Q超幾何級数
, b s ; q ) n ( ( − 1 ) n q ( n 2 ) ) s − r z n {\displaystyle {\begin{aligned}{}_{r}\phi _{s}\left[{\begin{matrix}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{s}\end{matrix}};q,z\right]&=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{s},q;q)_{n}}}\left((-1)^{n}q^{\binom {n}{2}}\right)^{s+1-r}z^{n}\\{}_{r}\psi _{s}\left[{\begin{matrix}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{s}\end{matrix}};q,z\right]&=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{s};q)_{n}}}\left((-1)^{n}q^{\binom {n}{2}}\right)^{s-r}z^{n}\\\end{aligned}}}
の形式で表される級数である。中でも
r φ r − 1 [ a 1 , a 2 , ... , a r b 1 , b 2 , ... , b r − 1 ; q , z ] = ∑ n = 0 ∞ ( a 1 , a 2 , ... , a r ; q ) n ( b 1 , b 2 , .