q-Karlsson-Mintonの和公式の証明

$r=0$のとき，Heineの和公式を用いることによって
$${}_2{\varphi }_1[\begin{array}{c}a,b\\ bq\end{array};\frac{q}{a}]=\frac{(q,bq/a;q{)}_{\mathrm{\infty }}}{(q/a,bq;q{)}_{\mathrm{\infty }}}$$
となることからわかる．$r-1$の時に成立するものとすると，$q$-Vandermondeの和公式によって
$$\frac{(b_r{q}^{m_r};q{)}_k}{(b_r;q{)}_k}=q^{m_{r}k}{}_2{\varphi }_1[\begin{array}{c}{q}^{-m_r},q^{-k}\\ b_r\end{array};q]$$
だから，
$$\begin{array}{rl}{}_{r+2}{\varphi }_{r+1}[\begin{array}{c}a,b,b_1{q}^{m_1}\dots ,b_r{q}^{m_r}\\ bq,b_1,\dots ,b_r\end{array};a^{-1}{q}^{1-m_1+\cdots +m_r}]& =\sum _{0\le k}\frac{(a,b,b_1{q}^{m_1},\dots ,b_{r-1}{q}^{m_{r-1}};q{)}_k}{(b_1,\dots ,b_r;q{)}_k}(a^{-1}{q}^{1-m_1+\cdots +m_r}{)}^k{q}^{m_{r}k}\sum _{0\le l}\frac{(q^{-m_r},q^{-k}{)}_l}{(bq,q;q{)}_l}\\ & =\sum _{0\le k,l}\frac{(a,b,b_1{q}^{m_1},\dots ,b_{r-1}{q}^{m_{r-1}};q{)}_k}{(b_1,\dots ,b_r;q{)}_k}(a^{-1}{q}^{1-m_1+\cdots +m_r}{)}^k{q}^{m_{r}k}\frac{(q^{-m_r};q{)}_l}{(c,q;q{)}_l(q;q{)}_{k-l}}(-1{)}^l{q}^{l(l-1)/2-kl}{q}^l\\ & =\sum _{0\le k,l}\frac{(q^{-m_r};q{)}_l}{(bq,q;q{)}_l}(-1{)}^l{q}^{l(l+1)/2}\frac{(b_1{q}^l,\dots ,b_r{q}^l;q{)}_{k+l}}{(b_1,\dots ,b_r{)}_{k+l}(q;q{)}_k}(a^{-1}{q}^{1-m_1+\cdots +m_r}{)}^{k+l}{q}^{n(k+l)-l(k+l)}\\ & =\sum _{0\le l}\frac{(a,b,b_1{q}^{m_1}\dots ,b_{r-1}{q}^{m_{r-1}},q^{-m_r};q{)}_l}{(bq,b_1,\dots ,b_r,q;q{)}_l}{q}^{-l(l-1)/2}{(-\frac{q^{1-m_1-\cdots -m_{r-1}}}{a})}^l{}_{r+1}{\varphi }_r[\begin{array}{c}a{q}^k,b{q}^k,b_1{q}^{m_1+k},\dots ,b_{r-1}{q}^{m_{r-1}+k}\\ b{q}^{k+1},b_1{q}^k,\dots ,b_{r-1}{q}^k\end{array};\frac{q^{1-m_1-\cdots -m_{r-1}-k}}{a}]\\ & =\sum _{0\le l}\frac{(a,b,b_1{q}^{m_1}\dots ,b_{r-1}{q}^{m_{r-1}},q^{-m_r};q{)}_l}{(bq,b_1,\dots ,b_r,q;q{)}_l}{q}^{-l(l-1)/2}{(-\frac{q^{1-m_1-\cdots -m_{r-1}}}{a})}^l\frac{(q,bq/a;q{)}_{\mathrm{\infty }}}{(b{q}^{1+l},q^{1-l}/a;q{)}_{\mathrm{\infty }}}\prod _{k=1}^{r-1}\frac{(b_k/b;q{)}_{m_k}}{(b_k{q}^l;q{)}_{m_k}}(b{q}^l{)}^{m_k}\\ & =\frac{(q,bq/a;q{)}_{\mathrm{\infty }}}{(bq,q/a;q{)}_{\mathrm{\infty }}}\sum _{0\le l}\frac{(a,b,b_1{q}^{m_1}\dots ,b_{r-1}{q}^{m_{r-1}},q^{-m_r};q{)}_l}{(bq,b_r,q;q{)}_l}{q}^{-l(l-1)/2}{\left(\frac{q}{a}\right)}^l(bq;q{)}_l(q/a;q{)}_{-l}\prod _{k=1}^{r-1}\frac{(b_k/b;q{)}_{m_k}}{(b_k;q{)}_{m_k+l}}{b}^{m_k}\\ & =\frac{(q,bq/a;q{)}_{\mathrm{\infty }}}{(bq,q/a;q{)}_{\mathrm{\infty }}}\prod _{k=0}^{r-1}\frac{(b_k/b;q{)}_{m_k}}{(b_k;q{)}_{m_k}}{b}^{m_k}\sum _{0\le l}\frac{(b,q^{-m_r};q{)}_l}{(b_r,q;q{)}_l}{q}^k\\ & =\frac{(q,bq/a;q{)}_{\mathrm{\infty }}}{(bq,q/a;q{)}_{\mathrm{\infty }}}\prod _{k=1}^r\frac{(b_k/b;q{)}_{m_k}}{(b_k;q{)}_{m_k}}{b}^{m_k}\end{array}$$
より示される．