σ-Functions: Old and New Results

We are considering multi-variable sigma function of genus g hyperelliptic curve as a function of two groups of variables - jacobian variables and parameters of the curve. In theta-functional representation of sigma-function the second group arises as periods of first and second kind differentials of the curve. We develop representation of periods in terms of theta-constants, for the first kind, period generalizations of Rosenhain type formulae are obtained whilst for the second kind, period theta-constant expressions are presented which are explicitly related to the fixed co-homology basis. We describe a method of constructing differentiation operators for hyperelliptic analogues of ζ- and $\wp$-functions on the parameters of the hyperelliptic curve. To demonstrate this method, we give the detailed construction of these operators in the case genus 1 and 2.