A new \(A_n\) extension of Ramanujan’s \(_1\psi_1\) summation with applications to multilateral \(A_n\) series. (English) Zbl 1059.33025
This paper proves several new identities for multilateral basic hypergeometric series associated with the root system \(A_n\) (or, equivalently, associated with the group \(U(n+1)\)).
Applying an argument of analytic continuation to an \(A_n\) \(q\)-binomial theorem of Milne, the authors derive an \(A_n\) extension of Ramanujan’s \(_1\psi_1\) summation theorem. Together with a \(_1\psi_1\) summation of Gustafson, this \(A_n\) \(_1\psi_1\) summation is used to prove two lemmas which provide an efficient method for deriving multilateral series identities in \(A_n\) from identities of the classical one-dimensional theory. The lemmas are shown to be related to Macdonald identities for \(A_n\). As examples of application of their method, the authors obtain several \(A_n\) extensions of Bailey’s \(_2\psi_2\) transformation formulas as well as many \(A_n\) extensions of a specific \(_2\psi_2\) summation.
Applying an argument of analytic continuation to an \(A_n\) \(q\)-binomial theorem of Milne, the authors derive an \(A_n\) extension of Ramanujan’s \(_1\psi_1\) summation theorem. Together with a \(_1\psi_1\) summation of Gustafson, this \(A_n\) \(_1\psi_1\) summation is used to prove two lemmas which provide an efficient method for deriving multilateral series identities in \(A_n\) from identities of the classical one-dimensional theory. The lemmas are shown to be related to Macdonald identities for \(A_n\). As examples of application of their method, the authors obtain several \(A_n\) extensions of Bailey’s \(_2\psi_2\) transformation formulas as well as many \(A_n\) extensions of a specific \(_2\psi_2\) summation.
Reviewer: Angela Pasquale (Metz)
MSC:
| 33D67 | Basic hypergeometric functions associated with root systems |
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
Keywords:
multilateral basic hypergeometric series; root system \(A_n\); Ramanujan’s \(_1\psi_1\) summation; Macdonald identities; Bailey’s \(_2\psi_2\) transformations; \(_2\psi_2\) summationReferences:
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