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

Pieri Integral Formula and Asymptotics of Jack Unitary Characters