© 2000 by London Mathematical Society
© The London Mathematical Society
Hypergeometric Series, a Barnes-Type Lemma, and Whittaker Functions
Department of Mathematics, University of Colorado Boulder, CO 80309, USA, stade{at}euclid.colorado.edu
Department of Mathematics, Rockford College Rockford, IL 61108, USA, taggarjl.faculty.rc{at}rockford.edu
Received 6 July 1998. Revision received 12 October 1998.
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 
Cn–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 
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 
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.)