= Solution
Taylor-expand the gain terms:
$$
\lambda P(n-1)
=\lambda\left(P-P_n+\frac12P_{nn}+\cdots\right),
$$
and, writing $Q(n)=\beta n^2P(n)$,
$$
Q(n+2)=Q+2Q_n+2Q_{nn}+\cdots.
$$
After cancellation of the loss terms, the second-order <Kramers-Moyal expansion> is
$$
\frac{\partial P}{\partial t}
=-\frac{\partial}{\partial n}
\left[(\lambda-2\beta n^2)P\right]
+\frac12\frac{\partial^2}{\partial n^2}
\left[(\lambda+4\beta n^2)P\right].
$$
Hence the <constant-birth pair-annihilation process> has
$$
\boxed{A(n)=\lambda-2\beta n^2},
\qquad
\boxed{B(n)=\lambda+4\beta n^2}.
$$
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