  In this paper, we obtain affine analogues of Gindikin-Karpelevich formula and
Casselman-Shalika formula as sums over Kashiwara-Lusztig's canonical bases.
Suggested by these formulas, we define natural $q$-deformation of arithmetical
functions such as (multi-)partition function and Ramanujan $\tau$-function, and
prove various identities among them. In some examples, we recover classical
identities by taking limits. We also consider $q$-deformation of Kostant's
function and study certain $q$-polynomials whose special values are weight
multiplicities.
