A multidimensional generalization of Shukla's \(_8\psi _8\) summation (Q1871747)

A new proof of a summation formula of \textit{H. S. Shukla} [Proc. Camb. Philos. Soc. 55, 262-266 (1959; Zbl 0092.29103)] for a very-well-poised \(_8\psi_8\) basic hypergeometric series is given, and a multidimensional generalisation for the root system \(A_{r-1}\) (equivalently, the unitary group \(U(r)\)) is established. The new proof of Shukla's theorem is in two steps. First, Rogers' \(_6\phi_5\) summation formula is used to establish a summation formula for a very-well-poised \(_8\phi_7\) series in a simple and clever way. Second, an analytic continuation process known as Ismail's argument is used to extend the result to the \(_8\psi_8\) series. The same method of proof is shown to work for the multivariate case. Just as Shukla's result contains Bailey's \(_6\psi_6\) summation formula when one of the parameters is specialised, the multivariate \(_8\psi_8\) formula presented in this paper reduces to a multivariate \(_6\psi_6\) summation formula of \textit{R. A. Gustafson} [SIAM J. Math. Anal. 18, 1576-1596 (1987; Zbl 0624.33012)] in a special case.