I'm reading about Fourier expansion of modular functions, but I have trouble understanding the following equation:
Is it some inherent property of the denominator, as it is?
I'm reading about Fourier expansion of modular functions, but I have trouble understanding the following equation:
Is it some inherent property of the denominator, as it is?
As user8268 explained in the comments, any power series $1+a_1x+a_2x^2+\dots$ with integral coefficients and constant term 1 has an inverse with integral coefficients. You can use the nice expansion for $(1+a_1x+a_2x^2+\dots)^{-1}$ given by $\frac{1}{1-(-a_1x-a_2x^2-\ldots)} = 1 + (-a_1x-a_2x^2-\ldots) + (-a_1x-a_2x^2-\ldots)^2 + \ldots$
We will not get an integral power series for $(a_0+a_1x+a_2x^2 + \ldots)^{-1}$ if the leading term $a_0$ is not invertible; we will have $\frac{1}{a_0} + \frac{(-a_1x-a_2x^2-\ldots)}{a_0{}^2} + \frac{(-a_1x-a_2x^2-\ldots)^2}{a_0{}^3} + \ldots$ But you can see that the only denominators we get are powers of $a_0$.