a-F interpolation with boundary

A recent derivation of the interpolation between the free energy and conformal anomaly for free fields on spheres is generalised to hemispheres with Neumann (N) and Dirichlet (D) conditions at the rim for GJMS scalar fields. It is shown that the N minus D interpolation is minus a quarter of that for a higher derivative fermion on the spherical rim. In particular, since, for ordinary bosons k=1, the related fermion is irregular propagating according to a second order (pseudo) operator. (2k is the derivative order in the equation of motion.) It is suggested that the relation has a role to play in the Type--B AdS/CFT mismatch. The DN boundary value problem is enlarged upon in the context of Branson and Gover's construction of the D to N operator. Contact is made with a result of Park and Wojciechowski which is then related to a duality relation of Barvinsky and Nesterov.