On the monomials of Poincaré Series of negative index
It is well known that an Eisenstein series $E_k$ cannot be written as a product of two lower-weight Eisenstein series, except for $E_{14}=E_4 E_{10}= E_4^2 E_6= E_6 E_8$. A similar result is also known for the Hecke eigenforms. It is also known that equalities among the monomials of the Eisenstein series can be reduced to the above-mentioned identities. Recently, the present author, along with E. Saha studied the possible monomial relations of the Poincaré cusp forms of index $1$. So far, the problem of studying the monomial relations has been restricted to the setup of holomorphic modular forms. For an even integer $k\ge 4$ and a negative integer $m$, the Poincaré series $G_{k}(z,m)$ of weight $k$ and index $m$ is a weakly holomorphic modular form having a pole of order $-m$ at $ι\infty$. It is immediate that the Poincaré series $G_k(z,m)$ for $m<0$ can not be written as a product of two lower-weight Poincaré series of the same index $m$. In this article, we investigate the possible equalities among the monomials of the Poincaré series $G_k(z,m)$ for arbitrary $m<0$. In particular, we show that for any $m<0$ such that $0.06\le\{-2mπ\}\le0.99$, two monomials composed of $G_k(z,m)$ of the weights $k\ge 50$ are never equal, where $\{x\}$ denotes the fractional part of a real number $x$. In view of Weyl's equidistribution criterion, at least $93\%$ of $m<0$ satisfies the above inequality.