Abstract The eigenvalue problem of the general anharmonic oscillator (Hamiltonian H2μ(k, λ) = -d2 / dx2 + kx2 + λx2μ, (k, λ) is investi­gated in this work. Very accurate eigenvalues are obtained in all régimes of the quantum number n and the anharmonicity constant λ. The eigenvalues, as functions of λ, exhibit crossings. The qualitative features of the actual crossing pattern are substantially reproduced in the W. K. B. approximation. Successive moments of any transition between two general anharmonic oscillator eigenstates satisfy exactly a linear recurrence relation. The asymptotic behaviour of this recursion and its consequences are examined.