17 q-Hypergeometric and Related Functions Properties 17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

§17.6 ${{}_{2}\phi_{1}}$ Function

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Referenced by:: Erratum (V1.2.1) for §17.6, Erratum (V1.2.5) for Chapter 17 Additions
Permalink:: http://dlmf.nist.gov/17.6
Addition (effective with 1.2.1):: Just above §17.6(i) a paragraph entitled “Analytic continuation” was inserted describing the analytic continuation of the formulas which follow.
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

Analytic Continuation

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Referenced by:: Erratum (V1.2.1) for §17.6
See also:: Annotations for §17.6 and Ch.17

Note that for several of the equations below, the constraints are included to guarantee that the infinite series representation (17.4.1) of the ${{}_{2}\phi_{1}}$ functions converges. These equations can also be used as analytic continuation of these ${{}_{2}\phi_{1}}$ functions.

§17.6(i) Special Values

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Notes:: See Gasper and Rahman (2004, pp. 13, 14, 18, 26, 27).
Referenced by:: §17.6, Erratum (V1.2.1) for §17.6
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

-Gauss Sum

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

17.6.1 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,\ifrac{c}{(ab)}\right)=\frac{\left(c/a,c/% b;q\right)_{\infty}}{\left(c,c/(ab);q\right)_{\infty}},$ $\left|c\right|<\left|ab\right|$ .

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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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol and : complex base
Referenced by:: Erratum (V1.2.0) for Equation (17.6.1)
Permalink:: http://dlmf.nist.gov/17.6.E1
Encodings:: TeX, pMML, png
Addition (effective with 1.2.0):: This constraint $\left|c\right|<\left|ab\right|$ was added.
See also:: Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

First -Chu–Vandermonde Sum

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

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

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, : complex base and : 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 -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:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, : complex base and : 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, -hypergeometric function
See also:: Annotations for §17.6(i), §17.6 and Ch.17

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

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol and : 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 ${{}_{2}\phi_{1}}\left({b^{2},\ifrac{b^{2}}{c}\atop cq^{2}};q^{2},\ifrac{cq^{3}% }{b^{2}}\right)=\frac{1}{2b}\frac{\left(b^{2},q;q^{2}\right)_{\infty}}{\left(% cq^{2},\ifrac{cq}{b^{2}};q^{2}\right)_{\infty}}\left(\frac{\left(\ifrac{cq}{b}% ;q\right)_{\infty}}{\left(b;q\right)_{\infty}}-\frac{\left(\ifrac{-cq}{b};q% \right)_{\infty}}{\left(-b;q\right)_{\infty}}\right),$ $\left|cq^{3}\right|<\left|b^{2}\right|$ .

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol and : 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 -Kummer Sum

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

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

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol and : 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), -hypergeometric function
See also:: Annotations for §17.6(ii), §17.6 and Ch.17

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

ⓘ

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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base and : 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 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(c/b,bz;q\right)_{% \infty}}{\left(c,z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({\ifrac{abz}{c},b% \atop bz};q,c/b\right),$ $\left|z\right|<1,\left|c\right|<\left|b\right|$ .

ⓘ

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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base and : 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 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(\ifrac{abz}{c};q% \right)_{\infty}}{\left(z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({c/a,c/b% \atop c};q,\ifrac{abz}{c}\right),$ $\left|z\right|<1,\left|abz\right|<\left|c\right|$ .

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, : complex base and : 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), -hypergeometric function
See also:: Annotations for §17.6(ii), §17.6 and Ch.17

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

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base, : nonnegative integer and : 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 $(1-z){{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=\sum_{n=0}^{\infty}\frac{% \left(b/a;q\right)_{n}(-az)^{n}q^{(n^{2}+n)/2}}{\left(bq,zq;q\right)_{n}},$ $\left|z\right|<1$ .

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base, : nonnegative integer and : 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), -hypergeometric function
See also:: Annotations for §17.6(ii), §17.6 and Ch.17

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

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base, : nonnegative integer and : 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, -hypergeometric function
See also:: Annotations for §17.6(ii), §17.6 and Ch.17

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

ⓘ

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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base, : nonnegative integer and : 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 -Vandermonde Sum

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Keywords:: Vandermonde sum, nonterminating -version, -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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol and : 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:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base, : nonnegative integer and : 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:: -hypergeometric function, three-term ${{}_{2}\phi_{1}}$ transformation
See also:: Annotations for §17.6(ii), §17.6 and Ch.17

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

ⓘ

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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base and : 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 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(b,c/a,az,q/(az);q% \right)_{\infty}}{\left(c,b/a,z,q/z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({a% ,aq/c\atop aq/b};q,cq/(abz)\right)+\frac{\left(a,c/b,bz,q/(bz);q\right)_{% \infty}}{\left(c,a/b,z,q/z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({b,bq/c% \atop bq/a};q,cq/(abz)\right),$ $\left|z\right|<1$ , $\left|cq\right|<\left|abz\right|$ .

ⓘ

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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base and : complex variable
Referenced by:: §17.6(ii), Erratum (V1.1.9) for Equation (17.6.16)
Permalink:: http://dlmf.nist.gov/17.6.E16
Encodings:: TeX, pMML, png
Correction (effective with 1.1.9):: The constraint originally given by $\left|abz\right|<\left|cq\right|$ has been corrected to read $\left|cq\right|<\left|abz\right|$ .
See also:: Annotations for §17.6(ii), §17.6(ii), §17.6 and Ch.17

For a similar result for -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

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

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

ⓘ

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 -hypergeometric) function, : complex base and : 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 ${{}_{2}\phi_{1}}\left({aq,b\atop c};q,z\right)-{{}_{2}\phi_{1}}\left({a,b\atop c% };q,z\right)=az\frac{1-b}{1-c}{{}_{2}\phi_{1}}\left({aq,bq\atop cq};q,z\right),$

ⓘ

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 -hypergeometric) function, : complex base and : 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 ${{}_{2}\phi_{1}}\left({aq,b\atop cq};q,z\right)-{{}_{2}\phi_{1}}\left({a,b% \atop c};q,z\right)=az\frac{(1-b)(1-(c/a))}{(1-c)(1-cq)}{{}_{2}\phi_{1}}\left(% {aq,bq\atop cq^{2}};q,z\right),$

ⓘ

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 -hypergeometric) function, : complex base and : 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 ${{}_{2}\phi_{1}}\left({aq,b/q\atop c};q,z\right)-{{}_{2}\phi_{1}}\left({a,b% \atop c};q,z\right)=az\frac{(1-b/(aq))}{1-c}{{}_{2}\phi_{1}}\left({aq,b\atop cq% };q,z\right),$

ⓘ

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 -hypergeometric) function, : complex base and : 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 $b(1-a){{}_{2}\phi_{1}}\left({aq,b\atop c};q,z\right)-a(1-b){{}_{2}\phi_{1}}% \left({a,bq\atop c};q,z\right)=(b-a){{}_{2}\phi_{1}}\left({a,b\atop c};q,z% \right),$

ⓘ

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 -hypergeometric) function, : complex base and : 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 $a\left(1-\frac{b}{c}\right){{}_{2}\phi_{1}}\left({a,b/q\atop c};q,z\right)-b% \left(1-\frac{a}{c}\right){{}_{2}\phi_{1}}\left({a/q,b\atop c};q,z\right)=(a-b% )\left(1-\frac{abz}{cq}\right){{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right),$

ⓘ

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 -hypergeometric) function, : complex base and : 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 -hypergeometric) function, : complex base and : 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 -hypergeometric) function, : complex base and : 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

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Keywords:: differential equations, -differential equations, -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 $\mathcal{D}_{q}^{n}{{}_{2}\phi_{1}}\left({a,b\atop c};q,zd\right)=\frac{\left(% a,b;q\right)_{n}d^{n}}{\left(c;q\right)_{n}(1-q)^{n}}{{}_{2}\phi_{1}}\left({aq% ^{n},bq^{n}\atop cq^{n}};q,dz\right),$

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base, : nonnegative integer, : complex variable and $\mathcal{D}_{q}$ : -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 $\mathcal{D}_{q}^{n}\left(\frac{\left(z;q\right)_{\infty}}{\left(abz/c;q\right)% _{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)\right)=\frac{\left(c/a% ,c/b;q\right)_{n}}{\left(c;q\right)_{n}(1-q)^{n}}\left(\frac{ab}{c}\right)^{n}% \frac{\left(zq^{n};q\right)_{\infty}}{\left(abz/c;q\right)_{\infty}}{{}_{2}% \phi_{1}}\left({a,b\atop cq^{n}};q,zq^{n}\right).$

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, : complex base, : nonnegative integer, : complex variable and $\mathcal{D}_{q}$ : -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

-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 -hypergeometric) function, : complex base, : complex variable and $\mathcal{D}_{q}$ : -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

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Keywords:: integral representations, -hypergeometric function
Notes:: See Gasper and Rahman (2004, pp. 23, 115).
Permalink:: http://dlmf.nist.gov/17.6.v
Errata (effective with 1.2.5):: This section was renamed (from “Integral Representations”) to clarify that it concerns specifically -generalizations.
See also:: Annotations for §17.6 and Ch.17

17.6.28 ${{}_{2}\phi_{1}}\left({q^{\alpha},q^{\beta}\atop q^{\gamma}};q,z\right)=\frac{% \Gamma_{q}\left(\gamma\right)}{\Gamma_{q}\left(\beta\right)\Gamma_{q}\left(% \gamma-\beta\right)}\qint_{0}^{1}\frac{t^{\beta-1}\left(tq;q\right)_{\gamma-% \beta-1}}{\left(zt;q\right)_{\alpha}}\,{\mathrm{d}}_{q}t,$ $\left|z\right|<1$ , $\Re{\beta}>0$ .

ⓘ

Symbols:: $\Gamma_{\NVar{q}}\left(\NVar{z}\right)$ : -gamma function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), $\,{\mathrm{d}}_{\NVar{q}}\NVar{x}$ : q-differential of , ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ 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 -hypergeometric) function, $\qint$ : q-integral, $\Re$ : real part, : complex base, : complex variable and : real variable
Referenced by:: Erratum (V1.2.5) for Equation (17.6.28)
Permalink:: http://dlmf.nist.gov/17.6.E28
Encodings:: TeX, pMML, png
Correction (effective with 1.2.5):: On the right-hand side in the denominator was replaced by . Also the constraints $\left|z\right|<1$ , $\Re{\beta}>0$ were added.
Suggested 2025-09-27 by Hans Volkmer
See also:: Annotations for §17.6(v), §17.6 and Ch.17

17.6.29 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\left(\frac{-1}{2\pi i}\right)% \frac{\left(a,b;q\right)_{\infty}}{\left(q,c;q\right)_{\infty}}\int_{-i\infty}% ^{i\infty}\frac{\left(q^{1+\zeta},cq^{\zeta};q\right)_{\infty}}{\left(aq^{% \zeta},bq^{\zeta};q\right)_{\infty}}\frac{\pi(-z)^{\zeta}}{\sin\left(\pi\zeta% \right)}\,\mathrm{d}\zeta,$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{i}$ : 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 -hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple -Pochhammer symbol, $\sin\NVar{z}$ : sine function, : complex base and : 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 $\left|z\right|<1$ , $|\operatorname{ph}\left(-z\right)|<\pi$ , 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

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Keywords:: continued fractions, -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).

17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions

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§17.6 Function

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