Low-temperature series expansions have been derived for the random field Ising model with a delta -function distribution on a Bethe lattice by two independent methods: (a) the finite-cluster method which uses graph embeddings and appropriate weighting functions; (b) the use of a recursion relation specific to the Bethe lattice. Numerical values have been evaluated when the coordination number q=3, 4 and the coefficients analysed to assess critical behaviour. For small fields, and temperatures near to Tco, the critical exponent of the magnetisation seems to retain its mean-field value. But there is clear evidence of a change in critical behaviour at some point on the critical curve. It is argued that when q>3 a tricritical point is indicated as found by Aharony in his mean-field solution.