  We consider the heat equations satisfied by the sigma function associated
with a planar curve, extending and developing earlier pioneering work of
Buchstaber and Leykin. These heat equations lead to useful {\em linear}
recursive relations for the coefficients of power series expansion of the sigma
function. In particular we exhibit explicit results for curves of genus 3, and
give a new constructive proof of an explicit expression for the main matrix in
the theory for {\em any} hyperelliptic curve. We also state and prove a new
explicit formula for the eigenvalues of the linear operators associated with
this matrix, as well as other practical formulae.
