  Normalized irreducible characters of the symmetric group S(n) can be
understood as zonal spherical functions of the Gelfand pair $(S(n)\times
S(n),\Diag S(n))$. They form an orthogonal basis in the space of the functions
on the group S(n) invariant with respect to conjugations by S(n). In this paper
we consider a different Gelfand pair connected with the symmetric group, that
is an ``unbalanced'' Gelfand pair $(S(n)\times S(n-1),\Diag S(n-1))$. Zonal
spherical functions of this Gelfand pair form an orthogonal basis in a larger
space of functions on S(n), namely in the space of functions invariant with
respect to conjugations by $S(n-1)$. We refer to these zonal spherical
functions as normalized \textit{generalized} characters of S(n). The main
discovery of the present paper is that these generalized characters can be
computed on the same level as the irreducible characters of the symmetric
group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type
formula, and an analogue of the determinantal formula for the generalized
characters of S(n).
