17 q-Hypergeometric and Related Functions Properties 17.8 Special Cases of

{{}_{r}\psi_{r}}

{{}_{r}\psi_{r}}

Functions

§17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

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Keywords:: $q$ -hypergeometric function, transformations
Referenced by:: Erratum (V1.0.12) for Section 17.9
Permalink:: http://dlmf.nist.gov/17.9
Correction (effective with 1.0.12):: The title for this section was changed from Transformations of Higher ${{}_{r}\phi_{r}}$ Functions to Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions.
Suggested 2016-07-16 by Howard Cohl
See also:: Annotations for Ch.17

§17.9(i) ${{}_{2}\phi_{1}}\to{{}_{2}\phi_{2}}$ , ${{}_{3}\phi_{1}}$ , or ${{}_{3}\phi_{2}}$

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Notes:: See Gasper and Rahman (2004, p. 351).
Permalink:: http://dlmf.nist.gov/17.9.i
Addition (effective with 1.1.0):: Equation (17.9.3_5) was added.
Suggested 2019-10-19 by Slobodan Damjanovic
See also:: Annotations for §17.9 and Ch.17

F. H. Jackson’s Transformations

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Keywords:: F. H. Jackson’s transformations, $q$ -hypergeometric function
See also:: Annotations for §17.9(i), §17.9 and Ch.17

17.9.1	$\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)$	$\displaystyle=\frac{\left(za;q\right)_{\infty}}{\left(z;q\right)_{\infty}}{{}_% {2}\phi_{2}}\left({a,c/b\atop c,az};q,bz\right),$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : 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 See also: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17
17.9.2	$\displaystyle{{}_{2}\phi_{1}}\left({q^{-n},b\atop c};q,z\right)$	$\displaystyle=\frac{\left(c/b;q\right)_{n}}{\left(c;q\right)_{n}}b^{n}{{}_{3}% \phi_{1}}\left({q^{-n},b,q/z\atop bq^{1-n}/c};q,z/c\right),$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable Referenced by: Erratum (V1.0.14) for Equation (17.9.2) Permalink: http://dlmf.nist.gov/17.9.E2 Encodings: TeX, pMML, png Correction (effective with 1.0.14): The entry $q/c$ in the first row of ${{}_{3}\phi_{1}}\left({q^{-n},b,q/c\atop bq^{1-n}/c};q,z/c\right)$ was replaced by $q/z$ . Suggested 2016-08-30 by Xinrong Ma See also: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17
17.9.3	$\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)$	$\displaystyle=\frac{\left(abz/c;q\right)_{\infty}}{\left(bz/c;q\right)_{\infty% }}{{}_{3}\phi_{2}}\left({a,c/b,0\atop c,cq/(bz)};q,q\right)+\frac{\left(a,bz,c% /b;q\right)_{\infty}}{\left(c,z,c/(bz);q\right)_{\infty}}{{}_{3}\phi_{2}}\left% ({z,abz/c,0\atop bz,bzq/c};q,q\right),$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable Referenced by: Erratum (V1.0.23) for Equation (17.9.3) Permalink: http://dlmf.nist.gov/17.9.E3 Encodings: TeX, pMML, png Correction (effective with 1.0.23): The second term on the right-hand side was missing. The form of the equation where the second term is missing is correct if the ${{}_{2}\phi_{1}}$ is terminating. It is this form which appeared in the first edition of Gasper and Rahman (1990). The more general version which appears now is what is reproduced in Gasper and Rahman (2004, (III.5)). Suggested 2019-04-26 by Roberto S. Costas-Santos See also: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17
17.9.3_5	$\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)$	$\displaystyle=\frac{\left(c/a,c/b;q\right)_{\infty}}{\left(c,c/(ab);q\right)_{% \infty}}{{}_{3}\phi_{2}}\left({a,b,abz/c\atop qab/c,0};q,q\right)+\frac{\left(% a,b,abz/c;q\right)_{\infty}}{\left(c,ab/c,z;q\right)_{\infty}}{{}_{3}\phi_{2}}% \left({c/a,c/b,z\atop qc/(ab),0};q,q\right),$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $z$ : complex variable Proof sketch: Derivable from (17.9.13) after replacing $a$ with $d/a$ and then taking $d\rightarrow 0$ . Then replace $a$ , $b$ , $c$ , $e$ with $abz/c$ , $a$ , $b$ , $c$ respectively. Referenced by: §17.9(i), Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/17.9.E3_5 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17
17.9.4	$\displaystyle{{}_{2}\phi_{1}}\left({q^{-n},b\atop c};q,z\right)$	$\displaystyle=\frac{\left(c/b;q\right)_{n}}{\left(c;q\right)_{n}}\left(\frac{% bz}{q}\right)^{n}{{}_{3}\phi_{2}}\left({q^{-n},q/z,q^{1-n}/c\atop bq^{1-n}/c,0% };q,q\right),$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : 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 See also: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17
17.9.5	$\displaystyle{{}_{2}\phi_{1}}\left({q^{-n},b\atop c};q,z\right)$	$\displaystyle=\frac{\left(c/b;q\right)_{n}}{\left(c;q\right)_{n}}{{}_{3}\phi_{% 2}}\left({q^{-n},b,bzq^{-n}/c\atop bq^{1-n}/c,0};q,q\right).$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : 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 See also: Annotations for §17.9(i), §17.9(i), §17.9 and Ch.17

§17.9(ii) ${{}_{3}\phi_{2}}\to{{}_{3}\phi_{2}}$

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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
See also:: Annotations for §17.9 and Ch.17

Transformations of ${{}_{3}\phi_{2}}$ -Series

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See also:: Annotations for §17.9(ii), §17.9 and Ch.17

17.9.6	$\displaystyle{{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,de/(abc)\right)$	$\displaystyle=\frac{\left(e/a,de/(bc);q\right)_{\infty}}{\left(e,de/(abc);q% \right)_{\infty}}{{}_{3}\phi_{2}}\left({a,d/b,d/c\atop d,de/(bc)};q,e/a\right),$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base Permalink: http://dlmf.nist.gov/17.9.E6 Encodings: TeX, pMML, png See also: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17
17.9.7	$\displaystyle{{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,de/(abc)\right)$	$\displaystyle=\frac{\left(b,de/(ab),de/(bc);q\right)_{\infty}}{\left(d,e,de/(% abc);q\right)_{\infty}}\*{{}_{3}\phi_{2}}\left({d/b,e/b,de/(abc)\atop de/(ab),% de/(bc)};q,b\right),$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base Permalink: http://dlmf.nist.gov/17.9.E7 Encodings: TeX, pMML, png See also: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17
17.9.8	$\displaystyle{{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,q\right)$	$\displaystyle=\frac{\left(de/(bc);q\right)_{n}}{\left(e;q\right)_{n}}\left(% \frac{bc}{d}\right)^{n}{{}_{3}\phi_{2}}\left({q^{-n},d/b,d/c\atop d,de/(bc)};q% ,q\right),$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : 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 See also: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17
17.9.9	$\displaystyle{{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,q\right)$	$\displaystyle=\frac{\left(e/c;q\right)_{n}}{\left(e;q\right)_{n}}c^{n}{{}_{3}% \phi_{2}}\left({q^{-n},c,d/b\atop d,cq^{1-n}/e};q,\frac{bq}{e}\right),$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : 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 See also: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17
17.9.10	$\displaystyle{{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,\frac{deq^{n}}{bc}\right)$	$\displaystyle=\frac{\left(e/c;q\right)_{n}}{\left(e;q\right)_{n}}{{}_{3}\phi_{% 2}}\left({q^{-n},c,d/b\atop d,cq^{1-n}/e};q,q\right).$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : 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 See also: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

$q$ -Sheppard Identity

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Keywords:: $q$ -Sheppard identity, $q$ -hypergeometric function
See also:: Annotations for §17.9(ii), §17.9 and Ch.17

17.9.11		${{}_{3}\phi_{2}}\left({q^{-n},b,c\atop d,e};q,q\right)=\frac{\left(e/c,d/c;q% \right)_{n}}{\left(e,d;q\right)_{n}}c^{n}{{}_{3}\phi_{2}}\left({q^{-n},c,% \ifrac{cbq^{1-n}}{(de)}\atop\ifrac{cq^{1-n}}{e},\ifrac{cq^{1-n}}{d}};q,q\right),$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $n$ : nonnegative integer Referenced by: §17.9(ii) Permalink: http://dlmf.nist.gov/17.9.E11 Encodings: TeX, pMML, png See also: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

17.9.12		${{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,\frac{de}{abc}\right)=\frac{\left(e/b% ,e/c,cq/a,q/d;q\right)_{\infty}}{\left(e,cq/d,q/a,e/(bc);q\right)_{\infty}}{{}% _{3}\phi_{2}}\left({c,d/a,cq/e\atop cq/a,bcq/e};q,\frac{bq}{d}\right)-\frac{% \left(q/d,eq/d,b,c,d/a,de/(bcq),bcq^{2}/(de);q\right)_{\infty}}{\left(d/q,e,bq% /d,cq/d,q/a,e/(bc),bcq/e;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({aq/d,bq/d,cq% /d\atop q^{2}/d,eq/d};q,\frac{de}{abc}\right),$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base Permalink: http://dlmf.nist.gov/17.9.E12 Encodings: TeX, pMML, png See also: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

17.9.13		${{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,\frac{de}{abc}\right)=\frac{\left(e/b% ,e/c;q\right)_{\infty}}{\left(e,e/(bc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left% ({d/a,b,c\atop d,bcq/e};q,q\right)+\frac{\left(d/a,b,c,de/(bc);q\right)_{% \infty}}{\left(d,e,bc/e,de/(abc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({e/b,% e/c,de/(abc)\atop de/(bc),eq/(bc)};q,q\right).$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base Referenced by: (17.9.3_5) Permalink: http://dlmf.nist.gov/17.9.E13 Encodings: TeX, pMML, png See also: Annotations for §17.9(ii), §17.9(ii), §17.9 and Ch.17

§17.9(iii) Further ${{}_{r}\phi_{s}}$ Functions

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Notes:: See Gasper and Rahman (2004, pp. 43, 50, 63–64, 70, 72, 361).
Referenced by:: Erratum (V1.1.12) for Subsection 17.9(iii)
Permalink:: http://dlmf.nist.gov/17.9.iii
See also:: Annotations for §17.9 and Ch.17

Sears’ Balanced ${{}_{4}\phi_{3}}$ Transformations

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Keywords:: Sears’ balanced ${{}_{4}\phi_{3}}$ transformation, $q$ -hypergeometric function
See also:: Annotations for §17.9(iii), §17.9 and Ch.17

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

17.9.14		${{}_{4}\phi_{3}}\left({q^{-n},a,b,c\atop d,e,f};q,q\right)=\frac{\left(e/a,f/a% ;q\right)_{n}}{\left(e,f;q\right)_{n}}a^{n}{{}_{4}\phi_{3}}\left({q^{-n},a,d/b% ,d/c\atop d,aq^{1-n}/e,aq^{1-n}/f};q,q\right)=\frac{\left(a,ef/(ab),ef/(ac);q% \right)_{n}}{\left(e,f,ef/(abc);q\right)_{n}}{{}_{4}\phi_{3}}\left({q^{-n},e/a% ,f/a,ef/(abc)\atop ef/(ab),ef/(ac),q^{1-n}/a};q,q\right).$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/17.9.E14 Encodings: TeX, pMML, png See also: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

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

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Keywords:: Watson’s $q$ -analog, Whipple’s theorem
See also:: Annotations for §17.9(iii), §17.9 and Ch.17

With $n$ a nonnegative integer

17.9.15		$\frac{\left(aq,aq/(de);q\right)_{n}}{\left(aq/d,aq/e;q\right)_{n}}{{}_{4}\phi_% {3}}\left({aq/(bc),d,e,q^{-n}\atop aq/b,aq/c,deq^{-n}/a};q,q\right)={{}_{8}% \phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,q^{-n}\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq^{n+1}};q,\frac{a^{2}q^{2+% n}}{bcde}\right).$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : complex base and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/17.9.E15 Encodings: TeX, pMML, png See also: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

Bailey’s Transformation of Very-Well-Poised ${{}_{8}\phi_{7}}$

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Keywords:: Bailey’s transformation of very-well-poised ${{}_{8}\phi_{7}}$ , $q$ -hypergeometric function
See also:: Annotations for §17.9(iii), §17.9 and Ch.17

17.9.16		${{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,f\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};q,\frac{a^{2}q^{2}}{% bcdef}\right)=\frac{\left(aq,aq/(de),aq/(df),aq/(ef);q\right)_{\infty}}{\left(% aq/d,aq/e,aq/f,aq/(def);q\right)_{\infty}}{{}_{4}\phi_{3}}\left({aq/(bc),d,e,f% \atop aq/b,aq/c,def/a};q,q\right)+\frac{\left(aq,aq/(bc),d,e,f,a^{2}q^{2}/(% bdef),a^{2}q^{2}/(cdef);q\right)_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,aq/f,a^{2% }q^{2}/(bcdef),def/(aq);q\right)_{\infty}}\*{{}_{4}\phi_{3}}\left({aq/(de),aq/% (df),aq/(ef),a^{2}q^{2}/(bcdef)\atop a^{2}q^{2}/(bdef),a^{2}q^{2}/(cdef),aq^{2% }/(def)};q,q\right).$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol and $q$ : complex base Permalink: http://dlmf.nist.gov/17.9.E16 Encodings: TeX, pMML, png See also: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

Sears–Carlitz Transformation

ⓘ

See also:: Annotations for §17.9(iii), §17.9 and Ch.17

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

17.9.17		${{}_{3}\phi_{2}}\left({a,b,c\atop aq/b,aq/c};q,\frac{aqz}{bc}\right)=\frac{% \left(az;q\right)_{\infty}}{\left(z;q\right)_{\infty}}\*{{}_{5}\phi_{4}}\left(% {a^{\frac{1}{2}},-a^{\frac{1}{2}},(aq)^{\frac{1}{2}},-(aq)^{\frac{1}{2}},aq/(% bc)\atop aq/b,aq/c,az,q/z};q,q\right).$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : 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 See also: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

Gasper’s $q$ -Analog of Clausen’s Formula (16.12.2)

ⓘ

See also:: Annotations for §17.9(iii), §17.9 and Ch.17

17.9.18		$\left({{}_{4}\phi_{3}}\left({a,b,abz,ab/z\atop abq^{\frac{1}{2}},-abq^{\frac{1% }{2}},-ab};q,q\right)\right)^{2}={{}_{5}\phi_{4}}\left({a^{2},b^{2},ab,abz,ab/% z\atop abq^{\frac{1}{2}},-abq^{\frac{1}{2}},-ab,a^{2}b^{2}};q,q\right),$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : 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 See also: Annotations for §17.9(iii), §17.9(iii), §17.9 and Ch.17

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

§17.9(iv) Bibasic Series

ⓘ

Keywords:: bibasic sums and series, $q$ -hypergeometric function
Notes:: See Andrews (1966b).
Permalink:: http://dlmf.nist.gov/17.9.iv
See also:: Annotations for §17.9 and Ch.17

Mixed-Base Heine-Type Transformations

ⓘ

Keywords:: mixed base Heine-type transformations, $q$ -hypergeometric function
See also:: Annotations for §17.9(iv), §17.9 and Ch.17

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: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (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 See also: Annotations for §17.9(iv), §17.9(iv), §17.9 and Ch.17

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: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (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 See also: Annotations for §17.9(iv), §17.9(iv), §17.9 and Ch.17

17.8 Special Cases of

{{}_{r}\psi_{r}}

Functions 17.10 Transformations of

{{}_{r}\psi_{r}}

Functions

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§17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

Contents

§17.9(i) ${{}_{2}\phi_{1}}\to{{}_{2}\phi_{2}}$ , ${{}_{3}\phi_{1}}$ , or ${{}_{3}\phi_{2}}$

F. H. Jackson’s Transformations

§17.9(ii) ${{}_{3}\phi_{2}}\to{{}_{3}\phi_{2}}$

Transformations of ${{}_{3}\phi_{2}}$ -Series

$q$ -Sheppard Identity

§17.9(iii) Further ${{}_{r}\phi_{s}}$ Functions

Sears’ Balanced ${{}_{4}\phi_{3}}$ Transformations

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

Bailey’s Transformation of Very-Well-Poised ${{}_{8}\phi_{7}}$

Sears–Carlitz Transformation

Gasper’s $q$ -Analog of Clausen’s Formula (16.12.2)

§17.9(iv) Bibasic Series

Mixed-Base Heine-Type Transformations

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§17.9 Further Transformations of ϕrr+1 Functions

Contents

§17.9(i) ϕ12→ϕ22, ϕ13, or ϕ23

F. H. Jackson’s Transformations

§17.9(ii) ϕ23→ϕ23

Transformations of ϕ23-Series

q-Sheppard Identity

§17.9(iii) Further ϕsr Functions

Sears’ Balanced ϕ34 Transformations

Watson’s q-Analog of Whipple’s Theorem

Bailey’s Transformation of Very-Well-Poised ϕ78

Sears–Carlitz Transformation

Gasper’s q-Analog of Clausen’s Formula (16.12.2)

§17.9(iv) Bibasic Series

Mixed-Base Heine-Type Transformations

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§17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions

§17.9(i) ${{}_{2}\phi_{1}}\to{{}_{2}\phi_{2}}$ , ${{}_{3}\phi_{1}}$ , or ${{}_{3}\phi_{2}}$

§17.9(ii) ${{}_{3}\phi_{2}}\to{{}_{3}\phi_{2}}$

Transformations of ${{}_{3}\phi_{2}}$ -Series

$q$ -Sheppard Identity

§17.9(iii) Further ${{}_{r}\phi_{s}}$ Functions

Sears’ Balanced ${{}_{4}\phi_{3}}$ Transformations

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

Bailey’s Transformation of Very-Well-Poised ${{}_{8}\phi_{7}}$

Gasper’s $q$ -Analog of Clausen’s Formula (16.12.2)