§17.5 $\mathop{{{}_{0}\phi_{0}}\/}\nolimits,\mathop{{{}_{1}\phi_{0}}\/}\nolimits,% \mathop{{{}_{1}\phi_{1}}\/}\nolimits$ Functions

Euler’s Second Sum

 17.5.1 $\mathop{{{}_{0}\phi_{0}}\/}\nolimits\!\left(-;-;q,z\right)=\sum_{n=0}^{\infty}% \frac{(-1)^{n}q^{\binom{n}{2}}z^{n}}{\left(q;q\right)_{n}}=\left(z;q\right)_{% \infty},$ $|z|<1$;

compare (17.3.2).

$q$-Binomial Series

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

compare (17.2.37).

$q$-Binomial Theorem

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

This is (17.2.35) reformulated.

Euler’s First Sum

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

compare (17.3.1).

Cauchy’s Sum

 17.5.5 $\mathop{{{}_{1}\phi_{1}}\/}\nolimits\!\left({a\atop c};q,c/a\right)=\frac{% \left(c/a;q\right)_{\infty}}{\left(c;q\right)_{\infty}},$ $|c|<|a|$.