基本超几何函数

基本超几何函数是广义超几何函数的q模拟。

第一类基本超几何函数

_{j} ϕ_{k} [\begin{matrix} a_{1} & a_{2} & \dots & a_{j} \\ b_{1} & b_{2} & \dots & b_{k} \end{matrix}; q, z] = \sum_{n = 0}^{\infty} \frac{(a_{1}, a_{2}, \dots, a_{j}; q)_{n}}{(b_{1}, b_{2}, \dots, b_{k}, q; q)_{n}} {((- 1)^{n} q^{(\binom{n}{2})})}^{1 + k - j} z^{n}

其中

(a_{1}, a_{2}, \dots, a_{m}; q)_{n} = (a_{1}; q)_{n} (a_{2}; q)_{n} \dots (a_{m}; q)_{n}

其中

(a; q)_{n} = \prod_{k = 0}^{n - 1} (1 - a q^{k}) = (1 - a) (1 - a q) (1 - a q^{2}) \dots (1 - a q^{n - 1}) .

.

_{j} ψ_{k} [\begin{matrix} a_{1} & a_{2} & \dots & a_{j} \\ b_{1} & b_{2} & \dots & b_{k} \end{matrix}; q, z] = \sum_{n = - \infty}^{\infty} \frac{(a_{1}, a_{2}, \dots, a_{j}; q)_{n}}{(b_{1}, b_{2}, \dots, b_{k}; q)_{n}} {((- 1)^{n} q^{(\binom{n}{2})})}^{k - j} z^{n} .

下列基本超几何函数在q->1时，化为超几何函数^[1]

\lim_{q \to 1}_{j} ϕ_{k} [\begin{matrix} q^{a_{1}} & q^{a_{2}} & \dots & q^{a_{j}} \\ q^{b_{1}} & q^{b_{2}} & \dots & q^{b_{k}} \end{matrix}; q, (q - 1) * z]

=

_{j} F_{k} [\begin{matrix} a_{1} & a_{2} & \dots & a_{j} \\ b_{1} & b_{2} & \dots & b_{k} \end{matrix}; q, z]

下列公式是二项式定理的q模拟：

_{1} Φ_{0} ([a], []; q; z) =

\sum_{n = 0}^{\infty}

\frac{(a; q)_{n}}{(q; q)_{n}}

↑ Roelof KoeKoek, Peter Lesky,Rene Swarttouw,Hypergeometric Orthogonal Polynomials and Their q-Analogues p15 Springer

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Eduard Heine, Theorie der Kugelfunctionen, (1878) 1, pp 97–125.
Eduard Heine, Handbuch die Kugelfunctionen. Theorie und Anwendung (1898) Springer, Berlin