Rankin-Cohen Operators for Jacobi and Siegel Forms

For any non-negative integer v we construct explicitly [v/2]+1 independent covariant bilinear differential operators from J_{k,m} x J_{k',m'} to J_{k+k'+v,m+m'}. As an application we construct a covariant bilinear differential operator mapping S_k^{(2)} x S^{(2)}_{k'} to S^{(2)}_{k+k'+v}. Here J_{k,m} denotes the space of Jacobi forms of weight k and index m and S^{(2)}_k the space of Siegel modular forms of degree 2 and weight k. The covariant bilinear differential operators constructed are analogous to operators already studied in the elliptic case by R. Rankin and H. Cohen and we call them Rankin-Cohen operators.