Basic Hypergeometric Series - Simple Series

Simple Series

Some simple series expressions include

\frac{z}{1-q} \;_{2}\phi_1 \left[\begin{matrix}
q \; q \\
q^2 \end{matrix}\; ; q,z \right] =
\frac{z}{1-q}
+ \frac{z^2}{1-q^2}
+ \frac{z^3}{1-q^3}
+ \ldots

and

\frac{z}{1-q^{1/2}} \;_{2}\phi_1 \left[\begin{matrix}
q \; q^{1/2} \\
q^{3/2} \end{matrix}\; ; q,z \right] =
\frac{z}{1-q^{1/2}}
+ \frac{z^2}{1-q^{3/2}}
+ \frac{z^3}{1-q^{5/2}}
+ \ldots

and

\;_{2}\phi_1 \left[\begin{matrix}
q \; -1 \\
-q \end{matrix}\; ; q,z \right] = 1+
\frac{2z}{1+q}
+ \frac{2z^2}{1+q^2}
+ \frac{2z^3}{1+q^3}
+ \ldots.

Read more about this topic:  Basic Hypergeometric Series

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