  In this article we show analytic properties of certain Rankin-Selberg type
Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of
half-integral weight. The numerators of these Dirichlet series are the inner
products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The
denominators and the range of summation of these Dirichlet series are like the
ones of the Koecher-Maass series. The meromorphic continuations and functional
equations of these Dirichlet series are obtained. Moreover, an identity between
the Petersson norms of Jacobi forms with respect to linear isomorphism between
Jacobi forms of integral weight and half-integral weight is also obtained.
