Mellin transforms of Whittaker functions on \(\text{GL} (4,\mathbb{R})\) and \(\text{GL} (4,\mathbb{C})\)
The author considers the Whittaker functionals associated with representations of \(\text{GL} (4, \mathbb{R})\) and \(\text{GL} (4, \mathbb{C})\), or rather their Mellin transforms. First of all he shows how the complex case reduces simply to the real one. The Mellin transform in the real case can be found in terms of the generalized hypergeometric functions of W. N. Bailey and L. J. Slater, more precisely, by a well-poised \({}_7 F_6\) function evaluated at 1. The author also proves a new formula in the theory of these functions. The formula for the Mellin transform is unexpected and interesting; it opens a number of further problems, both in itself and through the interactions with other investigations; these are discussed in the final section of this paper.
scientific article; zbMATH DE number 824176
scientific article