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We study spherical functions on the space isomorphic to $U(2n)/(U(n)\timesU(n))$ over a $p$-adic field; those functional equations with respect to theaction of the Weyl group, the location of possible poles and zeros, explicitformulas, and spherical Fourier transforms. Then, as an application, we give afunctional equation of $p$-adic local hermitian Siegel series.

Spherical functions on $U(n,n)/(U(n) \times U(n))$ and hermitian Siegel series