In this talk we treat two problems which are parts of an ongoing work. The first one is

a generalization of the canonical Rankin-Cohen structure to the graded algebras which

contain elements of negative degree, namely a generalization of Proposition 1 of

"D. Zagier, Modular forms and differential operators. Proceedings Mathematical Sciences,

104(1):57{75, 1994". The second problem is to give some sufficient conditions under which

a certain combination of a weight 2 quasi-modular form with itself or with an arbitrary modular

form and their derivatives will be again a modular form. Indeed, we solve this problem in

a more algebraic context, namely in Rankin-Cohen algebras.

The idea of these problems are coming from previous works of the speaker, where he is endowing

the space of Calabi-Yau modular forms, which is the space generated by solution components of a

certain vector field on a special moduli space of Calabi-Yau varieties, with a Rankin-Cohen algebra

structure. In fact, the space of Calabi-Yau modular forms that are defined in this way contains

elements of negative weights.

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