ガウスの超幾何級数
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{\displaystyle _{2}F_{1}\left[{\begin{matrix}a,b\\c\end{matrix}};z\right]=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}\;n!}}z^{n}}
に対し、その q -類似は
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{\displaystyle _{2}\phi _{1}\left[{\begin{matrix}a,b\\c\end{matrix}};q,z\right]=\sum _{n=0}^{\infty }{\frac {(a;q)_{n}(b;q)_{n}}{(q;q)_{n}(c;q)_{n}}}z^{n}}
で定義される。但し、ポッホハマー記号
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{\displaystyle (a)_{n}=a(a+1)\cdots (a+n-1)}
と q -ポッホハマー記号
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{\displaystyle (a;q)_{n}={\begin{cases}(1-aq)(1-aq^{2})\cdots (1-aq^{n-1})&(n>0)\\1&(n=0)\end{cases}}}
を用いた。
このとき、次の関係式をハイネの和公式と呼ぶ。
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{\displaystyle _{2}\phi _{1}\left[{\begin{matrix}a,b\\c\end{matrix}};q,{\frac {c}{ab}}\right]={\frac {\left({\frac {c}{a}};q\right)_{\infty }\left({\frac {c}{b}};q\right)_{\infty }}{\left(c;q\right)_{\infty }\left({\frac {c}{ab}};q\right)_{\infty }}}\qquad {\biggl (}{\bigl |}{\frac {c}{ab}}{\bigr |}<1{\biggr )}}
これはガウスの超幾何定理
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{\displaystyle _{2}F_{1}\left[{\begin{matrix}a,b\\c\end{matrix}};1\right]={\frac {\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}}\qquad (\Re {a}+\Re {b}<\Re {c},c\not \in \mathbb {Z} \setminus \mathbb {N} )}
の q -類似となっている。
ハイネの和公式は、次のハイネの変換式 (Heine's transformation)から導くことができる。
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{\displaystyle _{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}{\frac {c}{b}},z\\az\end{matrix}};q,b\right]}
ハイネの変換式はq二項定理 から導かれる。
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{\displaystyle {\begin{aligned}_{2}\phi _{1}\left[{\begin{matrix}a,b\\c\end{matrix}};q,z\right]&=\sum _{n=0}^{\infty }{\frac {(a;q)_{n}(b;q)_{n}}{(q;q)_{n}(c;q)_{n}}}z^{n}\\&=\sum _{n=0}^{\infty }{\frac {(a;q)_{n}(b;q)_{\infty }(cq^{n};q)_{\infty }}{(q;q)_{n}(c;q)_{\infty }(bq^{n};q)_{\infty }}}z^{n}\\&={\frac {(b;q)_{\infty }}{(c;q)_{\infty }}}\sum _{n=0}^{\infty }{\frac {(a;q)_{n}}{(q;q)_{n}}}\left({\frac {(cq^{n};q)_{\infty }}{(bq^{n};q)_{\infty }}}\right)z^{n}\\&={\frac {(b;q)_{\infty }}{(c;q)_{\infty }}}\sum _{n=0}^{\infty }{\frac {(a;q)_{n}}{(q;q)_{n}}}\left(\sum _{m=0}^{\infty }{\frac {({\frac {c}{b}};q)_{m}}{(q;q)_{m}}}(bq^{n})^{m}\right)z^{n}\\&={\frac {(b;q)_{\infty }}{(c;q)_{\infty }}}\sum _{m=0}^{\infty }{\frac {({\frac {c}{b}};q)_{m}}{(q;q)_{m}}}\left(\sum _{n=0}^{\infty }{\frac {(a;q)_{n}}{(q;q)_{n}}}(zq^{m})^{n}\right)b^{m}\\&={\frac {(b;q)_{\infty }}{(c;q)_{\infty }}}\sum _{m=0}^{\infty }{\frac {({\frac {c}{b}};q)_{m}}{(q;q)_{m}}}\left({\frac {(azq^{m};q)_{\infty }}{(zq^{m};q)_{\infty }}}\right)b^{n}\\&={\frac {(b;q)_{\infty }}{(c;q)_{\infty }}}\sum _{m=0}^{\infty }{\frac {({\frac {c}{b}};q)_{m}(z;q)_{m}(az;q)_{\infty }}{(q;q)_{m}(az;q)_{m}(z;q)_{\infty }}}\\&={\frac {(b;q)_{\infty }(az;q)_{\infty }}{(c;q)_{\infty }(z;q)_{\infty }}}{_{2}\phi _{1}}\left[{\begin{matrix}{\frac {c}{b}},z\\az\end{matrix}};q,b\right]\\\end{aligned}}}
ハイネの和公式はハイネの変換式に
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{\displaystyle {\begin{aligned}_{2}\phi _{1}\left[{\begin{matrix}a,b\\c\end{matrix}};q,{\frac {c}{ab}}\right]&={\frac {(b;q)_{\infty }({\frac {c}{b}};q)_{\infty }}{(c;q)_{\infty }({\frac {c}{ab}};q)_{\infty }}}{_{2}\phi _{1}}\left[{\begin{matrix}{\frac {c}{b}},{\frac {c}{ab}}\\{\frac {c}{b}}\end{matrix}};q,b\right]\\&={\frac {(b;q)_{\infty }({\frac {c}{b}};q)_{\infty }}{(c;q)_{\infty }({\frac {c}{ab}};q)_{\infty }}}\left(\sum _{n=0}^{\infty }{\frac {({\frac {c}{b}};q)_{n}({\frac {c}{ab}};q)_{n}}{(q;q)_{n}({\frac {c}{b}};q)_{n}}}b^{n}\right)\\&={\frac {(b;q)_{\infty }({\frac {c}{b}};q)_{\infty }}{(c;q)_{\infty }({\frac {c}{ab}};q)_{\infty }}}\left(\sum _{n=0}^{\infty }{\frac {({\frac {c}{ab}};q)_{n}}{(q;q)_{n}}}b^{n}\right)\\&={\frac {(b;q)_{\infty }({\frac {c}{b}};q)_{\infty }}{(c;q)_{\infty }({\frac {c}{ab}};q)_{\infty }}}\left({\frac {({\frac {c}{a}};q)_{\infty }}{(b;q)_{\infty }}}\right)\\&={\frac {({\frac {c}{a}};q)_{\infty }({\frac {c}{b}};q)_{\infty }}{(c;q)_{\infty }({\frac {c}{ab}};q)_{\infty }}}\\\end{aligned}}}
Andrews, George E. (1986). q -Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra . CBMS. American Mathematical Society. ISBN 978-0821807163 
![{\displaystyle _{2}F_{1}\left[{\begin{matrix}a,b\\c\end{matrix}};z\right]=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}\;n!}}z^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c72fec9be6ae9b6caa4fc405b668a4940af89fb0)
![{\displaystyle _{2}\phi _{1}\left[{\begin{matrix}a,b\\c\end{matrix}};q,z\right]=\sum _{n=0}^{\infty }{\frac {(a;q)_{n}(b;q)_{n}}{(q;q)_{n}(c;q)_{n}}}z^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1577d7097306a597e30c171677a149ddc4b47784)
![{\displaystyle _{2}\phi _{1}\left[{\begin{matrix}a,b\\c\end{matrix}};q,{\frac {c}{ab}}\right]={\frac {\left({\frac {c}{a}};q\right)_{\infty }\left({\frac {c}{b}};q\right)_{\infty }}{\left(c;q\right)_{\infty }\left({\frac {c}{ab}};q\right)_{\infty }}}\qquad {\biggl (}{\bigl |}{\frac {c}{ab}}{\bigr |}<1{\biggr )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0df05f6180c330c99949d3f7e0a9be05818fc8a9)
![{\displaystyle _{2}F_{1}\left[{\begin{matrix}a,b\\c\end{matrix}};1\right]={\frac {\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}}\qquad (\Re {a}+\Re {b}<\Re {c},c\not \in \mathbb {Z} \setminus \mathbb {N} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c4dd2a4d2f240d8586f42a15b1218e2f60baf2d)
![{\displaystyle _{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}{\frac {c}{b}},z\\az\end{matrix}};q,b\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36acd99fcc8d69dd4cd982818459f557554b52bb)
![{\displaystyle {\begin{aligned}_{2}\phi _{1}\left[{\begin{matrix}a,b\\c\end{matrix}};q,z\right]&=\sum _{n=0}^{\infty }{\frac {(a;q)_{n}(b;q)_{n}}{(q;q)_{n}(c;q)_{n}}}z^{n}\\&=\sum _{n=0}^{\infty }{\frac {(a;q)_{n}(b;q)_{\infty }(cq^{n};q)_{\infty }}{(q;q)_{n}(c;q)_{\infty }(bq^{n};q)_{\infty }}}z^{n}\\&={\frac {(b;q)_{\infty }}{(c;q)_{\infty }}}\sum _{n=0}^{\infty }{\frac {(a;q)_{n}}{(q;q)_{n}}}\left({\frac {(cq^{n};q)_{\infty }}{(bq^{n};q)_{\infty }}}\right)z^{n}\\&={\frac {(b;q)_{\infty }}{(c;q)_{\infty }}}\sum _{n=0}^{\infty }{\frac {(a;q)_{n}}{(q;q)_{n}}}\left(\sum _{m=0}^{\infty }{\frac {({\frac {c}{b}};q)_{m}}{(q;q)_{m}}}(bq^{n})^{m}\right)z^{n}\\&={\frac {(b;q)_{\infty }}{(c;q)_{\infty }}}\sum _{m=0}^{\infty }{\frac {({\frac {c}{b}};q)_{m}}{(q;q)_{m}}}\left(\sum _{n=0}^{\infty }{\frac {(a;q)_{n}}{(q;q)_{n}}}(zq^{m})^{n}\right)b^{m}\\&={\frac {(b;q)_{\infty }}{(c;q)_{\infty }}}\sum _{m=0}^{\infty }{\frac {({\frac {c}{b}};q)_{m}}{(q;q)_{m}}}\left({\frac {(azq^{m};q)_{\infty }}{(zq^{m};q)_{\infty }}}\right)b^{n}\\&={\frac {(b;q)_{\infty }}{(c;q)_{\infty }}}\sum _{m=0}^{\infty }{\frac {({\frac {c}{b}};q)_{m}(z;q)_{m}(az;q)_{\infty }}{(q;q)_{m}(az;q)_{m}(z;q)_{\infty }}}\\&={\frac {(b;q)_{\infty }(az;q)_{\infty }}{(c;q)_{\infty }(z;q)_{\infty }}}{_{2}\phi _{1}}\left[{\begin{matrix}{\frac {c}{b}},z\\az\end{matrix}};q,b\right]\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d62eb2e9ee8695cbb15f8ae49e9b254a8d82c6)
![{\displaystyle {\begin{aligned}_{2}\phi _{1}\left[{\begin{matrix}a,b\\c\end{matrix}};q,{\frac {c}{ab}}\right]&={\frac {(b;q)_{\infty }({\frac {c}{b}};q)_{\infty }}{(c;q)_{\infty }({\frac {c}{ab}};q)_{\infty }}}{_{2}\phi _{1}}\left[{\begin{matrix}{\frac {c}{b}},{\frac {c}{ab}}\\{\frac {c}{b}}\end{matrix}};q,b\right]\\&={\frac {(b;q)_{\infty }({\frac {c}{b}};q)_{\infty }}{(c;q)_{\infty }({\frac {c}{ab}};q)_{\infty }}}\left(\sum _{n=0}^{\infty }{\frac {({\frac {c}{b}};q)_{n}({\frac {c}{ab}};q)_{n}}{(q;q)_{n}({\frac {c}{b}};q)_{n}}}b^{n}\right)\\&={\frac {(b;q)_{\infty }({\frac {c}{b}};q)_{\infty }}{(c;q)_{\infty }({\frac {c}{ab}};q)_{\infty }}}\left(\sum _{n=0}^{\infty }{\frac {({\frac {c}{ab}};q)_{n}}{(q;q)_{n}}}b^{n}\right)\\&={\frac {(b;q)_{\infty }({\frac {c}{b}};q)_{\infty }}{(c;q)_{\infty }({\frac {c}{ab}};q)_{\infty }}}\left({\frac {({\frac {c}{a}};q)_{\infty }}{(b;q)_{\infty }}}\right)\\&={\frac {({\frac {c}{a}};q)_{\infty }({\frac {c}{b}};q)_{\infty }}{(c;q)_{\infty }({\frac {c}{ab}};q)_{\infty }}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d48caad5821c6db79f048228a491f779da0c5f0)