17 q-Hypergeometric and Related Functions Properties 17.5

{{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}

{{}_{r}\phi_{s}}

Functions

§17.6 ${{}_{2}\phi_{1}}$ Function

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Permalink:: http://dlmf.nist.gov/17.6
See also:: Annotations for Ch.17

§17.6(i) Special Values
§17.6(ii) ${{}_{2}\phi_{1}}$ 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

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Notes:: See Gasper and Rahman (2004, pp. 13, 14, 18, 26, 27).
Permalink:: http://dlmf.nist.gov/17.6.i
Addition (effective with 1.1.0):: Equation (17.6.4_5) was added.
Suggested 2019-10-19 by Slobodan Damjanovic
See also:: Annotations for §17.6 and Ch.17

$q$ -Gauss Sum

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See also:: Annotations for §17.6(i), §17.6 and Ch.17

17.6.1		${{}_{2}\phi_{1}}\left({a,b\atop c};q,\ifrac{c}{(ab)}\right)=\frac{\left(c/a,c/% b;q\right)_{\infty}}{\left(c,c/(ab);q\right)_{\infty}}.$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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/17.6.E1 Encodings: TeX, pMML, png See also: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

First $q$ -Chu–Vandermonde Sum

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Keywords:: Chu–Vandermonde sums (first and second), $q$ -analogs, $q$ -hypergeometric function
See also:: Annotations for §17.6(i), §17.6 and Ch.17

17.6.2		${{}_{2}\phi_{1}}\left({a,q^{-n}\atop c};q,\ifrac{cq^{n}}{a}\right)=\frac{\left% (c/a;q\right)_{n}}{\left(c;q\right)_{n}}.$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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: §17.6(i) Permalink: http://dlmf.nist.gov/17.6.E2 Encodings: TeX, pMML, png See also: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

Second $q$ -Chu–Vandermonde Sum

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See also:: Annotations for §17.6(i), §17.6 and Ch.17

This reverses the order of summation in (17.6.2):

17.6.3		${{}_{2}\phi_{1}}\left({a,q^{-n}\atop c};q,q\right)=\frac{a^{n}\left(c/a;q% \right)_{n}}{\left(c;q\right)_{n}}.$
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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/17.6.E3 Encodings: TeX, pMML, png See also: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

Andrews–Askey Sum

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Keywords:: Andrews–Askey sum, $q$ -hypergeometric function
See also:: Annotations for §17.6(i), §17.6 and Ch.17

17.6.4		${{}_{2}\phi_{1}}\left({b^{2},\ifrac{b^{2}}{c}\atop c};q^{2},\ifrac{cq}{b^{2}}% \right)=\frac{1}{2}\frac{\left(b^{2},q;q^{2}\right)_{\infty}}{\left(c,cq/b^{2}% ;q^{2}\right)_{\infty}}\left(\frac{\left(c/b;q\right)_{\infty}}{\left(b;q% \right)_{\infty}}+\frac{\left(-c/b;q\right)_{\infty}}{\left(-b;q\right)_{% \infty}}\right),$
$\|cq\|<\|b^{2}\|$ .
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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: §17.7(i), §17.7(i), §17.8 Permalink: http://dlmf.nist.gov/17.6.E4 Encodings: TeX, pMML, png See also: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

Related formulas are (17.7.3), (17.8.8) and

17.6.4_5		${{}_{2}\phi_{1}}\left({b^{2},\ifrac{b^{2}}{c}\atop cq^{2}};q^{2},\ifrac{cq^{3}% }{b^{2}}\right)=\frac{1}{2b}\frac{\left(b^{2},q;q^{2}\right)_{\infty}}{\left(% cq^{2},\ifrac{cq}{b^{2}};q^{2}\right)_{\infty}}\left(\frac{\left(\ifrac{cq}{b}% ;q\right)_{\infty}}{\left(b;q\right)_{\infty}}-\frac{\left(\ifrac{-cq}{b};q% \right)_{\infty}}{\left(-b;q\right)_{\infty}}\right),$
$\left\|cq^{3}\right\|<\left\|b^{2}\right\|$ .
ⓘ 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 Source: Verma and Jain (1983, (3.6)) Referenced by: §17.6(i), Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/17.6.E4_5 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. Suggested 2019-10-19 by Slobodan Damjanovic See also: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

For similar formulas see Verma and Jain (1983).

Bailey–Daum $q$ -Kummer Sum

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Keywords:: Bailey–Daum $q$ -Kummer sum, $q$ -hypergeometric function
See also:: Annotations for §17.6(i), §17.6 and Ch.17

17.6.5		${{}_{2}\phi_{1}}\left({a,b\atop aq/b};q,-q/b\right)=\frac{\left(-q;q\right)_{% \infty}\left(aq,\ifrac{aq^{2}}{b^{2}};q^{2}\right)_{\infty}}{\left(-q/b,aq/b;q% \right)_{\infty}},$
$\|b\|>\|q\|$ .
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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/17.6.E5 Encodings: TeX, pMML, png See also: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

§17.6(ii) ${{}_{2}\phi_{1}}$ Transformations

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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).
Referenced by:: Erratum (V1.0.10) for References
Permalink:: http://dlmf.nist.gov/17.6.ii
Addition (effective with 1.0.10):: A sentence and reference to Morita (2013) were added after (17.6.16).
See also:: Annotations for §17.6 and Ch.17

Heine’s First Transformation

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Keywords:: Heine’s transformations (first, second, third), $q$ -hypergeometric function
See also:: Annotations for §17.6(ii), §17.6 and Ch.17

17.6.6		${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(b,az;q\right)_{% \infty}}{\left(c,z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({c/b,z\atop az};q,b% \right),$
$\|z\|<1,\|b\|<1$ .
ⓘ 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 $z$ : complex variable Permalink: http://dlmf.nist.gov/17.6.E6 Encodings: TeX, pMML, png See also: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

Heine’s Second Tranformation

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See also:: Annotations for §17.6(ii), §17.6 and Ch.17

17.6.7		${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(c/b,bz;q\right)_{% \infty}}{\left(c,z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({\ifrac{abz}{c},b% \atop bz};q,c/b\right),$
$\|z\|<1,\|c\|<\|b\|$ .
ⓘ 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 Permalink: http://dlmf.nist.gov/17.6.E7 Encodings: TeX, pMML, png See also: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

Heine’s Third Transformation

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See also:: Annotations for §17.6(ii), §17.6 and Ch.17

17.6.8		${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(\ifrac{abz}{c};q% \right)_{\infty}}{\left(z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({c/a,c/b% \atop c};q,\ifrac{abz}{c}\right),$
$\|z\|<1,\|abz\|<\|c\|$ .
ⓘ 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/17.6.E8 Encodings: TeX, pMML, png See also: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

Fine’s First Transformation

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Keywords:: Fine’s transformations (first, second, third), $q$ -hypergeometric function
See also:: Annotations for §17.6(ii), §17.6 and Ch.17

17.6.9		${{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=-\frac{(1-b)(aq/b)}{(1-(\ifrac% {aq}{b}))}\sum_{n=0}^{\infty}\frac{\left(aq,azq/b;q\right)_{n}q^{n}}{\left(azq% ^{2}/b;q\right)_{n}}+\frac{\left(aq,azq/b;q\right)_{\infty}}{\left(aq/b;q% \right)_{\infty}}{{}_{2}\phi_{1}}\left({q,0\atop bq};q,z\right),$
$\|z\|<1$ .
ⓘ 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, $q$ : 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 See also: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

Fine’s Second Transformation

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See also:: Annotations for §17.6(ii), §17.6 and Ch.17

17.6.10		$(1-z){{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=\sum_{n=0}^{\infty}\frac{% \left(b/a;q\right)_{n}(-az)^{n}q^{(n^{2}+n)/2}}{\left(bq,zq;q\right)_{n}},$
$\|z\|<1$ .
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/17.6.E10 Encodings: TeX, pMML, png See also: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

Fine’s Third Transformation

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Keywords:: Fine’s transformations (first, second, third), $q$ -hypergeometric function
See also:: Annotations for §17.6(ii), §17.6 and Ch.17

17.6.11		$\frac{1-z}{1-b}{{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=\sum_{n=0}^{% \infty}\frac{\left(aq;q\right)_{n}\left(azq/b;q\right)_{2n}b^{n}}{\left(zq,aq/% b;q\right)_{n}}-aq\sum_{n=0}^{\infty}\frac{\left(aq;q\right)_{n}\left(azq/b;q% \right)_{2n+1}(bq)^{n}}{\left(zq;q\right)_{n}\left(aq/b;q\right)_{n+1}},$
$\|z\|<1,\|b\|<1$ .
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/17.6.E11 Encodings: TeX, pMML, png See also: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

Rogers–Fine Identity

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Keywords:: Rogers–Fine identity, $q$ -hypergeometric function
See also:: Annotations for §17.6(ii), §17.6 and Ch.17

17.6.12		$(1-z){{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=\sum_{n=0}^{\infty}\frac{% \left(aq,azq/b;q\right)_{n}}{\left(bq,zq;q\right)_{n}}(1-azq^{2n+1})(bz)^{n}q^% {n^{2}},$
$\|z\|<1$ .
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §17.6(ii) Permalink: http://dlmf.nist.gov/17.6.E12 Encodings: TeX, pMML, png See also: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

Nonterminating Form of the $q$ -Vandermonde Sum

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Keywords:: Vandermonde sum, nonterminating $q$ -version, $q$ -hypergeometric function
See also:: Annotations for §17.6(ii), §17.6 and Ch.17

17.6.13		${{}_{2}\phi_{1}}\left(a,b;c;q,q\right)+\frac{\left(q/c,a,b;q\right)_{\infty}}{% \left(c/q,aq/c,bq/c;q\right)_{\infty}}{{}_{2}\phi_{1}}\left(aq/c,bq/c;q^{2}/c;% q,q\right)=\frac{\left(q/c,abq/c;q\right)_{\infty}}{\left(aq/c,bq/c;q\right)_{% \infty}},$
ⓘ Symbols: ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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/17.6.E13 Encodings: TeX, pMML, png See also: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

17.6.14		$\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}\left(b;q^{2}\right)_{n}z^{n}}{% \left(q;q\right)_{n}\left(azb;q^{2}\right)_{n}}=\frac{\left(az,bz;q^{2}\right)% _{\infty}}{\left(z,azb;q^{2}\right)_{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop bz% };q^{2},zq\right).$
ⓘ 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, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §17.6(ii) Permalink: http://dlmf.nist.gov/17.6.E14 Encodings: TeX, pMML, png See also: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

Three-Term ${{}_{2}\phi_{1}}$ Transformations

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Keywords:: $q$ -hypergeometric function, three-term ${{}_{2}\phi_{1}}$ transformation
See also:: Annotations for §17.6(ii), §17.6 and Ch.17

17.6.15		${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(abz/c,q/c;q\right)_{% \infty}}{\left(az/c,q/a;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({c/a,cq/(abz)% \atop cq/(az)};q,bq/c\right)-\frac{\left(b,q/c,c/a,az/q,q^{2}/(az);q\right)_{% \infty}}{\left(c/q,bq/c,q/a,az/c,cq/(az);q\right)_{\infty}}{{}_{2}\phi_{1}}% \left({aq/c,bq/c\atop q^{2}/c};q,z\right),$
$\|z\|<1,\|bq\|<\|c\|$ .
ⓘ 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 Permalink: http://dlmf.nist.gov/17.6.E15 Encodings: TeX, pMML, png See also: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

17.6.16		${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(b,c/a,az,q/(az);q% \right)_{\infty}}{\left(c,b/a,z,q/z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({a% ,aq/c\atop aq/b};q,cq/(abz)\right)+\frac{\left(a,c/b,bz,q/(bz);q\right)_{% \infty}}{\left(c,a/b,z,q/z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({b,bq/c% \atop bq/a};q,cq/(abz)\right),$
$\|z\|<1$ , $\|abz\|<\|cq\|$ .
ⓘ 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 Referenced by: §17.6(ii) Permalink: http://dlmf.nist.gov/17.6.E16 Encodings: TeX, pMML, png See also: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

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

§17.6(iii) Contiguous Relations

ⓘ

Notes:: See Gasper and Rahman (2004, pp. 26–27).
Permalink:: http://dlmf.nist.gov/17.6.iii
See also:: Annotations for §17.6 and Ch.17

Heine’s Contiguous Relations

ⓘ

Keywords:: contiguous relations (Heine’s), $q$ -hypergeometric function
See also:: Annotations for §17.6(iii), §17.6 and Ch.17

17.6.17	$\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c/q};q,z\right)-{{}_{2}\phi_{1}}% \left({a,b\atop c};q,z\right)$	$\displaystyle=cz\frac{(1-a)(1-b)}{(q-c)(1-c)}{{}_{2}\phi_{1}}\left({aq,bq\atop cq% };q,z\right),$
ⓘ 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/17.6.E17 Encodings: TeX, pMML, png See also: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17
17.6.18	$\displaystyle{{}_{2}\phi_{1}}\left({aq,b\atop c};q,z\right)-{{}_{2}\phi_{1}}% \left({a,b\atop c};q,z\right)$	$\displaystyle=az\frac{1-b}{1-c}{{}_{2}\phi_{1}}\left({aq,bq\atop cq};q,z\right),$
ⓘ 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/17.6.E18 Encodings: TeX, pMML, png See also: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17
17.6.19	$\displaystyle{{}_{2}\phi_{1}}\left({aq,b\atop cq};q,z\right)-{{}_{2}\phi_{1}}% \left({a,b\atop c};q,z\right)$	$\displaystyle=az\frac{(1-b)(1-(c/a))}{(1-c)(1-cq)}{{}_{2}\phi_{1}}\left({aq,bq% \atop cq^{2}};q,z\right),$
ⓘ 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/17.6.E19 Encodings: TeX, pMML, png See also: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17
17.6.20	$\displaystyle{{}_{2}\phi_{1}}\left({aq,b/q\atop c};q,z\right)-{{}_{2}\phi_{1}}% \left({a,b\atop c};q,z\right)$	$\displaystyle=az\frac{(1-b/(aq))}{1-c}{{}_{2}\phi_{1}}\left({aq,b\atop cq};q,z% \right),$
ⓘ 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/17.6.E20 Encodings: TeX, pMML, png See also: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17

17.6.21	$\displaystyle b(1-a){{}_{2}\phi_{1}}\left({aq,b\atop c};q,z\right)-a(1-b){{}_{% 2}\phi_{1}}\left({a,bq\atop c};q,z\right)$	$\displaystyle=(b-a){{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right),$
ⓘ 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/17.6.E21 Encodings: TeX, pMML, png See also: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17
17.6.22	$\displaystyle a\left(1-\frac{b}{c}\right){{}_{2}\phi_{1}}\left({a,b/q\atop c};% q,z\right)-b\left(1-\frac{a}{c}\right){{}_{2}\phi_{1}}\left({a/q,b\atop c};q,z\right)$	$\displaystyle=(a-b)\left(1-\frac{abz}{cq}\right){{}_{2}\phi_{1}}\left({a,b% \atop c};q,z\right),$
ⓘ 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/17.6.E22 Encodings: TeX, pMML, png See also: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17

17.6.23		$q\left(1-\frac{a}{c}\right){{}_{2}\phi_{1}}\left({a/q,b\atop c};q,z\right)+(1-% a)\left(1-\frac{abz}{c}\right){{}_{2}\phi_{1}}\left({aq,b\atop c};q,z\right)=% \left(1+q-a-\frac{aq}{c}+\frac{a^{2}z}{c}-\frac{abz}{c}\right){{}_{2}\phi_{1}}% \left({a,b\atop c};q,z\right),$
ⓘ 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/17.6.E23 Encodings: TeX, pMML, png See also: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17

17.6.24		$(1-c)(q-c)(abz-c){{}_{2}\phi_{1}}\left({a,b\atop c/q};q,z\right)+z(c-a)(c-b){{% }_{2}\phi_{1}}\left({a,b\atop cq};q,z\right)=(c-1)(c(q-c)+z(ca+cb-ab-abq)){{}_% {2}\phi_{1}}\left({a,b\atop c};q,z\right).$
ⓘ 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/17.6.E24 Encodings: TeX, pMML, png See also: Annotations for §17.6(iii), §17.6(iii), §17.6 and Ch.17

§17.6(iv) Differential Equations

ⓘ

Keywords:: differential equations, $q$ -differential equations, $q$ -hypergeometric function
Notes:: See Gasper and Rahman (2004, pp. 26–28).
Referenced by:: §17.2(iv)
Permalink:: http://dlmf.nist.gov/17.6.iv
See also:: Annotations for §17.6 and Ch.17

Iterations of $\mathcal{D}$

ⓘ

See also:: Annotations for §17.6(iv), §17.6 and Ch.17

17.6.25	$\displaystyle\mathcal{D}_{q}^{n}{{}_{2}\phi_{1}}\left({a,b\atop c};q,zd\right)$	$\displaystyle=\frac{\left(a,b;q\right)_{n}d^{n}}{\left(c;q\right)_{n}(1-q)^{n}% }{{}_{2}\phi_{1}}\left({aq^{n},bq^{n}\atop cq^{n}};q,dz\right),$
ⓘ 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, $n$ : nonnegative integer, $z$ : complex variable and $\mathcal{D}_{q}$ : $q$ -differential operator Permalink: http://dlmf.nist.gov/17.6.E25 Encodings: TeX, pMML, png See also: Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17
17.6.26	$\displaystyle\mathcal{D}_{q}^{n}\left(\frac{\left(z;q\right)_{\infty}}{\left(% abz/c;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)\right)$	$\displaystyle=\frac{\left(c/a,c/b;q\right)_{n}}{\left(c;q\right)_{n}(1-q)^{n}}% \left(\frac{ab}{c}\right)^{n}\frac{\left(zq^{n};q\right)_{\infty}}{\left(abz/c% ;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop cq^{n}};q,zq^{n}\right).$
ⓘ 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, $n$ : nonnegative integer, $z$ : complex variable and $\mathcal{D}_{q}$ : $q$ -differential operator Permalink: http://dlmf.nist.gov/17.6.E26 Encodings: TeX, pMML, png See also: Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17

$q$ -Differential Equation

ⓘ

See also:: Annotations for §17.6(iv), §17.6 and Ch.17

17.6.27		$z(c-abqz)\mathcal{D}_{q}^{2}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)+% \left(\frac{1-c}{1-q}+\frac{(1-a)(1-b)-(1-abq)}{1-q}z\right)\mathcal{D}_{q}{{}% _{2}\phi_{1}}\left({a,b\atop c};q,z\right)-\frac{(1-a)(1-b)}{(1-q)^{2}}{{}_{2}% \phi_{1}}\left({a,b\atop c};q,z\right)=0.$
ⓘ 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, $z$ : complex variable and $\mathcal{D}_{q}$ : $q$ -differential operator Referenced by: §17.6(iv) Permalink: http://dlmf.nist.gov/17.6.E27 Encodings: TeX, pMML, png See also: Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17

(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

ⓘ

Keywords:: integral representations, $q$ -hypergeometric function
Notes:: See Gasper and Rahman (2004, pp. 23, 115).
Permalink:: http://dlmf.nist.gov/17.6.v
See also:: Annotations for §17.6 and Ch.17

17.6.28	$\displaystyle{{}_{2}\phi_{1}}\left({q^{\alpha},q^{\beta}\atop q^{\gamma}};q,z\right)$	$\displaystyle=\frac{\Gamma_{q}\left(\gamma\right)}{\Gamma_{q}\left(\beta\right% )\Gamma_{q}\left(\gamma-\beta\right)}\int_{0}^{1}\frac{t^{\beta-1}\left(tq;q% \right)_{\gamma-\beta-1}}{\left(xt;q\right)_{\alpha}}{\mathrm{d}}_{q}t.$
ⓘ Symbols: $\int$ : integral, $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : $q$ -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${\mathrm{d}}_{\NVar{q}}\NVar{x}$ : $q$ -differential, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $z$ : complex variable and $x$ : real variable Permalink: http://dlmf.nist.gov/17.6.E28 Encodings: TeX, pMML, png See also: Annotations for §17.6(v), §17.6 and Ch.17
17.6.29	$\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)$	$\displaystyle=\left(\frac{-1}{2\pi i}\right)\frac{\left(a,b;q\right)_{\infty}}% {\left(q,c;q\right)_{\infty}}\int_{-i\infty}^{i\infty}\frac{\left(q^{1+\zeta},% cq^{\zeta};q\right)_{\infty}}{\left(aq^{\zeta},bq^{\zeta};q\right)_{\infty}}% \frac{\pi(-z)^{\zeta}}{\sin\left(\pi\zeta\right)}\mathrm{d}\zeta,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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, $\sin\NVar{z}$ : sine function, $q$ : complex base and $z$ : complex variable Permalink: http://dlmf.nist.gov/17.6.E29 Encodings: TeX, pMML, png See also: Annotations for §17.6(v), §17.6 and Ch.17

where $|z|<1$ , $|\operatorname{ph}\left(-z\right)|<\pi$ , and the contour of integration separates the poles of $\left(q^{1+\zeta},cq^{\zeta};q\right)_{\infty}/\sin\left(\pi\zeta\right)$ 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
See also:: Annotations for §17.6 and Ch.17

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

{{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}

Functions 17.7 Special Cases of Higher

{{}_{r}\phi_{s}}

Functions

§17.6 ϕ12 Function

Contents