  In the first part of this article, we consider a Groebner basis of the
differential ideal {x_1^2} with respect to "the" weighted lexicographical
monomial order and show that its computation is related with an identity
involving the partitions that appear in the first Rogers-Ramanujan identity. We
then prove that a Grobener basis of this ideal is not differentially finite in
contrary with the case of "the" weighted reverse lexicographical order. In the
second part, we give a simple and direct proof of a theorem of Nguyen Duc Tam
about the Groaner basis of the differential ideal {x_1y_1}; we then obtain
identities involving partitions with 2 colors.
