q阶乘幂是阶乘幂的Q-模拟[1]。与阶乘幂在广义超几何函数中的作用类似,q阶乘幂也是定义基本超几何函数的基础。
- 当n为正整数时,q阶乘幂定义为
![{\displaystyle (a;q)_{n}=\prod _{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots (1-aq^{n-1}),}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Eatw2zAdFz2sQytdEnge3zNdCatnBaNK1aDGNajvAate5ngw0njsO)
- 当n为0时,q阶乘幂定义为
![{\displaystyle (a;q)_{0}=1.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzgsQyjs4ajo0ygaPzNCOzAoQaDw1zjnAnDG3zNnCoNK4zqzByjvA)
- 与一般的阶乘幂不同的是,q阶乘幂可以扩展成一个无穷乘积
![{\displaystyle (a;q)_{\infty }=\prod _{k=0}^{\infty }(1-aq^{k}),}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnDJEz2vCaNaQntGPaNoOoNnDnjs4oDvCyqs5ztC1zDzFntCNati4)
- 这时它是一个关于q在单位圆盘内的解析函数,也可以考虑为一个关于q的形式幂级数。其中一个特殊情况
![{\displaystyle \phi (q)=(q;q)_{\infty }=\prod _{k=1}^{\infty }(1-q^{k})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CaDi1oqePzNGOztm3zAeQnjrFoqe0oqa1aAeQoAw3zNsNyjo3otm5)
- 被称为欧拉函数。
- 有限q阶乘幂可以用无穷q阶乘幂表示
![{\displaystyle (a;q)_{n}={\frac {(a;q)_{\infty }}{(aq^{n};q)_{\infty }}},}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81oNJAytm3yta2nDo2ygzDoqzCygoOoqe4yqw0oqvEaDnBzjePzji4)
- 这样就能把q阶乘幂扩展到n为负整数的情况:对于非负整数n,有
![{\displaystyle (a;q)_{-n}={\frac {1}{(aq^{-n};q)_{n}}}=\prod _{k=1}^{n}{\frac {1}{(1-a/q^{k})}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82ytoOoDJByta0o2zDygo2ntK0aNBAoNwNaDnAztlFzjoQyjJEajs0)
- 以及
![{\displaystyle (a;q)_{-n}={\frac {(-q/a)^{n}q^{n(n-1)/2}}{(q/a;q)_{n}}}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnDnAnjC5a2wPatwOyqa3nDvBzte1zjm2nja3otwNzAzFaAzAago1)
因为很多关于q阶乘幂的等式都含有多个q阶乘幂相乘,因此在标准写法中用一个含有多个变量的q阶乘幂来表示这个乘积:
![{\displaystyle (a_{1},a_{2},\ldots ,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FyjdEzDiQythDaNnBa2a0ajK0zjG5ajG4ygs1aAePntJEygiPaqw4)
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![{\displaystyle (a;b)_{2}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Eoqe2ythCoNa1aqwNytC1zAi3agwQzDBEzAaPytCQytrCnAs1njeO)
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![{\displaystyle (a;b)_{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzNJAztK1aja0nthAztvDotG2oqdFnjFEyga3zDC4oNvDnArAnja3)
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![{\displaystyle (a;b)_{4}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CaNJFotKPotw1njm4otK3o2nDzDC3zNzCoteNzqaQztK1yjJCygrE)
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![{\displaystyle (a;b)_{5}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81aga4ageQzAo1zgaQz2rAoNrCyjGQyjGNnjJBngo2otK0ygsNntFE)
- ^ Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538