Sextic and octic anharmonic oscillators: connection between strong-coupling and weak-coupling expansions

For the eigenvalues of the Hamiltonians 1/2 (p) over cap(2) + 1/2x(2) + 1/2 beta x(2m) With m = 3, 4..., we propose representations reproducing the well known weak-coupling asymptotic perturbation and strong-coupling expansions as limiting cases. Some analytical relations which the strong-coupling coefficients must satisfy are presented.