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The aim of this paper is to give expressions for modular linear differential operators (MLDOs) of any order. In particular, we show that they can all be described in terms of Rankin-Cohen brackets and a modified Rankin-Cohen bracket found by Kaneko and Koike. We also give more uniform descriptions of MLDOs in terms of canonically defined higher Serre derivatives and an extension of Rankin-Cohen brackets, as well as in terms of quasimodular forms and almost holomorphic modular forms. The last of these descriptions involves the holomorphic projection map. The paper also includes some general results on the theory of quasimodular forms on both cocompact and non-cocompact subgroups of SL_2(\mathbb {R}), as well as a slight sharpening of a theorem of Martin and Royer on Rankin-Cohen brackets of quasimodular forms. [ABSTRACT FROM AUTHOR]

In this paper, we find some conditions under which a sequence \{\alpha (n)\} will satisfy the Laguerre inequality of any order asymptotically. Using this method, we prove that for any r and some constant c, the Maclaurin coefficients \gamma (n) of the Riemann Xi-function satisfy the Laguerre inequality of order r when n>cr^3, which provides a necessary condition for the Riemann hypothesis. We also prove that the partition function satisfies the Laguerre inequality of order r\geq 5 when n\geq 6r^4. As a consequence, it gives an affirmative answer to Wagner's conjecture on the threshold for the Laguerre inequalities of order no more than 10 for the partition function. Moreover, motivated by the study of Craven and Csordas on the complete monotonicity of the Maclaurin coefficients of entire functions in Laguerre-Pólya class, we consider the complete monotonicity of the sequences \{\alpha (n)\}. We give the criteria for the asymptotically complete monotonicity of the sequence \{\alpha (n)\} and \{\log \alpha (n)\}, respectively. With this criteria, we show that (-1)^r \Delta ^r \gamma (n)>0 for n>ce^{r^3} and (-1)^{r-1} \Delta ^r \log \gamma (n)>0 for n>cr^2. Furthermore, we propose some open problems. [ABSTRACT FROM AUTHOR]