---
abstract: |
    We consider multi-variable sigma function of a genus $g$ hyperelliptic curve as a function of two group of variables - jacobian variables and parameters of the curve. In the 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 periods, generalizations of Rosenhain type formulae are obtained, whilst for the second kind periods 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 $\zeta$- and $\wp$-functions on the parameters of the hyperelliptic curve. To demonstrate this method, we gave the detailed construction of these operators in the cases of genus 1 and 2.
address:
- 'Steklov Mathematical Institute, Moscow'
- 'National University of Kyiv-Mohyla Academy'
- Institute of Magnetism NASU
author:
- 'V.M.Buchstaber'
- 'V.Z. Enolski'
- 'D.V.Leykin'
title: 'Multi-variable sigma-functions: old and new results'
---

..

Introduction
============

Our note belongs to an area in which Emma Previato took active part in the development. Since the time of first publication of the present authors [@bel97] she has inspired them, 