Abelian functions for cyclic trigonal curves of genus 4

We discuss the theory of generalized Weierstrass σ$\sigma$ and ℘$\wp$-functions defined on a trigonal curve of genus 4, following earlier work on the genus 3 case. The specific example of the “purely trigonal” (or “cyclic trigonal”) curve y³=x⁵+_λ4x⁴+_λ3x³+_λ2x²+_λ1x+_λ0$y^3=x^5+{\lambda }_4{x}^4+{\lambda }_3{x}^3+{\lambda }_2{x}^2+{\lambda }_{1}x+{\lambda }_0$ is discussed in detail, including a list of some of the associated partial differential equations satisfied by the ℘$\wp$-functions, and the derivation of addition formulae.