Summation theorems for multidimensional basic hypergeometric series by determinant evaluations
As a lemma for establishing summation theorems for \(r\)-dimensional series, the author derives the formula \[ \begin{multlined} \det\left( {[A X_s]_{r-t} [AC/X_s]_{r-t} \over[BX_s]_{r-t} [BC/X_s]_{r-t}} \right)=A^{r \choose s}q^{r\choose 3}\prod_{1\leq i<j\leq r}\left\{ (X_j-X_i) \left(1- {C\over X_iX_j} \right)\right\}\\ \times\prod^r_{i=1} {[B/A]_{i-1} [ABCq^{2r-2i}]_{i-1} \over [BX_i]_{r-1} [BC/X_i]_{r-1}}, \end{multlined} \] where \((s,t)\) are determinant indices and \([\alpha]_\mu\) is the basic Pochhammer symbol; the base \(q\) is understood throughout. Using this lemma and Ramanujan's \(_1\psi_1\) summation theorem, the author obtains \[ \begin{aligned}\sum^\infty_{k_1=-\infty} & \cdots \sum^\infty_{k_r= -\infty} \left( \prod_{1\leq i<j\leq r}{1-q^{k_i-k_j}x_i/x_j \over 1-x_i/x_j} \prod^r_{i=1} {[ax_i]_{k_i} z^kq^K \over [b_i]_{k_i}} \right)\\ & =q^{-{r \choose 2}}\prod_{1\leq i<j\leq r}\left({b_i/x_i-b_j/x_j \over 1/x_i-1/x_j} \right) \prod^r_{i=1} {[q]_\infty [b_i/ax_i]_\infty [ax_iz]_\infty [q/a x_iz]_\infty \over[b_i]_\infty [q/ax_i]_\infty [zq^{1-i}]_\infty [b_i/a x_iz ]_\infty}, \end{aligned} \] where \(k=k_1+ \cdots +k_r\), and \(K=\sum_{i=1}^r (i-r)k_i\), and two more theorems of a similar appearance; they all reduce to Ramanujan's theorem for \(r=1\). By a similar approach, other generalizations of well known theorems are obtained. Among these are the summation theorems for very-well-poised series due to Bailey \((_6\psi_6)\) and Jackson \((_8\psi_7)\). Some limiting and particular cases are also considered.
scientific article; zbMATH DE number 1389516