Confluent hypergeometric functions on an exceptional domain (Q2266148)

A recent paper of \textit{G. Shimura} [Math. Ann. 260, 269-302 (1982; Zbl 0502.10013)] contained a comprehensive investigation (including the analytic continuation and functional equation) of generalized confluent hypergeometric functions defined in a natural manner for four major families of tube domains associated with formally real Jordan algebras; an important objective of the analysis was an application in a subsequent paper of \textit{G. Shimura}, Duke Math. J. 50, 417-476 (1983; Zbl 0519.10019) to an interesting problem concerning the analytic continuation of Eisenstein series attached to (discrete subgroups of) the symplectic and unitary groups. In this paper, the author describes (his analogous results on confluent hypergeometric functions for the tube domain arising from Cayley's octonion algebra over \({\mathbb{R}}\). (The exceptional tube domain of type \(E_ 7\) arises in this manner.) The author's main theorem deals with the analytic continuation (to the whole of \({\mathbb{C}}^ 2\) of these confluent hypergeometric functions and their functional equation; analogously to Shimura's, results concerning (related) series which generalizes \(\sum_{n\in {\mathbb{Z}}}(z+n)^{- \alpha}(\bar z+n)^{-\beta}\) for complex z with \(Im(z)>0\) and \(\alpha\),\(\beta\in {\mathbb{C}}\) with \(Re(\alpha +\beta)>1\) are also stated. Further, by way of an applications, the author states a conjecture on the meromorphic continuation and functional equation of Eisenstein series for the exceptional arithmetic group studied in a well-known paper of \textit{W. L. Baily} [Ann. Math. 91, 512-549 (1970; Zbl 0202.079)].