Closed-form expression for the Dirichlet-series function g(ξ,η) in the nonlinear adjoint solution

Derive a closed-form expression for the function g(ξ,η) defined by equation (41) as the eigenfunction expansion entering the analytic adjoint solutions (equations (35)–(36)) for the Blasius boundary layer, beyond the linear Oseen case where g reduces to an error function.

Background

The analytic adjoint solution is expressed through a function g(ξ,η) given by an infinite eigenfunction expansion, which behaves as a generalized Dirichlet series. While the authors establish convergence, boundedness, and some identities, they explicitly note the lack of a closed-form expression for g except in the linearized case, leaving a compact analytic representation open.