Qザールシュッツの和公式 From Wikipedia, the free encyclopedia qザールシュッツの和公式(q-Saalschütz summation formula)はザールシュッツの定理のq-類似であり、q超幾何級数 3 ϕ 2 {\displaystyle {_{3}\phi _{2}}} の和を与える公式である[1]。 3 ϕ 2 [ a , b , q − n c , a b c q − n + 1 ; q , q ] = ( c a ; q ) n ( c b ; q ) n ( c a b ; q ) n ( c ; q ) n {\displaystyle {_{3}\phi _{2}}\left[{\begin{matrix}a,b,q^{-n}\\c,{\frac {ab}{c}}q^{-n+1}\end{matrix}};q,q\right]={\frac {({\frac {c}{a}};q)_{n}({\frac {c}{b}};q)_{n}}{({\frac {c}{ab}};q)_{n}(c;q)_{n}}}} 但し、 ( a ; q ) n {\displaystyle (a;q)_{n}} はqポッホハマー記号である。 qザールシュッツの和公式はハイネの変換式から導かれる。ハイネの変換式を反復すると 2 ϕ 1 [ a , b c ; q , z ] = ( b ; q ) ∞ ( a z ; q ) ∞ ( c ; q ) ∞ ( z ; q ) ∞ 2 ϕ 1 [ z , c b a z ; q , b ] = ( b ; q ) ∞ ( a z ; q ) ∞ ( c ; q ) ∞ ( z ; q ) ∞ ⋅ ( b c ; q ) ∞ ( b z ; q ) ∞ ( a z ; q ) ∞ ( b ; q ) ∞ 2 ϕ 1 [ b , a b z c b z ; q , c b ] = ( b ; q ) ∞ ( a z ; q ) ∞ ( c ; q ) ∞ ( z ; q ) ∞ ⋅ ( b c ; q ) ∞ ( b z ; q ) ∞ ( a z ; q ) ∞ ( b ; q ) ∞ ⋅ ( a b z c ; q ) ∞ ( c ; q ) ∞ ( b z ; q ) ∞ ( c b ; q ) ∞ 2 ϕ 1 [ c b , c a c ; q , a b z c ] = 2 ϕ 1 [ c a , c b c ; q , a b z c ] ( a b z c ; q ) ∞ ( z ; q ) ∞ {\displaystyle {\begin{aligned}{_{2}\phi _{1}}\left[{\begin{matrix}a,b\\c\end{matrix}};q,z\right]&={\frac {(b;q)_{\infty }(az;q)_{\infty }}{(c;q)_{\infty }(z;q)_{\infty }}}{_{2}\phi _{1}}\left[{\begin{matrix}z,{\frac {c}{b}}\\az\end{matrix}};q,b\right]\\&={\frac {(b;q)_{\infty }(az;q)_{\infty }}{(c;q)_{\infty }(z;q)_{\infty }}}\cdot {\frac {({\frac {b}{c}};q)_{\infty }(bz;q)_{\infty }}{(az;q)_{\infty }(b;q)_{\infty }}}{_{2}\phi _{1}}\left[{\begin{matrix}b,{\frac {abz}{c}}\\bz\end{matrix}};q,{\frac {c}{b}}\right]\\&={\frac {(b;q)_{\infty }(az;q)_{\infty }}{(c;q)_{\infty }(z;q)_{\infty }}}\cdot {\frac {({\frac {b}{c}};q)_{\infty }(bz;q)_{\infty }}{(az;q)_{\infty }(b;q)_{\infty }}}\cdot {\frac {({\frac {abz}{c}};q)_{\infty }(c;q)_{\infty }}{(bz;q)_{\infty }({\frac {c}{b}};q)_{\infty }}}{_{2}\phi _{1}}\left[{\begin{matrix}{\frac {c}{b}},{\frac {c}{a}}\\c\end{matrix}};q,{\frac {abz}{c}}\right]\\&={_{2}\phi _{1}}\left[{\begin{matrix}{\frac {c}{a}},{\frac {c}{b}}\\c\end{matrix}};q,{\frac {abz}{c}}\right]{\frac {({\frac {abz}{c}};q)_{\infty }}{(z;q)_{\infty }}}\\\end{aligned}}} となり、q二項定理を用いて 2 ϕ 1 ∑ n = 0 ∞ ( a ; q ) n ( b ; q ) n ( q ; q ) n ( c ; q ) n z n = ∑ m = 0 ∞ ( c a ; q ) m ( c b ; q ) m ( q ; q ) m ( c ; q ) m ( a b z c ) m ∑ k = 0 ∞ ( a b c ; q ) k ( q ; q ) k z k {\displaystyle {\begin{aligned}{_{2}\phi _{1}}\sum _{n=0}^{\infty }{\frac {(a;q)_{n}(b;q)_{n}}{(q;q)_{n}(c;q)_{n}}}z^{n}&=\sum _{m=0}^{\infty }{\frac {({\frac {c}{a}};q)_{m}({\frac {c}{b}};q)_{m}}{(q;q)_{m}(c;q)_{m}}}\left({\frac {abz}{c}}\right)^{m}\sum _{k=0}^{\infty }{\frac {({\frac {ab}{c}};q)_{k}}{(q;q)_{k}}}z^{k}\\\end{aligned}}} となる。 z n {\displaystyle z^{n}} の係数を比べると ( 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 ) m ( a b c ; q ) n − m ( q ; q ) n − m = ∑ m = 0 ∞ ( c a ; q ) m ( c b ; q ) m ( q ; q ) m ( c ; q ) m ( a b c ) m ( a b c ; q ) n ( q 1 + n − m ; q ) m ( q ; q ) n ( a b c q n − m ; q ) m {\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}}}\left({\frac {ab}{c}}\right)^{m}{\frac {({\frac {ab}{c}};q)_{n-m}}{(q;q)_{n-m}}}\\&=\sum _{m=0}^{\infty }{\frac {({\frac {c}{a}};q)_{m}({\frac {c}{b}};q)_{m}}{(q;q)_{m}(c;q)_{m}}}\left({\frac {ab}{c}}\right)^{m}{\frac {({\frac {ab}{c}};q)_{n}(q^{1+n-m};q)_{m}}{(q;q)_{n}({\frac {ab}{c}}q^{n-m};q)_{m}}}\end{aligned}}} であるが、qポッホハマー記号の変換式 ( a q − m + 1 ; q ) m = ( − a ) m q − m ( m − 1 ) / 2 ( a − 1 ; q ) m {\displaystyle (aq^{-m+1};q)_{m}=(-a)^{m}q^{-m(m-1)/2}\left(a^{-1};q\right)_{m}} を用いて、 ( 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}}} を得る。 a , b {\displaystyle a,b} を c a , c b {\displaystyle {\tfrac {c}{a}},{\tfrac {c}{b}}} に置き換えて 3 ϕ 2 [ a , b , q − n c , a b c q − n + 1 ; q , q ] = ( c a ; q ) n ( c b ; q ) n ( c a b ; q ) n ( c ; q ) n {\displaystyle {_{3}\phi _{2}}\left[{\begin{matrix}a,b,q^{-n}\\c,{\frac {ab}{c}}q^{-n+1}\end{matrix}};q,q\right]={\frac {({\frac {c}{a}};q)_{n}({\frac {c}{b}};q)_{n}}{({\frac {c}{ab}};q)_{n}(c;q)_{n}}}} を得る。 出典 ↑ Wolfram Mathworld: q-Saalschütz Sum この項目は、数学に関連した書きかけの項目です。この項目を加筆・訂正などしてくださる協力者を求めています（プロジェクト:数学／Portal:数学）。表示編集 Related Articles
![{\displaystyle {\begin{aligned}{_{2}\phi _{1}}\left[{\begin{matrix}a,b\\c\end{matrix}};q,z\right]&={\frac {(b;q)_{\infty }(az;q)_{\infty }}{(c;q)_{\infty }(z;q)_{\infty }}}{_{2}\phi _{1}}\left[{\begin{matrix}z,{\frac {c}{b}}\\az\end{matrix}};q,b\right]\\&={\frac {(b;q)_{\infty }(az;q)_{\infty }}{(c;q)_{\infty }(z;q)_{\infty }}}\cdot {\frac {({\frac {b}{c}};q)_{\infty }(bz;q)_{\infty }}{(az;q)_{\infty }(b;q)_{\infty }}}{_{2}\phi _{1}}\left[{\begin{matrix}b,{\frac {abz}{c}}\\bz\end{matrix}};q,{\frac {c}{b}}\right]\\&={\frac {(b;q)_{\infty }(az;q)_{\infty }}{(c;q)_{\infty }(z;q)_{\infty }}}\cdot {\frac {({\frac {b}{c}};q)_{\infty }(bz;q)_{\infty }}{(az;q)_{\infty }(b;q)_{\infty }}}\cdot {\frac {({\frac {abz}{c}};q)_{\infty }(c;q)_{\infty }}{(bz;q)_{\infty }({\frac {c}{b}};q)_{\infty }}}{_{2}\phi _{1}}\left[{\begin{matrix}{\frac {c}{b}},{\frac {c}{a}}\\c\end{matrix}};q,{\frac {abz}{c}}\right]\\&={_{2}\phi _{1}}\left[{\begin{matrix}{\frac {c}{a}},{\frac {c}{b}}\\c\end{matrix}};q,{\frac {abz}{c}}\right]{\frac {({\frac {abz}{c}};q)_{\infty }}{(z;q)_{\infty }}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e47065abfabb09b9924a6fecbc188f7b763988cf)
の係数を比べると
を用いて、![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16091f320112abd3d6e0a72c96c1b71efbac11a0)
を
に置き換えて![{\displaystyle {_{3}\phi _{2}}\left[{\begin{matrix}a,b,q^{-n}\\c,{\frac {ab}{c}}q^{-n+1}\end{matrix}};q,q\right]={\frac {({\frac {c}{a}};q)_{n}({\frac {c}{b}};q)_{n}}{({\frac {c}{ab}};q)_{n}(c;q)_{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4e649bcf552c3e994d1e74871c17696ad72d3b)
