Cylinder kernel expansion of Casimir energy with a Robin boundary

We compute the Casimir energy of a massless scalar field obeying the Robin boundary condition on one plate and the Dirichlet boundary condition on another plate for two parallel plates with a separation of alpha. The Casimir energy densities for general dimensions (D = d + 1) are obtained as functions of alpha and beta by studying the cylinder kernel. We construct an infinite-series solution as a sum over classical paths. The multiple-reflection analysis continues to apply. We show that finite Casimir energy can be obtained by subtracting from the total vacuum energy of a single plate the vacuum energy in the region (0,??????)x R^d-1. In comparison with the work of Romeo and Saharian(2002), the relation between Casimir energy and the coeffcient beta agrees well.