  We introduce Jack (unitary) characters and prove two kinds of formulas that
are suitable for their asymptotics, as the lengths of the signatures that
parametrize them go to infinity. The first kind includes several integral
representations for Jack characters of one variable. The second identity we
prove is the Pieri integral formula for Jack characters which, in a sense, is
dual to the well known Pieri rule for Jack polynomials. The Pieri integral
formula can also be seen as a functional equation for irreducible spherical
functions of virtual Gelfand pairs.
  As an application of our formulas, we study the asymptotics of Jack
characters as the corresponding signatures grow to infinity in the sense of
Vershik-Kerov. We prove the existence of a small $\delta > 0$ such that the
Jack characters of $m$ variables have a uniform limit on the
$\delta$-neighborhood of the $m$-dimensional torus. Our result specializes to a
theorem of Okounkov and Olshanski.
