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17 q-Hypergeometric and Related Functions Properties 17.6

{{}_{2}\phi_{1}}

Function 17.8 Special Cases of

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

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

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Keywords:: $q$ -hypergeometric function, special cases
Permalink:: http://dlmf.nist.gov/17.7
See also:: Annotations for Ch.17

Contents

§17.7(i) ${{}_{2}\phi_{2}}$ Functions
§17.7(ii) ${{}_{3}\phi_{2}}$ Functions
§17.7(iii) Other ${{}_{r}\phi_{s}}$ Functions

§17.7(i) ${{}_{2}\phi_{2}}$ Functions

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Notes:: See Gasper and Rahman (2004, pp. 28–29). For (17.7.3) combine Gasper and Rahman (2004, p. 359, Equation (III.4)) with (17.6.4).
Permalink:: http://dlmf.nist.gov/17.7.i
See also:: Annotations for §17.7 and Ch.17

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

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Keywords:: Bailey’s ${{}_{2}F_{1}}(-1)$ sum, $q$ -analog, $q$ -hypergeometric function
See also:: Annotations for §17.7(i), §17.7 and Ch.17

17.7.1		${{}_{2}\phi_{2}}\left({a,q/a\atop-q,b};q,-b\right)=\frac{\left(ab,bq/a;q^{2}% \right)_{\infty}}{\left(b;q\right)_{\infty}},$
$\|b\|<1$ .
ⓘ 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 and $q$ : complex base Permalink: http://dlmf.nist.gov/17.7.E1 Encodings: TeX, pMML, png See also: Annotations for §17.7(i), §17.7(i), §17.7 and Ch.17

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

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Keywords:: Gauss’s ${{}_{2}F_{1}}(-1)$ sum, $q$ -analog, $q$ -hypergeometric function
See also:: Annotations for §17.7(i), §17.7 and Ch.17

17.7.2		${{}_{2}\phi_{2}}\left({a^{2},b^{2}\atop abq^{\frac{1}{2}},-abq^{\frac{1}{2}}};% q,-q\right)=\frac{\left(a^{2}q,b^{2}q;q^{2}\right)_{\infty}}{\left(q,a^{2}b^{2% }q;q^{2}\right)_{\infty}}.$
ⓘ 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/17.7.E2 Encodings: TeX, pMML, png See also: Annotations for §17.7(i), §17.7(i), §17.7 and Ch.17

Sum Related to (17.6.4)

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Keywords:: Andrews–Askey sum, $q$ -hypergeometric function
See also:: Annotations for §17.7(i), §17.7 and Ch.17

17.7.3		${{}_{2}\phi_{2}}\left({\ifrac{c^{2}}{b^{2}},b^{2}\atop c,cq};q^{2},q\right)=% \frac{1}{2}\frac{\left(b^{2},q;q^{2}\right)_{\infty}}{\left(c,cq;q^{2}\right)_% {\infty}}{\left(\frac{\left(c/b;q\right)_{\infty}}{\left(b;q\right)_{\infty}}+% \frac{\left(-c/b;q\right)_{\infty}}{\left(-b;q\right)_{\infty}}\right)}.$
ⓘ 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 and $q$ : complex base Referenced by: §17.6(i), §17.7(i) Permalink: http://dlmf.nist.gov/17.7.E3 Encodings: TeX, pMML, png See also: Annotations for §17.7(i), §17.7(i), §17.7 and Ch.17

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

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Notes:: See Gasper and Rahman (2004, pp. 17, 58, 356).
Permalink:: http://dlmf.nist.gov/17.7.ii
See also:: Annotations for §17.7 and Ch.17

$q$ -Pfaff–Saalschütz Sum

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Keywords:: $q$ -Pfaff–Saalschütz sum, $q$ -hypergeometric function
See also:: Annotations for §17.7(ii), §17.7 and Ch.17

17.7.4		${{}_{3}\phi_{2}}\left({a,b,q^{-n}\atop c,abq^{1-n}/c};q,q\right)=\frac{\left(c% /a,c/b;q\right)_{n}}{\left(c,c/(ab);q\right)_{n}}.$
ⓘ 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/17.7.E4 Encodings: TeX, pMML, png See also: Annotations for §17.7(ii), §17.7(ii), §17.7 and Ch.17

Nonterminating Form of the $q$ -Saalschütz Sum

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Keywords:: nonterminating form, $q$ -Saalschütz sum, $q$ -hypergeometric function
See also:: Annotations for §17.7(ii), §17.7 and Ch.17

17.7.5		${{}_{3}\phi_{2}}\left({a,b,c\atop e,f};q,q\right)+\frac{\left(q/e,a,b,c,qf/e;q% \right)_{\infty}}{\left(e/q,aq/e,bq/e,cq/e,f;q\right)_{\infty}}\*{{}_{3}\phi_{% 2}}\left({aq/e,bq/e,cq/e\atop q^{2}/e,qf/e};q,q\right)=\frac{\left(q/e,f/a,f/b% ,f/c;q\right)_{\infty}}{\left(aq/e,bq/e,cq/e,f;q\right)_{\infty}},$
ⓘ 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/17.7.E5 Encodings: TeX, pMML, png See also: Annotations for §17.7(ii), §17.7(ii), §17.7 and Ch.17

where $ef=abcq$ .

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

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Keywords:: Dixon’s sum, F. H. Jackson’s $q$ -analog, $q$ -hypergeometric function
See also:: Annotations for §17.7(ii), §17.7 and Ch.17

17.7.6		${{}_{3}\phi_{2}}\left({q^{-2n},b,c\atop q^{1-2n}/b,q^{1-2n}/c};q,\frac{q^{2-n}% }{bc}\right)=\frac{\left(b,c;q\right)_{n}\left(q,bc;q\right)_{2n}}{\left(q,bc;% q\right)_{n}\left(b,c;q\right)_{2n}}.$
ⓘ 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/17.7.E6 Encodings: TeX, pMML, png See also: Annotations for §17.7(ii), §17.7(ii), §17.7 and Ch.17

Continued Fractions

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Keywords:: continued fractions, $q$ -hypergeometric function
See also:: Annotations for §17.7(ii), §17.7 and Ch.17

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

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

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Notes:: See Gasper and Rahman (2004, pp. 19, 43–44, 61–62, 81–83, 355–356) and Andrews (1996). See also Gao and Guo (2013).
Referenced by:: Erratum (V1.0.10) for References
Permalink:: http://dlmf.nist.gov/17.7.iii
Addition (effective with 1.0.10):: A citation to Gao and Guo (2013) was added to the metadata.
See also:: Annotations for §17.7 and Ch.17

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

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Keywords:: Dixon’s ${{}_{3}F_{2}}(1)$ sum, $q$ -analog, $q$ -hypergeometric function
See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.7		${{}_{4}\phi_{3}}\left({a,-qa^{\frac{1}{2}},b,c\atop-a^{\frac{1}{2}},aq/b,aq/c}% ;q,\frac{qa^{\frac{1}{2}}}{bc}\right)=\frac{\left(aq,qa^{\frac{1}{2}}/b,qa^{% \frac{1}{2}}/c,aq/(bc);q\right)_{\infty}}{\left(aq/b,aq/c,qa^{\frac{1}{2}},qa^% {\frac{1}{2}}/(bc);q\right)_{\infty}}.$
ⓘ 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/17.7.E7 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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

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Keywords:: Gasper–Rahman $q$ -analog, Watson’s ${{}_{3}F_{2}}$ sum
See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.8		${{}_{8}\phi_{7}}\left({\lambda,q\lambda^{\frac{1}{2}},-q\lambda^{\frac{1}{2}},% a,b,c,-c,\lambda q/c^{2}\atop\lambda^{\frac{1}{2}},-\lambda^{\frac{1}{2}},% \lambda q/a,\lambda q/b,\lambda q/c,-\lambda q/c,c^{2}};q,-\frac{\lambda q}{ab% }\right)=\frac{\left(\lambda q,c^{2}/\lambda;q\right)_{\infty}\left(aq,bq,c^{2% }q/a,c^{2}q/b;q^{2}\right)_{\infty}}{\left(\lambda q/a,\lambda q/b;q\right)_{% \infty}\left(q,abq,c^{2}q,c^{2}q/(ab);q^{2}\right)_{\infty}},$
ⓘ 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: §17.7(iii) Permalink: http://dlmf.nist.gov/17.7.E8 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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

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

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Keywords:: Andrews’ terminating $q$ -analog, Watson’s ${{}_{3}F_{2}}$ sum
See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.9		${{}_{4}\phi_{3}}\left({q^{-n},aq^{n},c,-c\atop(aq)^{\frac{1}{2}},-(aq)^{\frac{% 1}{2}},c^{2}};q,q\right)=\begin{cases}0,&\mbox{$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}$
ⓘ 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/17.7.E9 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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

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Keywords:: Gasper–Rahman $q$ -analog, Whipple’s ${{}_{3}F_{2}}$ sum
See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.10		${{}_{8}\phi_{7}}\left({-c,q(-c)^{\frac{1}{2}},-q(-c)^{\frac{1}{2}},a,q/a,c,-d,% -q/d\atop(-c)^{\frac{1}{2}},-(-c)^{\frac{1}{2}},-cq/a,-ac,-q,cq/d,cd};q,c% \right)=\frac{\left(-c,-cq;q\right)_{\infty}\left(acd,acq/d,cdq/a,cq^{2}/(ad);% q^{2}\right)_{\infty}}{\left(cd,cq/d,-ac,-cq/a;q\right)_{\infty}}.$
ⓘ 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/17.7.E10 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

Andrews’ Terminating $q$ -Analog

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See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.11		${{}_{4}\phi_{3}}\left({q^{-n},q^{n+1},c,-c\atop e,c^{2}q/e,-q};q,q\right)=% \frac{\left(eq^{-n},eq^{n+1},c^{2}q^{1-n}/e,c^{2}q^{n+2}/e;q^{2}\right)_{% \infty}}{\left(e,c^{2}q/e;q\right)_{\infty}}.$
ⓘ 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/17.7.E11 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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

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Keywords:: Bailey’s ${{}_{4}F_{3}}(1)$ sum, $q$ -analogs (first and second), $q$ -analogs (first and second), $q$ -hypergeometric function
See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.12		${{}_{4}\phi_{3}}\left({a,aq,b^{2}q^{2n},q^{-2n}\atop b,bq,a^{2}q^{2}};q^{2},q^% {2}\right)=\frac{a^{n}\left(-q,b/a;q\right)_{n}}{\left(-aq,b;q\right)_{n}}.$
ⓘ 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/17.7.E12 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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

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See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.13		${{}_{4}\phi_{3}}\left({a,aq,b^{2}q^{2n-2},q^{-2n}\atop b,bq,a^{2}};q^{2},q^{2}% \right)=\frac{a^{n}\left(-q,b/a;q\right)_{n}(1-bq^{n-1})}{\left(-a,b;q\right)_% {n}(1-bq^{2n-1})}.$
ⓘ 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/17.7.E13 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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

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Keywords:: Dougall’s ${{}_{7}F_{6}}(1)$ sum, F. H. Jackson’s $q$ -analog, $q$ -hypergeometric function
See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.14		${{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,q^{-n}% \atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq^{n+1}};q,q\right% )=\frac{\left(aq,aq/(bc),aq/(bd),aq/(cd);q\right)_{n}}{\left(aq/b,aq/c,aq/d,aq% /(bcd);q\right)_{n}},$
ⓘ 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: §17.7(iii) Permalink: http://dlmf.nist.gov/17.7.E14 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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

Limiting Cases of (17.7.14)

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See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.15		${{}_{6}\phi_{5}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d};q,\frac{aq}{bcd}\right)=\frac{% \left(aq,aq/(bc),aq/(bd),aq/(cd);q\right)_{\infty}}{\left(aq/b,aq/c,aq/d,aq/(% bcd);q\right)_{\infty}},$
ⓘ 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/17.7.E15 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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

17.7.16		${{}_{6}\phi_{5}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,q^{-n}\atop a^% {\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq^{n+1}};q,\frac{aq^{n+1}}{bc}\right% )=\frac{\left(aq,aq/(bc);q\right)_{n}}{\left(aq/b,aq/c;q\right)_{n}}.$
ⓘ 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/17.7.E16 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

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

ⓘ

See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.17		${{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e,f\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e,aq/f};q,q\right)-\frac{b}{a}% \frac{\left(aq,c,d,e,f,bq/a,bq/c,bq/d,bq/e,bq/f;q\right)_{\infty}}{\left(aq/b,% aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a,b^{2}q/a;q\right)_{\infty}}\*{{}_{8}% \phi_{7}}\left({b^{2}/a,qba^{-\frac{1}{2}},-qba^{-\frac{1}{2}},b,bc/a,bd/a,be/% a,bf/a\atop ba^{-\frac{1}{2}},-ba^{-\frac{1}{2}},bq/a,bq/c,bq/d,bq/e,bq/f};q,q% \right)=\frac{\left(aq,b/a,aq/(cd),aq/(ce),aq/(cf),aq/(de),aq/(df),aq/(ef);q% \right)_{\infty}}{\left(aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a;q\right)_{% \infty}},$
ⓘ 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/17.7.E17 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

where $qa^{2}=bcdef$ .

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

ⓘ

See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.18		${{}_{r+2}\phi_{r+1}}\left({a,b,b_{1}q^{m_{1}},\dots,b_{r}q^{m_{r}}\atop bq,b_{% 1},\dots,b_{r}};q,a^{-1}q^{1-(m_{1}+\cdots+m_{r})}\right)=\frac{\left(q,bq/a;q% \right)_{\infty}\left(b_{1}/b;q\right)_{m_{1}}\cdots\left(b_{r}/b;q\right)_{m_% {r}}}{\left(bq,q/a;q\right)_{\infty}\left(b_{1};q\right)_{m_{1}}\cdots\left(b_% {r};q\right)_{m_{r}}}b^{m_{1}+\cdots+m_{r}},$
ⓘ 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, $m$ : nonnegative integer and $r$ : nonnegative integer Permalink: http://dlmf.nist.gov/17.7.E18 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

and

17.7.19		${{}_{r+1}\phi_{r}}\left({a,b_{1}q^{m_{1}},\dots,b_{r}q^{m_{r}}\atop b_{1},% \dots,b_{r}};q,a^{-1}q^{1-(m_{1}+\cdots+m_{r})}\right)=0,$
ⓘ 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, $m$ : nonnegative integer and $r$ : nonnegative integer Permalink: http://dlmf.nist.gov/17.7.E19 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

where $m_{1},m_{2},\dots,m_{r}$ are arbitrary nonnegative integers.

Gosper’s Bibasic Sum

ⓘ

Keywords:: bibasic sums and series, $q$ -hypergeometric function
See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.20		$\sum_{k=0}^{n}\frac{1-ap^{k}q^{k}}{1-a}\frac{\left(a;p\right)_{k}\left(c;q% \right)_{k}}{\left(q;q\right)_{k}\left(ap/c;p\right)_{k}}c^{-k}=\frac{\left(ap% ;p\right)_{n}\left(cq;q\right)_{n}}{\left(q;q\right)_{n}\left(ap/c;p\right)_{n% }}c^{-n}.$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $k$ : nonnegative integer, $q$ : complex base and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/17.7.E20 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

Gasper’s Extensions of Gosper’s Bibasic Sum

ⓘ

See also:: Annotations for §17.7(iii), §17.7 and Ch.17

17.7.21		$\sum_{k=0}^{n}\frac{(1-ap^{k}q^{k})(1-bp^{k}q^{-k})}{(1-a)(1-b)}\frac{\left(a,% b;p\right)_{k}\left(c,a/(bc);q\right)_{k}}{\left(q,aq/b;q\right)_{k}\left(ap/c% ,bcp;p\right)_{k}}q^{k}=\frac{\left(ap,bp;p\right)_{n}\left(cq,aq/(bc);q\right% )_{n}}{\left(q,aq/b;q\right)_{n}\left(ap/c,bcp;p\right)_{n}},$
ⓘ Symbols: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $k$ : nonnegative integer, $q$ : complex base and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/17.7.E21 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

17.7.22		$\sum_{k=-m}^{n}\frac{(1-adp^{k}q^{k})(1-bp^{k}/(dq^{k}))}{(1-ad)(1-(b/d))}% \frac{\left(a,b;p\right)_{k}\left(c,ad^{2}/(bc);q\right)_{k}}{\left(dq,adq/b;q% \right)_{k}\left(adp/c,bcp/d;p\right)_{k}}q^{k}=\frac{(1-a)(1-b)(1-c)(1-(ad^{2% }/(bc)))}{d(1-ad)(1-(b/d))(1-(c/d))(1-(ad/(bc)))}\left(\frac{\left(ap,bp;p% \right)_{n}\left(cq,ad^{2}q/(bc);q\right)_{n}}{\left(dq,adq/b;q\right)_{n}% \left(adp/c,bcp/d;p\right)_{n}}-\frac{\left(c/(ad),d/(bc);p\right)_{m+1}\left(% 1/d,b/(ad);q\right)_{m+1}}{\left(1/c,bc/(ad^{2});q\right)_{m+1}\left(1/a,1/b;p% \right)_{m+1}}\right),$
ⓘ Symbols: $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $k$ : nonnegative integer, $q$ : complex base, $m$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/17.7.E22 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

and $n$ -th difference generalization:

17.7.23		$\left(1-\frac{a}{q}\right)\left(1-\frac{b}{q}\right)\sum_{k=0}^{n}\frac{\left(% ap^{k},bp^{-k};q\right)_{n-1}(1-(ap^{2k}/b))}{\left(p;p\right)_{n}\left(p;p% \right)_{n-k}\left(ap^{k}/b;q\right)_{n+1}}(-1)^{k}p^{\genfrac{(}{)}{0.0pt}{}{% k}{2}}=\delta_{n,0}.$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $k$ : nonnegative integer, $q$ : complex base and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/17.7.E23 Encodings: TeX, pMML, png See also: Annotations for §17.7(iii), §17.7(iii), §17.7 and Ch.17

© 2010–2020 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.1.0; Release date 2020-12-15. A printed companion is available. 17.6

{{}_{2}\phi_{1}}

Function 17.8 Special Cases of

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