Qザールシュッツの和公式
( a ; q ) n ( b ; q ) n ( q ; q ) n ( c ; q ) n = ∑ m = 0 ∞ ( c a ; q ) m ( c b ; q ) m ( q ; q ) m ( c ; q ) m ( a b c ; q ) n ( q − n ; q ) m ( q ; q ) n ( c a b q − n + 1 ; q ) m q m ( a ; q ) n ( b ; q ) n ( a b c ; q ) n ( c ; q ) n = ∑ m = 0 ∞ ( c a ; q ) m ( c b ; q ) m ( q ; q ) m ( c ; q ) m ( q − n ; q ) m ( c a b q − n + 1 ; q ) m q m = 3 φ 2 [ c a , c b , q − n c , c a b q − n + 1 ; q , q ] {\displaystyle {\begin{aligned}&{\frac {(a;q)_{n}(b;q)_{n}}{(q;q)_{n}(c;q)_{n}}}=\sum _{m=0}^{\infty }{\frac {({\frac {c}{a}};q)_{m}({\frac {c}{b}};q)_{m}}{(q;q)_{m}(c;q)_{m}}}{\frac {({\frac {ab}{c}};q)_{n}(q^{-n};q)_{m}}{(q;q)_{n}({\frac {c}{ab}}q^{-n+1};q)_{m}}}q^{m}\\&{\frac {(a;q)_{n}(b;q)_{n}}{({\frac {ab}{c}};q)_{n}(c;q)_{n}}}=\sum _{m=0}^{\infty }{\frac {({\frac {c}{a}};q)_{m}({\frac {c}{b}};q)_{m}}{(q;q)_{m}(c;q)_{m}}}{\frac {(q^{-n};q)_{m}}{({\frac {c}{ab}}q^{-n+1};q)_{m}}}q^{m}={_{3}\phi _{2}}\left[{\begin{matrix}{\frac {c}{a}},{\frac {c}{b}},q^{-n}\\c,{\frac {c}{ab}}q^{-n+1}\end{matrix}};q,q\right]\end{aligned}}}