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

§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), $q$ -hypergeometric function
See also:: Annotations for §17.5 and Ch.17

17.5.1		${{}_{0}\phi_{0}}\left(-;-;q,z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{% \genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(q;q\right)_{n}}=\left(z;q\right)_{% \infty};$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §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).

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

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

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

17.5.4		${{}_{1}\phi_{0}}\left(0;-;q,z\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{\left(q;q% \right)_{n}}=\frac{1}{\left(z;q\right)_{\infty}},$
$\left\|z\right\|<1$ ;
ⓘ Symbols: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §17.2(iii), §17.5, Erratum (V1.2.1) for §17.5 Permalink: http://dlmf.nist.gov/17.5.E4 Encodings: TeX, pMML, png See also: Annotations for §17.5, §17.5 and Ch.17

compare (17.3.1). This equation can be used as the analytic continuation for this ${{}_{1}\phi_{0}}$ .

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17.5.5		${{}_{1}\phi_{1}}\left({a\atop c};q,c/a\right)=\frac{\left(c/a;q\right)_{\infty% }}{\left(c;q\right)_{\infty}}.$
ⓘ 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 and $q$ : complex base Referenced by: Erratum (V1.1.10) for Equation (17.5.5) Permalink: http://dlmf.nist.gov/17.5.E5 Encodings: TeX, pMML, png Correction (effective with 1.1.10): The constraint $\left\|c\right\|<\left\|a\right\|$ is not necessary and was removed. See also: Annotations for §17.5, §17.5 and Ch.17

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