The Weierstrass P-function, $\wp(z;g_2,g_3)$ and its derivative $\wp'(z;g_2,g_3)$ are the building blocks of all elliptic functions of a given period lattice. The differential equation $$\wp''=6\wp^2-\frac{1}{2} g_2 $$ implies that all derivatives $\wp^{(n)}$ can be expressed as polynomials in $\wp,\wp'$. Recently, I've tried exploring antiderivatives.
The Mathematica command
Integrate[WeierstrassP[z,{g2,g3}],z]
shows that $$\int \wp(z;g_2,g_3) \mathrm{d} z=-\zeta(z;g_2,g_3)+A $$ where $A$ is an arbitrary constant, and $\zeta$ is the Weierstrass Zeta function. Moreover, the command
Integrate[WeierstrassP[z,{g2,g3}],z,z]
shows that
$$\int - \zeta(z; g_2,g_3) \mathrm{d} z=- \log \sigma(z;g_2,g_3)+B $$ where $B$ is an arbitrary constant, and $\sigma$ is the Weierstrass Sigma function. Trying a third time, the command
Integrate[WeierstrassP[z,{g2,g3}],z,z,z]
returns the unevaluated integral $$-\int \log \sigma(z;g_2,g_3) \mathrm{d} z. $$
My questions are: is it really not possible to express $\wp^{(-n)}$ in terms of other known functions for $n \geq 3$? Are there any other known expressions for $\wp^{(-n)}$ with $n \geq 3$?
Thank you!
There could be...the integral$$\int x^2\wp(x+z)\,dx\tag1$$has two periods as a function of $z$, and using IBP twice, will produce$$-x^2\zeta(x+z)+2x\ln\sigma(x+z)-2\int \ln\sigma(x+z)\,dx$$
But the integrand in $(1)$ has second-order poles; thus its integral can be evaluated as a series of residues, implying it can be rewritten as a sum of Weierstrass zeta functions (in $z$). Finally, set $z=0$.
Higher-order antiderivatives can be handled similarly by increasing the exponent in $(1)$.