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

## Euler’s Second Sum

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

compare (17.3.2).

## $q$-Binomial Series

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

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

## $q$-Binomial Theorem

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

This is (17.2.35) reformulated.

## Euler’s First Sum

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

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

## Cauchy’s Sum

 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 $|c|<|a|$ is not necessary and was removed. See also: Annotations for §17.5, §17.5 and Ch.17