Power Series with Integral Coefficients in the Sense of Hurwitz

If $f(z)$ is a power series with H-integral coefficients, then

are also power series with H-integral coefficients.

If $f(z)$ and $g(z)$ are power series with H-integral coefficients, then

are also power series with H-integral coefficients, and under the further assumption that $g(0)=\pm 1$ also

If $f(z)$ is a power series with H-integral coefficients and $f(0)=0$, then

is also a power series with H-integral coefficients, $n=1,2,3,\ldots$ [174].

If $f(z)$ and $g(z)$ are power series with H-integral coefficients and $f(0)=0$, then $g[f(z)]$ is also a power series with H-integral coefficients.

The function $\varphi (z)$, well defined by the differential equation

and the initial conditions $\varphi (0)=0$, $\varphi '(0)>0$, may be expanded in a power series with H-integral coefficients.

For every prime number $p$, we have

For every composite number $m$ greater than 4 we have

The Bernoulli numbers $B_n$ are defined as the coefficients in the power series expansion

(Problem 154 of Part I). We have

In general we have

where $G_n$ is an integer and $2,3,\alpha ,\beta ,\gamma ,\ldots$ denote those prime numbers that exceed by one the divisors of $2n$. [Expand first $z$ and then $z/(e^z-1)$ in ascending powers of $(e^z-1)$ and apply 179--181.]

The coefficients $C_n$ in the series

where $\varphi (z)$ denotes the function defined in 178, are rational numbers and moreover the denominator of $C_n$ is not divisible by any square other than 1 and is divisible only by such prime numbers as are $\equiv 1 \pmod4$. [Expand first $z$ and then $z/\varphi (z)$ in ascending powers of $\varphi (z)$.]

The differential equation

has a well-defined solution of the form

($\wp (z)$ is a particular elliptic function, termed a ``lemniscate function''; its period-parallelogram is a square.) The numbers $D_n$ are rational, and moreover the denominator of $D_n$ is not divisible by any square other than 1, and is divisible only by such prime numbers as are $\equiv 1 \pmod4$. [Show that $\wp (z)=[\varphi (z)]^{-2}$, cf. 183. Expand first $z^2$ and then $z^2[\varphi (z)]^{-2}$ in ascending powers of $\varphi (z)$; $z^2$ as a function of $\varphi (z)$ satisfies a linear differential equation of second order.]

If a power series with H-integral coefficients $\sum_{n=0}^\infty (a_n/n!)z^n$ satisfies a homogeneous linear differential equation with constant coefficients, then the sequence of coefficients $a_0,a_1,a_2,\ldots ,a_n,\ldots$ is periodic mod $m$ after a certain number of terms, where $m$ is an arbitrary integer. [Examples of this are found already in 179, 180.]

The power series

shows that theorem 185 cannot without further assumptions be extended to homogeneous linear differential equations whose coefficients are polynomials.

If a transcendental entire function $g(z)$ has a power series expansion with H-integral coefficients, then we have for the maximum modulus (Chapter 1 of Part IV) of $g(z)$