be a nonconstant function, then any $h\in\mathcal{F}$ can be expressed rationally in terms of $(f,\partial_{u_1}f,\dots,\partial_{u_g}f)$;

\(4) there exists an entire function $\vartheta\colon\mathbb{C}^g\to\mathbb{C}$ such that $\partial_{u_i,u_j}\log\vartheta \in \mathcal{F}$, $i,j=1,\dots,g$.

For example, any elliptic function $f\in \mathcal{F}_1$ is a rational function in the Weierstrass functions $\wp(u; g_2, g_3)$ and $\partial_u\wp(u; g_2, g_3)$, where $g_2$ and $g_3$ are parameters of elliptic curve $$V = \{ (x,y) \in \mathbb{C}^2\;|\; y^2 = 4 x^3 - g_2 x - g_3 \}.$$ It is easy to see that the function $\frac{\partial}{\partial g_2}\wp(u; g_2, g_3)$ will no longer be elliptic. This is due to the fact that the period lattice $\Gamma$ is a function of the parameters $g_2$ and $g_3$. In [@fs882] Frobenius and Stickelberger described all the differential operators $L$ in the variables $u,\;g_2$ and $g_3$, such that $Lf\in \mathcal{F}_1$ for any function $f\in \mathcal{F}_1$ (see below Section 7.3).

In [@bl07; @bl08] the classical problem of differentiation of Abelian functions over parameters for families of $(n,s)$-curves was solved. In the case of hyperelliptic curves this problem was solved more explicitly.

All genus $2$ curves are hyperelliptic. We denote by $\pi\colon \mathcal{U}_g \to \mathcal{B}_g$ the universal bundle of Jacobian varieties $Jac(\overline{V}_\lambda)$ of hyperelliptic curves. Let us consider the 