Central to the Fourier development of automorphic forms on GL(n, ℝ) are the
class 1 principal series Whittaker functions Wn,a(z), which were first studied
systematically by Jacquet [13]. (See Section 2 below for the definition of Wn,a(z).)
Of particular interest are the Mellin transforms Mn,a(s) (s ∈ [Copf ]n−1) of Wn,a(z). (See
equation (2.11) below.) For example, such transforms, and analogous Mellin
transforms of products of Whittaker functions, arise as archimedean Euler factors for
certain automorphic L-functions (see [5, 6, 8, 21, 22] for discussions and examples.)
Moreover, Mn,a(s) has relevance to the problem of special values of Whittaker
functions; cf. [7] in the case n = 3.
Friedberg and Goldfeld [11] have shown Mn,a(s), for general n, to have analytic
continuation and to satisfy certain recurrence relations. However, explicit formulae
for these transforms have, until the present work, been deduced only for n [les ] 4. In
particular, both M2,a(s) (cf.
[4, 25]) and M3,a(s) (cf. [7, 9, 19]) are expressible as (some
powers of 2 and π times) ratios of gamma functions. On the other hand, M4,a(s)
(cf. [22]) may be realized essentially as a hypergeometric series of type 7F6(1), or
equivalently as a sum of two series of type 4F3(1). (See Section 2 below for a brief
general discussion of hypergeometric series.)