文献あり

Andrews-Askeyの和公式

Andrews-Askey(1977)

$\begin{aligned} _{2} ϕ_{1} [\begin{array}{c} b^{2}, b^{2} / c \\ c \end{array}; q^{2}; \frac{c q}{b^{2}}] & = \frac{1}{2} \frac{(b^{2}, q; q^{2})_{\infty}}{(c, c q / b^{2}; q^{2})_{\infty}} (\frac{(c / b; q)_{\infty}}{(b; q)_{\infty}} + \frac{(- c / b; q)_{\infty}}{(- b; q)_{\infty}}) \end{aligned}$

Heineの変換公式
$\begin{aligned} _{2} ϕ_{1} [\begin{array}{c} a, b \\ c \end{array}; x] & = \frac{(b, a x; q)_{\infty}}{(c, x; q)_{\infty}}_{2} ϕ_{1} [\begin{array}{c} c / b, x \\ a x \end{array}; b] \end{aligned}$
において, $q \mapsto q^{2}$ として $a \mapsto b^{2} / c, b \mapsto b^{2}, x \mapsto \frac{c q}{b}$ として,
$\begin{aligned} _{2} ϕ_{1} [\begin{array}{c} b^{2}, b^{2} / c \\ c \end{array}; q^{2}; \frac{c q}{b^{2}}] & = \frac{(b^{2}, q; q^{2})_{\infty}}{(c, c q / b^{2}; q^{2})_{\infty}}_{2} ϕ_{1} [\begin{array}{c} c / b^{2}, c q / b^{2} \\ q \end{array}; q^{2}; b^{2}] \\ = \frac{(b^{2}, q; q^{2})_{\infty}}{(c, c q / b^{2}; q^{2})_{\infty}} \sum_{0 \leq n} \frac{(c / b^{2}; q)_{2 n}}{(q; q)_{2 n}} b^{2 n} \\ = \frac{(b^{2}, q; q^{2})_{\infty}}{2 (c, c q / b^{2}; q^{2})_{\infty}} \sum_{0 \leq n} \frac{(c / b^{2}; q)_{n}}{(q; q)_{n}} (b^{n} + (- b)^{n}) \\ = \frac{(b^{2}, q; q^{2})_{\infty}}{2 (c, c q / b^{2}; q^{2})_{\infty}} (\frac{(c / b; q)_{\infty}}{(b; q)_{\infty}} + \frac{(- c / b; q)_{\infty}}{(- b; q)_{\infty}}) \end{aligned}$
となって示される. 最後の等号は $q$ 二項定理による.

類似として, 以下のような公式もある.

$\begin{aligned} _{2} ϕ_{1} [\begin{array}{c} b^{2}, b^{2} / c \\ c q^{2} \end{array}; q^{2}, \frac{c q^{3}}{b^{2}}] & = \frac{1}{2 b} \frac{(b^{2}, q; q^{2})_{\infty}}{(c q^{2}, c q / b^{2}; q^{2})_{\infty}} (\frac{(c q / b; q)_{\infty}}{(b; q)_{\infty}} - \frac{(- c q / b; q)_{\infty}}{(- b; q)_{\infty}}) \end{aligned}$

先ほどと同様にHeineの変換公式を用いると,
$\begin{aligned} _{2} ϕ_{1} [\begin{array}{c} b^{2}, b^{2} / c \\ c q^{2} \end{array}; q^{2}, \frac{c q^{3}}{b^{2}}] & = \frac{(b^{2}, q^{3}; q^{2})_{\infty}}{(c q^{2}, c q^{3} / b^{2}; q^{2})_{\infty}}_{2} ϕ_{1} [\begin{array}{c} c q^{3} / b^{2}, c q^{2} / b^{2} \\ q^{3} \end{array}; b^{2}] \\ = \frac{(b^{2}, q; q^{2})_{\infty}}{b (c q^{2}, c q / b^{2}; q^{2})_{\infty}} \sum_{0 \leq n} \frac{(c q / b^{2}; q)_{2 n + 1}}{(q; q)_{2 n + 1}} b^{2 n + 1} \\ = \frac{(b^{2}, q; q^{2})_{\infty}}{2 b (c q^{2}, c q / b^{2}; q^{2})_{\infty}} \sum_{0 \leq n} \frac{(c q / b^{2}; q)_{n}}{(q; q)_{n}} (b^{n} - (- b)^{n}) \\ = \frac{(b^{2}, q; q^{2})_{\infty}}{2 b (c q^{2}, c q / b^{2}; q^{2})_{\infty}} (\frac{(c q / b; q)_{\infty}}{(b; q)_{\infty}} - \frac{(- c q / b; q)_{\infty}}{(- b; q)_{\infty}}) \end{aligned}$
となって示される.

参考文献

[1]

G. E. Andrews, R. Askey, Enumeration of partitions: the role of Eulerian series and q-orthogonal polynomials, NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., 31, 1977, 3-26

投稿日：3月27日