17 q-Hypergeometric and Related Functions Properties 17.4 Basic Hypergeometric Functions 17.6 ${{}_{2}\phi_{1}}$ Function

§17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions

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Notes:: See Andrews (1976, pp. 17, 19, 36) and Gasper and Rahman (2004, pp. 25–26).
Referenced by:: Erratum (V1.2.1) for §17.5
Permalink:: http://dlmf.nist.gov/17.5
Clarification (effective with 1.2.1):: Below (17.5.2) and (17.5.4), a sentence was added indicating that these equations can be used as the analytic continuation for the ${{}_{1}\phi_{0}}$ contained in those equations.
See also:: Annotations for Ch.17

Euler’s Second Sum

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Keywords:: Euler’s sums, Euler’s sums (first, second, third), -hypergeometric function
See also:: Annotations for §17.5 and Ch.17

17.5.1 ${{}_{0}\phi_{0}}\left(-;-;q,z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{% \genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(q;q\right)_{n}}=\left(z;q\right)_{% \infty};$

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Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -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, : nonnegative integer and : complex variable
Referenced by:: §17.2(iii), Erratum (V1.1.11) for Equation (17.5.1)
Permalink:: http://dlmf.nist.gov/17.5.E1
Encodings:: TeX, pMML, png
Correction (effective with 1.1.11):: The constraint $\left|z\right|<1$ is not necessary and was removed.
See also:: Annotations for §17.5, §17.5 and Ch.17

compare (17.3.2).

-Binomial Series

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Keywords:: -binomial series
See also:: Annotations for §17.5 and Ch.17

17.5.2 ${{}_{1}\phi_{0}}\left(a;-;q,z\right)=\frac{\left(az;q\right)_{\infty}}{\left(z% ;q\right)_{\infty}},$ $\left|z\right|<1$ ;

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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 : complex variable
Referenced by:: §17.5, Erratum (V1.2.1) for §17.5
Permalink:: http://dlmf.nist.gov/17.5.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §17.5, §17.5 and Ch.17

compare (17.2.37). This equation can be used as the analytic continuation for this ${{}_{1}\phi_{0}}$ .

-Binomial Theorem

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Keywords:: -binomial theorem
See also:: Annotations for §17.5 and Ch.17

17.5.3 ${{}_{1}\phi_{0}}\left(q^{-n};-;q,z\right)=\left(zq^{-n};q\right)_{n}.$

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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, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/17.5.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §17.5, §17.5 and Ch.17

This is (17.2.35) reformulated.

Euler’s First Sum

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Keywords:: Euler’s sums, Euler’s sums (first, second, third), -hypergeometric function
See also:: Annotations for §17.5 and Ch.17

17.5.4 ${{}_{1}\phi_{0}}\left(0;-;q,z\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{\left(q;q% \right)_{n}}=\frac{1}{\left(z;q\right)_{\infty}},$ $\left|z\right|<1$ ;

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