-n}},c,d/b\atop d,cq^{{1-n}}/e};q,\frac{bq}{e}\right),$

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 and $n$ : nonnegative integer
Referenced by:: §17.9(ii)
Permalink:: http://dlmf.nist.gov/17.9.E9
Encodings:: TeX, pMML, png

17.9.10 $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({q^{{-n}},b,c\atop d,e};q,\frac{deq^{n}}{bc}\right)=\frac{\left(e/c;q\right)_{{n}}}{\left(e;q\right)_{{n}}}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({q^{{-n}},c,d/b\atop d,cq^{{1-n}}/e};q,q\right).$

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 and $n$ : nonnegative integer
Referenced by:: §17.9(ii)
Permalink:: http://dlmf.nist.gov/17.9.E10
Encodings:: TeX, pMML, png

¶ $q$ -Sheppard Identity

Keywords:: $q$ -hypergeometric function

17.9.11 $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({q^{{-n}},b,c\atop d,e};q,q\right)=\frac{\left(e/c,d/c;q\right)_{{n}}}{\left(e,d;q\right)_{{n}}}c^{n}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({q^{{-n}},c,\ifrac{cbq^{{1-n}}}{(de)}\atop\ifrac{cq^{{1-n}}}{e},\ifrac{cq^{{1-n}}}{d}};q,q\right),$

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 and $n$ : nonnegative integer
Referenced by:: §17.9(ii)
Permalink:: http://dlmf.nist.gov/17.9.E11
Encodings:: TeX, pMML, png

17.9.12 $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({a,b,c\atop d,e};q,\frac{de}{abc}\right)=\frac{\left(e/b,e/c,cq/a,q/d;q\right)_{{\infty}}}{\left(e,cq/d,q/a,e/(bc);q\right)_{{\infty}}}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({c,d/a,cq/e\atop cq/a,bcq/e};q,\frac{bq}{d}\right)-\frac{\left(q/d,eq/d,b,c,d/a,de/(bcq),bcq^{2}/(de);q\right)_{{\infty}}}{\left(d/q,e,bq/d,cq/d,q/a,e/(bc),bcq/e;q\right)_{{\infty}}}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({aq/d,bq/d,cq/d\atop q^{2}/d,eq/d};q,\frac{de}{abc}\right),$

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 and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.9.E12
Encodings:: TeX, pMML, png

17.9.13 $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({a,b,c\atop d,e};q,\frac{de}{abc}\right)=\frac{\left(e/b,e/c;q\right)_{{\infty}}}{\left(e,e/(bc);q\right)_{{\infty}}}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({d/a,b,c\atop d,bcq/e};q,q\right)+\frac{\left(d/a,b,c,de/(bc);q\right)_{{\infty}}}{\left(d,e,bc/e,de/(abc);q\right)_{{\infty}}}\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({e/b,e/c,de/(abc)\atop de/(bc),eq/(bc)};q,q\right).$

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 and $q$ : complex base
Permalink:: http://dlmf.nist.gov/17.9.E13
Encodings:: TeX, pMML, png

§17.9(iii) Further $\mathop{{{}_{{r}}\phi _{{s}}}\/}\nolimits$ Functions

Notes:: See Gasper and Rahman (2004, pp. 43, 50, 63–64, 70, 72, 361).
Permalink:: http://dlmf.nist.gov/17.9.iii

¶ Sears’ Balanced $\mathop{{{}_{{4}}\phi _{{3}}}\/}\nolimits$ Transformations

Keywords:: $q$ -hypergeometric function

With $def=abcq^{{1-n}}$

17.9.14 $\mathop{{{}_{{4}}\phi _{{3}}}\/}\nolimits\!\left({q^{{-n}},a,b,c\atop d,e,f};q,q\right)=\frac{\left(e/a,f/a;q\right)_{{n}}}{\left(e,f;q\right)_{{n}}}a^{n}\mathop{{{}_{{4}}\phi _{{3}}}\/}\nolimits\!\left({q^{{-n}},a,d/b,d/c\atop d,aq^{{1-n}}/e,aq^{{1-n}}/f};q,q\right)=\frac{\left(a,ef/(ab),ef/(ac);q\right)_{{n}}}{\left(e,f,ef/(abc);q\right)_{{n}}}\mathop{{{}_{{4}}\phi _{{3}}}\/}\nolimits\!\left({q^{{-n}},e/a,f/a,ef/(abc)\atop ef/(ab),ef/(ac),q^{{1-n}}/a};q,q\right).$

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 and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.9.E14
Encodings:: TeX, pMML, png

¶ Watson’s $q$ -Analog of Whipple’s Theorem

Keywords:: Whipple’s theorem

With $n$ a nonnegative integer

17.9.15 $\frac{\left(aq,aq/(de);q\right)_{{n}}}{\left(aq/d,aq/e;q\right)_{{n}}}\mathop{{{}_{{4}}\phi _{{3}}}\/}\nolimits\!\left({aq/(bc),d,e,q^{{-n}}\atop aq/b,aq/c,deq^{{-n}}/a};q,q\right)=\mathop{{{}_{{8}}\phi _{{7}}}\/}\nolimits\!\left({a,qa^{{\frac{1}{2}}},-qa^{{\frac{1}{2}}},b,c,d,e,q^{{-n}}\atop a^{{\frac{1}{2}}},-a^{{\frac{1}{2}}},aq/b,aq/c,aq/d,aq/e,aq^{{n+1}}};q,\frac{a^{2}q^{{2+n}}}{bcde}\right).$

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 and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.9.E15
Encodings:: TeX, pMML, png

¶ Bailey’s Transformation of Very-Well-Poised $\mathop{{{}_{{8}}\phi _{{7}}}\/}\nolimits$

Keywords:: $q$ -hypergeometric function

17.9.16 $\mathop{{{}_{{8}}\phi _{{7}}}\/}\nolimits\!\left({a,qa^{{\frac{1}{2}}},-qa^{{\frac{1}{2}}},b,c,d,e,f\atop a^{{\frac{1}{2}}},-a^{{\frac{1}{2}}},aq/b,aq/c,aq/d,aq/e,aq/f};q,\frac{a^{2}q^{2}}{bcdef}\right)=\frac{\left(aq,aq/(de),aq/(df),aq/(ef);q\right)_{{\infty}}}{\left(aq/d,aq/e,aq/f,aq/(def);q\right)_{{\infty}}}\mathop{{{}_{{4}}\phi _{{3}}}\/}\nolimits\!\left({aq/(bc),d,e,f\atop aq/b,aq/c,def/a};q,q\right)+\frac{\left(aq,aq/(bc),d,e,f,a^{2}q^{2}/(bdef),a^{2}q^{2}/(cdef);q\right)_{{\infty}}}{\left(aq/b,aq/c,aq/d,aq/e,aq/f,a^{2}q^{2}/(bcdef),def/(aq);q\right)_{{\infty}}}\*\mathop{{{}_{{4}}\phi _{{3}}}\/}\nolimits\!\left({aq/(de),aq/(df),aq/(ef),a^{2}q^{2}/(bcdef)\atop a^{2}q^{2}/(bdef),a^{2}q^{2}/(cdef),aq^{2}/(def)};q,q\right).$

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/17.9.E16
Encodings:: TeX, pMML, png

¶ Sears–Carlitz Transformation

With $a=q^{{-n}}$ and $n$ a nonnegative integer,

17.9.17 $\mathop{{{}_{{3}}\phi _{{2}}}\/}\nolimits\!\left({a,b,c\atop aq/b,aq/c};q,\frac{aqz}{bc}\right)=\frac{\left(az;q\right)_{{\infty}}}{\left(z;q\right)_{{\infty}}}\*\mathop{{{}_{{5}}\phi _{{4}}}\/}\nolimits\!\left({a^{{\frac{1}{2}}},-a^{{\frac{1}{2}}},(aq)^{{\frac{1}{2}}},-(aq)^{{\frac{1}{2}}},aq/(bc)\atop aq/b,aq/c,az,q/z};q,q\right).$

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/17.9.E17
Encodings:: TeX, pMML, png

¶ Gasper’s $q$ -Analog of Clausen’s Formula

17.9.18 $\left(\mathop{{{}_{{4}}\phi _{{3}}}\/}\nolimits\!\left({a,b,abz,ab/z\atop abq^{{\frac{1}{2}}},-abq^{{\frac{1}{2}}},-ab};q,q\right)\right)^{2}=\mathop{{{}_{{5}}\phi _{{4}}}\/}\nolimits\!\left({a^{2},b^{2},ab,abz,ab/z\atop abq^{{\frac{1}{2}}},-abq^{{\frac{1}{2}}},-ab,a^{2}b^{2}};q,q\right),$

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/17.9.E18
Encodings:: TeX, pMML, png

provided that the series expansions of both $\phi$ ’s terminate.

§17.9(iv) Bibasic Series

Notes:: See Andrews (1966b).
Keywords:: $q$ -hypergeometric function
Permalink:: http://dlmf.nist.gov/17.9.iv

¶ Mixed-Base Heine-Type Transformations

Keywords:: $q$ -hypergeometric function

17.9.19 $\sum _{{n=0}}^{{\infty}}\frac{\left(a;q^{2}\right)_{{n}}\left(b;q\right)_{{n}}}{\left(q^{2};q^{2}\right)_{{n}}\left(c;q\right)_{{n}}}z^{n}=\frac{\left(b;q\right)_{{\infty}}\left(az;q^{2}\right)_{{\infty}}}{\left(c;q\right)_{{\infty}}\left(z;q^{2}\right)_{{\infty}}}\sum _{{n=0}}^{{\infty}}\frac{\left(c/b;q\right)_{{2n}}\left(z;q^{2}\right)_{{n}}b^{{2n}}}{\left(q;q\right)_{{2n}}\left(az;q^{2}\right)_{{n}}}+\frac{\left(b;q\right)_{{\infty}}\left(azq;q^{2}\right)_{{\infty}}}{\left(c;q\right)_{{\infty}}\left(zq;q^{2}\right)_{{\infty}}}\sum _{{n=0}}^{{\infty}}\frac{\left(c/b;q\right)_{{2n+1}}\left(zq;q^{2}\right)_{{n}}b^{{2n+1}}}{\left(q;q\right)_{{2n+1}}\left(azq;q^{2}\right)_{{n}}}.$

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E19
Encodings:: TeX, pMML, png

17.9.20 $\sum _{{n=0}}^{{\infty}}\frac{\left(a;q^{k}\right)_{{n}}\left(b;q\right)_{{kn}}z^{n}}{\left(q^{k};q^{k}\right)_{{n}}\left(c;q\right)_{{kn}}}=\frac{\left(b;q\right)_{{\infty}}\left(az;q^{k}\right)_{{\infty}}}{\left(c;q\right)_{{\infty}}\left(z;q^{k}\right)_{{\infty}}}\sum _{{n=0}}^{{\infty}}\frac{\left(c/b;q\right)_{{n}}\left(z;q^{k}\right)_{{n}}b^{n}}{\left(q;q\right)_{{n}}\left(az;q^{k}\right)_{{n}}},$ $k=1,2,3,\dots$ .

Symbols:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $k$ : nonnegative integer, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.9.E20
Encodings:: TeX, pMML, png

§17.9 Transformations of Higher Functions

Contents

§17.9(i) , , or

¶ F. H. Jackson’s Transformations

§17.9(ii)

¶ Transformations of -Series

¶ -Sheppard Identity

§17.9(iii) Further Functions

¶ Sears’ Balanced Transformations

¶ Watson’s -Analog of Whipple’s Theorem

¶ Bailey’s Transformation of Very-Well-Poised