  We distinguish two classifications of bidifferential operators: between (A)
spaces of modular forms and (B) spaces of weighted densities.
  (A) The invariant under the projective action of $\text{SL}(2;\mathbb{Z})$
binary differential operators between spaces of modular forms of integer or
half-integer weight on the 1-dimensional manifold were found by Gordan (called
transvectants), rediscovered and classified by Rankin and Cohen (called
brackets), and, in still another context, by Janson and Peetre. The invariant
under the algebraic supergroup $\text{OSp}(1|2; \mathbb{Z})$ super modular
forms of integer and half-integer weight on $(1|1)$-dimensional superstrings
with contact structure were introduced, bidifferential operators between them
classified and further studied by Gieres-Theisen, Cohen-Manin-Zagier, and
Gargoubi-Ovsienko.
  (B) For any complex weights, we classify the analogs of Gordan-Rankin-Cohen
(briefly: GRC) binary differential operators between spaces of weighted
densities invariant under $\mathfrak{pgl}(2)$. For any complex weights, we
classify the analogs of GRC-operators between spaces of weighted densities
invariant under the Lie superalgebra $\mathfrak{osp}(1|2)$. In the case of
$(1|1)$-dimensional superstring without any additional structure, we also
classify the analogs of GRC-operators between spaces of any weighted densities
invariant under the Lie superalgebra $\mathfrak{pgl}(1|2)$.
