  We introduce the (global) q-Whittaker function as the limit at t=0 of the
q,t-spherical function extending the symmetric Macdonald polynomials to
arbitrary eigenvalues. The construction heavily depends on the technique of the
q-Gaussians developed by the author (and Stokman in the non-reduced case). In
this approach, the q-Whittaker function is given by a series convergent
everywhere, a kind of generating function for multi-dimensional q-Hermite
polynomials (closely related to the level 1 Demazure characters). One of the
applications is a q-version of the Shintani- Casselman- Shalika formula, which
appeared directly connected with q-Mehta- Macdonald identities in terms of the
Jackson integrals. This formula generalizes that of type A due to Gerasimov et
al. to arbitrary reduced root systems. At the end of the paper, we obtain a
q,t-counterpart of the Harish-Chandra asymptotic formula for the spherical
functions, including the Whittaker limit.
